  
  [1X6 [33X[0;0YSolvable Subgroups of Maximal Order in Sporadic Simple Groups[133X[101X
  
  [33X[0;0YDate: May 14th, 2012[133X
  
  [33X[0;0YWe  determine the orders of solvable subgroups of maximal orders in sporadic
  simple  groups  and  their automorphism groups, using the information in the
  [5XAtlas[105X  of  Finite  Groups [CCN+85] and the [5XGAP[105X system [GAP21], in particular
  its   Character   Table   Library [Bre22]  and  its  library  of  Tables  of
  Marks [NMP18].[133X
  
  [33X[0;0YWe  also  determine the conjugacy classes of these solvable subgroups in the
  big group, and the maximal overgroups.[133X
  
  [33X[0;0YA  first  version  of this document, which was based on [5XGAP[105X 4.4.10, had been
  accessible  in  the  web  since  August 2006. The differences to the current
  version are as follows.[133X
  
  [30X    [33X[0;6YThe  format of the [5XGAP[105X output was adjusted to the changed behaviour of
        [5XGAP[105X 4.5.[133X
  
  [30X    [33X[0;6YThe  (too  wide) table of results was split into two tables, the first
        one  lists  the  orders  and  indices of the subgroups, the second one
        lists the structure of subgroups and the maximal overgroups.[133X
  
  [30X    [33X[0;6YThe  distribution  of  the solvable subgroups of maximal orders in the
        Baby  Monster  group and the Monster group to conjugacy classes is now
        proved.[133X
  
  [30X    [33X[0;6YThe  sporadic  simple  Monster  group has exactly one class of maximal
        subgroups  of  the  type  PSL[22X(2,  41)[122X (see [NW13]), and has no maximal
        subgroups which have the socle PSL[22X(2, 27)[122X (see [Wil10]). This does not
        affect  the  arguments  in Section [14X6.4-14[114X, but some statements in this
        section had to be corrected.[133X
  
  
  [1X6.1 [33X[0;0YThe Result[133X[101X
  
  [33X[0;0YThe  tables I  and II  list  information about solvable subgroups of maximal
  order  in  sporadic  simple  groups and their automorphism groups. The first
  column  in  each  table  gives  the  names of the almost simple groups [22XG[122X, in
  alphabetical  order.  The remaining columns of Table I contain the order and
  the  index  of  a  solvable  subgroup  [22XS[122X  of  maximal  order in [22XG[122X, the value
  [22Xlog_|G|(|S|)[122X,   and   the  page  number  in  the  [5XAtlas[105X [CCN+85]  where  the
  information  about  maximal  subgroups  of [22XG[122X is listed. The second and third
  columns  of Table II show a structure description of [22XS[122X and the structures of
  the  maximal subgroups that contain [22XS[122X; the value [21X[22XS[122X[121X in the third column means
  that  [22XS[122X  is itself maximal in [22XG[122X. The fourth and fifth columns list the pages
  in  the  [5XAtlas[105X with the information about the maximal subgroups of [22XG[122X and the
  section  in  this note with the proof of the table row, respectively. In the
  fourth  column,  page  numbers  in  brackets  refer  to the [5XAtlas[105X pages with
  information  about  the  maximal  subgroups  of nonsolvable quotients of the
  maximal subgroups of [22XG[122X listed in the third column.[133X
  
  [33X[0;0YNote that in the case of nonmaximal subgroups [22XS[122X, we do not claim to describe
  the [13Xmodule[113X structure of [22XS[122X in the third column of the table; we have kept the
  [5XAtlas[105X  description  of  the normal subgroups of the maximal overgroups of [22XS[122X.
  For  example,  the  subgroup  [22XS[122X  listed  for  [22XCo_2[122X  is  contained in maximal
  subgroups of the types [22X2^1+8_+:S_6(2)[122X and [22X2^4+10(S_4 × S_3)[122X, so [22XS[122X has normal
  subgroups  of  the  orders [22X2[122X, [22X2^4[122X, [22X2^9[122X, [22X2^14[122X, and [22X2^16[122X; more [5XAtlas[105X conformal
  notations would be [22X2^[14](S_4 × S_3)[122X or [22X2^[16](S_3 × S_3)[122X.[133X
  
  [33X[0;0YAs a corollary (see Section [14X6.5[114X), we read off the following.[133X
  
  [33X[0;0YCorollary:[133X
  
  [33X[0;0YExactly  the  following  almost  simple  groups [22XG[122X with sporadic simple socle
  contain a solvable subgroup [22XS[122X with the property [22X|S|^2 ≥ |G|[122X.[133X
  
  
  [24X[33X[0;6YFi_23, J_2, J_2.2, M_11, M_12, M_22.2.[133X[124X
  
  [33X[0;0YThe  existence  of  the  subgroups  [22XS[122X  of [22XG[122X with the structure and the order
  stated  in Table I and II follows from the [5XAtlas[105X: It is obvious in the cases
  where [22XS[122X is maximal in [22XG[122X, and in the other cases, the [5XAtlas[105X information about
  a nonsolvable factor group of a maximal subgroup of [22XG[122X suffices.[133X
  
  [33X[0;0YIn order to show that the table rows for the group [22XG[122X are correct, we have to
  show the following.[133X
  
  [30X    [33X[0;6Y[22XG[122X does not contain solvable subgroups of order larger than [22X|S|[122X.[133X
  
  [30X    [33X[0;6Y[22XG[122X contain exactly the conjugacy classes of solvable subgroups of order
        [22X|S|[122X that are listed in the second column of Table II.[133X
  
  [30X    [33X[0;6Y[22XS[122X  is  contained  exactly in the maximal subgroups listed in the third
        column of Table II.[133X
  
  [33X[0;0Y[13XRemark:[113X[133X
  
  [30X    [33X[0;6YEach  of  the  groups  [22XM_12[122X  and [22XHe[122X contains two classes of isomorphic
        solvable subgroups of maximal order.[133X
  
  [30X    [33X[0;6YEach of the groups [22XRu[122X, [22XTh[122X, and [22XM[122X contains two classes of nonisomorphic
        solvable subgroups of maximal order.[133X
  
  [30X    [33X[0;6YThe  solvable  subgroups  of maximal order in [22XMcL.2[122X have the structure
        [22X3^1+4_+:4S_4[122X,  the  subgroups  are maximal in the maximal subgroups of
        the  structures  [22X3^1+4_+:4S_5[122X  and  [22XU_4(3).2_3[122X in [22XMcL.2[122X. Note that the
        [5XAtlas[105X   claims  another  structure  for  these  maximal  subgroups  of
        [22XU_4(3).2_3[122X, see [CCN+85, p. 52].[133X
  
  [30X    [33X[0;6YThe solvable subgroups of maximal order in [22XCo_3[122X are the normalizers of
        Sylow [22X3[122X-subgroups of [22XCo_3[122X.[133X
  
      ┌──────────┬──────────────────┬───────────────────────┬──────────────┬─────┬──┐
      │ [22XG[122X        │              [22X|S|[122X │                 [22X|G/S|[122X │ [22Xlog_|G|(|S|)[122X │  p. │  │ 
      ├──────────┼──────────────────┼───────────────────────┼──────────────┼─────┼──┤
      ├──────────┼──────────────────┼───────────────────────┼──────────────┼─────┼──┤
      │ [22XM_11[122X     │              [22X144[122X │                    [22X55[122X │       [22X0.5536[122X │  [22X18[122X │  │ 
      │ [22XM_12[122X     │              [22X432[122X │                   [22X220[122X │       [22X0.5294[122X │  [22X33[122X │  │ 
      │ [22XM_12.2[122X   │              [22X432[122X │                   [22X440[122X │       [22X0.4992[122X │  [22X33[122X │  │ 
      │ [22XJ_1[122X      │              [22X168[122X │                  [22X1045[122X │       [22X0.4243[122X │  [22X36[122X │  │ 
      │ [22XM_22[122X     │              [22X576[122X │                   [22X770[122X │       [22X0.4888[122X │  [22X39[122X │  │ 
      │ [22XM_22.2[122X   │             [22X1152[122X │                   [22X770[122X │       [22X0.5147[122X │  [22X39[122X │  │ 
      │ [22XJ_2[122X      │             [22X1152[122X │                   [22X525[122X │       [22X0.5295[122X │  [22X42[122X │  │ 
      │ [22XJ_2.2[122X    │             [22X2304[122X │                   [22X525[122X │       [22X0.5527[122X │  [22X42[122X │  │ 
      │ [22XM_23[122X     │             [22X1152[122X │                  [22X8855[122X │       [22X0.4368[122X │  [22X71[122X │  │ 
      │ [22XHS[122X       │             [22X2000[122X │                 [22X22176[122X │       [22X0.4316[122X │  [22X80[122X │  │ 
      │ [22XHS.2[122X     │             [22X4000[122X │                 [22X22176[122X │       [22X0.4532[122X │  [22X80[122X │  │ 
      │ [22XJ_3[122X      │             [22X1944[122X │                 [22X25840[122X │       [22X0.4270[122X │  [22X82[122X │  │ 
      │ [22XJ_3.2[122X    │             [22X3888[122X │                 [22X25840[122X │       [22X0.4486[122X │  [22X82[122X │  │ 
      │ [22XM_24[122X     │            [22X13824[122X │                 [22X17710[122X │       [22X0.4935[122X │  [22X96[122X │  │ 
      │ [22XMcL[122X      │            [22X11664[122X │                 [22X77000[122X │       [22X0.4542[122X │ [22X100[122X │  │ 
      │ [22XMcL.2[122X    │            [22X23328[122X │                 [22X77000[122X │       [22X0.4719[122X │ [22X100[122X │  │ 
      │ [22XHe[122X       │            [22X13824[122X │                [22X291550[122X │       [22X0.4310[122X │ [22X104[122X │  │ 
      │ [22XHe.2[122X     │            [22X18432[122X │                [22X437325[122X │       [22X0.4305[122X │ [22X104[122X │  │ 
      │ [22XRu[122X       │            [22X49152[122X │               [22X2968875[122X │       [22X0.4202[122X │ [22X126[122X │  │ 
      │ [22XSuz[122X      │           [22X139968[122X │               [22X3203200[122X │       [22X0.4416[122X │ [22X131[122X │  │ 
      │ [22XSuz.2[122X    │           [22X279936[122X │               [22X3203200[122X │       [22X0.4557[122X │ [22X131[122X │  │ 
      │ [22XO'N[122X      │            [22X25920[122X │              [22X17778376[122X │       [22X0.3784[122X │ [22X132[122X │  │ 
      │ [22XO'N.2[122X    │            [22X51840[122X │              [22X17778376[122X │       [22X0.3940[122X │ [22X132[122X │  │ 
      │ [22XCo_3[122X     │            [22X69984[122X │               [22X7084000[122X │       [22X0.4142[122X │ [22X134[122X │  │ 
      │ [22XCo_2[122X     │          [22X2359296[122X │              [22X17931375[122X │       [22X0.4676[122X │ [22X154[122X │  │ 
      │ [22XFi_22[122X    │          [22X5038848[122X │              [22X12812800[122X │       [22X0.4853[122X │ [22X163[122X │  │ 
      │ [22XFi_22.2[122X  │         [22X10077696[122X │              [22X12812800[122X │       [22X0.4963[122X │ [22X163[122X │  │ 
      │ [22XHN[122X       │          [22X2000000[122X │             [22X136515456[122X │       [22X0.4364[122X │ [22X166[122X │  │ 
      │ [22XHN.2[122X     │          [22X4000000[122X │             [22X136515456[122X │       [22X0.4479[122X │ [22X166[122X │  │ 
      │ [22XLy[122X       │           [22X900000[122X │           [22X57516865560[122X │       [22X0.3562[122X │ [22X174[122X │  │ 
      │ [22XTh[122X       │           [22X944784[122X │           [22X96049408000[122X │       [22X0.3523[122X │ [22X177[122X │  │ 
      │ [22XFi_23[122X    │       [22X3265173504[122X │            [22X1252451200[122X │       [22X0.5111[122X │ [22X177[122X │  │ 
      │ [22XCo_1[122X     │         [22X84934656[122X │           [22X48952653750[122X │       [22X0.4258[122X │ [22X183[122X │  │ 
      │ [22XJ_4[122X      │         [22X28311552[122X │         [22X3065023459190[122X │       [22X0.3737[122X │ [22X190[122X │  │ 
      │ [22XFi_24'[122X   │      [22X29386561536[122X │        [22X42713595724800[122X │       [22X0.4343[122X │ [22X207[122X │  │ 
      │ [22XFi_24'.2[122X │      [22X58773123072[122X │        [22X42713595724800[122X │       [22X0.4413[122X │ [22X207[122X │  │ 
      │ [22XB[122X        │   [22X29686813949952[122X │ [22X139953768303693093750[122X │       [22X0.4007[122X │ [22X217[122X │  │ 
      │ [22XM[122X        │ [22X2849934139195392[122X │ [22X283521437805098363752[122X │              │     │  │ 
      │          │                  │    [22X344287234566406250[122X │       [22X0.2866[122X │ [22X234[122X │  │ 
      └──────────┴──────────────────┴───────────────────────┴──────────────┴─────┴──┘
  
       [1XTable:[101X  Table  I: Solvable subgroups of maximal order – orders and
       indices
  
  
      ┌─────────┬──────────────────────────────┬─────────────────────┬─────────   ─────┬────────┐
      │ [22XG[122X       │ [22XS[122X                            │ Max. overgroups     │ [CCN+85]        │ see    │ 
      ├─────────┼──────────────────────────────┼─────────────────────┼─────────   ─────┼────────┤
      ├─────────┼──────────────────────────────┼─────────────────────┼─────────   ─────┼────────┤
      │ [22XM_11[122X    │ [22X3^2:Q_8.2[122X                    │ [22XS[122X                   │       18        │ [14X6.3[114X    │ 
      │ [22XM_12[122X    │ [22X3^2:2S_4[122X                     │ [22XS[122X                   │       33        │ [14X6.3[114X    │ 
      │         │ [22X3^2:2S_4[122X                     │ [22XS[122X                   │       33        │ [14X6.3[114X    │ 
      │ [22XM_12.2[122X  │ [22X3^2:2S_4[122X                     │ [22XM_12[122X                │       33        │ [14X6.3[114X    │ 
      │ [22XJ_1[122X     │ [22X2^3:7:3[122X                      │ [22XS[122X                   │       36        │ [14X6.3[114X    │ 
      │ [22XM_22[122X    │ [22X2^4:3^2:4[122X                    │ [22X2^4:A_6[122X             │       39   (4)  │ [14X6.3[114X    │ 
      │ [22XM_22.2[122X  │ [22X2^4:3^2:D_8[122X                  │ [22X2^4:S_6[122X             │       39   (4)  │ [14X6.3[114X    │ 
      │ [22XJ_2[122X     │ [22X2^2+4:(3 × S_3)[122X              │ [22XS[122X                   │       42        │ [14X6.3[114X    │ 
      │ [22XJ_2.2[122X   │ [22X2^2+4:(S_3 × S_3)[122X            │ [22XS[122X                   │       42        │ [14X6.3[114X    │ 
      │ [22XM_23[122X    │ [22X2^4:(3 × A_4):2[122X              │ [22X2^4:(3 × A_5):2[122X,    │       71   (2)  │ [14X6.3[114X    │ 
      │         │                              │ [22X2^4:A_7[122X             │            (10) │        │ 
      │ [22XHS[122X      │ [22X5^1+2_+:8:2[122X                  │ [22XU_3(5).2[122X            │       80   (34) │ [14X6.3[114X    │ 
      │         │                              │ [22XU_3(5).2[122X            │                 │ [14X6.3[114X    │ 
      │ [22XHS.2[122X    │ [22X5^1+2_+:[2^5][122X                │ [22XS[122X                   │       80   (34) │ [14X6.3[114X    │ 
      │ [22XJ_3[122X     │ [22X3^2.3^1+2_+:8[122X                │ [22XS[122X                   │       82        │ [14X6.3[114X    │ 
      │ [22XJ_3.2[122X   │ [22X3^2.3^1+2_+:QD_16[122X            │ [22XS[122X                   │       82        │ [14X6.3[114X    │ 
      │ [22XM_24[122X    │ [22X2^6:3^1+2_+:D_8[122X              │ [22X2^6:3.S_6[122X           │       96   (4)  │ [14X6.3[114X    │ 
      │ [22XMcL[122X     │ [22X3^1+4_+:2S_4[122X                 │ [22X3^1+4_+:2S_5[122X,       │      100   (2)  │ [14X6.3[114X    │ 
      │         │                              │ [22XU_4(3)[122X              │            (52) │ [14X6.3[114X    │ 
      │ [22XMcL.2[122X   │ [22X3^1+4_+:4S_4[122X                 │ [22X3^1+4_+:4S_5[122X,       │      100   (2)  │ [14X6.3[114X    │ 
      │         │                              │ [22XU_4(3).2_3[122X          │            (52) │ [14X6.3[114X    │ 
      │ [22XHe[122X      │ [22X2^6:3^1+2_+:D_8[122X              │ [22X2^6:3.S_6[122X           │      104   (4)  │ [14X6.3[114X    │ 
      │         │ [22X2^6:3^1+2_+:D_8[122X              │ [22X2^6:3.S_6[122X           │            (4)  │ [14X6.3[114X    │ 
      │ [22XHe.2[122X    │ [22X2^4+4.(S_3 × S_3).2[122X          │ [22XS[122X                   │      104        │ [14X6.3[114X    │ 
      │ [22XRu[122X      │ [22X2.2^4+6:S_4[122X                  │ [22X2^3+8:L_3(2)[122X,       │      126   (3)  │ [14X6.4-1[114X  │ 
      │         │                              │ [22X2.2^4+6:S_5[122X         │            (2)  │        │ 
      │         │ [22X2^3+8:S_4[122X                    │ [22X2^3+8:L_3(2)[122X,       │            (3)  │ [14X6.4-1[114X  │ 
      │ [22XSuz[122X     │ [22X3^2+4:2(A_4 × 2^2).2[122X         │ [22XS[122X                   │      131        │ [14X6.4-2[114X  │ 
      │ [22XSuz.2[122X   │ [22X3^2+4:2(S_4 × D_8)[122X           │ [22XS[122X                   │      131        │ [14X6.4-2[114X  │ 
      │ [22XO'N[122X     │ [22X3^4:2^1+4_-D_10[122X              │ [22XS[122X                   │      132        │ [14X6.4-3[114X  │ 
      │ [22XO'N.2[122X   │ [22X3^4:2^1+4_-.(5:4)[122X            │ [22XS[122X                   │      132        │ [14X6.4-3[114X  │ 
      │ [22XCo_3[122X    │ [22X3^1+4_+:4.3^2:D_8[122X            │ [22X3^1+4_+:4S_6[122X        │      134   (4)  │ [14X6.3[114X    │ 
      │         │                              │ [22X3^5:(2 × M_11)[122X      │            (18) │        │ 
      │ [22XCo_2[122X    │ [22X2^4+10(S_4 × S_3)[122X            │ [22X2^1+8_+:S_6(2)[122X,     │      154   (46) │ [14X6.4-4[114X  │ 
      │         │                              │ [22X2^4+10(S_5 × S_3)[122X   │            (2)  │        │ 
      │ [22XFi_22[122X   │ [22X3^1+6_+:2^3+4:3^2:2[122X          │ [22XS[122X                   │      163        │ [14X6.4-5[114X  │ 
      │ [22XFi_22.2[122X │ [22X3^1+6_+:2^3+4:(S_3 × S_3)[122X    │ [22XS[122X                   │      163        │ [14X6.4-5[114X  │ 
      │ [22XHN[122X      │ [22X5^1+4_+:2^1+4_-.5.4[122X          │ [22XS[122X                   │      166        │ [14X6.4-6[114X  │ 
      │ [22XHN.2[122X    │ [22X5^1+4_+:(4 Y 2^1+4_-.5.4)[122X    │ [22XS[122X                   │      166        │ [14X6.4-6[114X  │ 
      │ [22XLy[122X      │ [22X5^1+4_+:4.3^2:D_8[122X            │ [22X5^1+4_+:4S_6[122X        │      174   (4)  │ [14X6.4-7[114X  │ 
      │ [22XTh[122X      │ [22X[3^9].2S_4[122X                   │ [22XS[122X                   │      177        │ [14X6.4-8[114X  │ 
      │         │ [22X3^2.[3^7].2S_4[122X               │ [22XS[122X                   │                 │        │ 
      │ [22XFi_23[122X   │ [22X3^1+8_+.2^1+6_-.3^1+2_+.2S_4[122X │ [22XS[122X                   │      177        │ [14X6.4-9[114X  │ 
      │ [22XCo_1[122X    │ [22X2^4+12.(S_3 × 3^1+2_+:D_8)[122X   │ [22X2^4+12.(S_3 × 3S_6)[122X │      183        │ [14X6.4-10[114X │ 
      └─────────┴──────────────────────────────┴─────────────────────┴─────────   ─────┴────────┘
  
       [1XTable:[101X  Table II: Solvable subgroups of maximal order – structures
       and overgroups
  
  
      ┌──────────┬─────────────────────────────────────┬──────────────────────────┬─────────   ───────┬────────┐
      │ [22XG[122X        │ [22XS[122X                                   │ Max. overgroups          │ [CCN+85]          │ see    │ 
      ├──────────┼─────────────────────────────────────┼──────────────────────────┼─────────   ───────┼────────┤
      ├──────────┼─────────────────────────────────────┼──────────────────────────┼─────────   ───────┼────────┤
      │ [22XJ_4[122X      │ [22X2^11:2^6:3^1+2_+:D_8[122X                │ [22X2^11:M_24[122X,               │      190   (96)   │ [14X6.4-11[114X │ 
      │          │                                     │ [22X2^1+12_+.3M_22:2[122X         │            (39)   │        │ 
      │ [22XFi_24'[122X   │ [22X3^1+10_+:2^1+6_-:3^1+2_+:2S_4[122X       │ [22X3^1+10_+:U_5(2):2[122X        │      207   (73)   │ [14X6.4-12[114X │ 
      │ [22XFi_24'.2[122X │ [22X3^1+10_+:(2 × 2^1+6_-:3^1+2_+:2S_4)[122X │ [22X3^1+10_+:(2 × U_5(2):2)[122X  │      207   (73)   │ [14X6.4-12[114X │ 
      │ [22XB[122X        │ [22X2^2+10+20(2^4:3^2:D_8 × S_3)[122X        │ [22X2^2+10+20(M_22:2 × S_3)[122X, │      217   (39)   │ [14X6.4-13[114X │ 
      │          │                                     │ [22X2^9+16S_8(2)[122X             │            (123)  │        │ 
      │ [22XM[122X        │ [22X2^1+2+6+12+18.(S_4 × 3^1+2_+:D_8)[122X   │ [22X2^[39].(L_3(2) × 3S_6)[122X,  │      234   (3, 4) │ [14X6.4-14[114X │ 
      │          │                                     │ [22X2^1+24_+.Co_1[122X            │            (183)  │        │ 
      │          │ [22X2^2+1+6+12+18.(S_4 × 3^1+2_+:D_8)[122X   │ [22X2^[39].(L_3(2) × 3S_6)[122X,  │            (3, 4) │ [14X6.4-14[114X │ 
      │          │                                     │ [22X2^2+11+22.(M_24 × S_3)[122X   │            (96)   │        │ 
      └──────────┴─────────────────────────────────────┴──────────────────────────┴─────────   ───────┴────────┘
  
       [1XTable:[101X  Table II: Solvable subgroups of maximal order – structures
       and overgroups (continued)
  
  
  
  [1X6.2 [33X[0;0YThe Approach[133X[101X
  
  [33X[0;0YWe  combine the information in the [5XAtlas[105X [CCN+85] with explicit computations
  using   the   [5XGAP[105X   system [GAP21],   in   particular  its  Character  Table
  Library [Bre22]  and  its  library of Tables of Marks [NMP18]. First we load
  these two packages.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CTblLib", "1.2", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "TomLib", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  orders  of  solvable  subgroups of maximal order will be collected in a
  global record [10XMaxSolv[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv:= rec();;[127X[104X
  [4X[32X[104X
  
  
  [1X6.2-1 [33X[0;0YUse the Table of Marks[133X[101X
  
  [33X[0;0YIf the [5XGAP[105X library of Tables of Marks [NMP18] contains the table of marks of
  a  group  [22XG[122X then we can easily inspect all conjugacy classes of subgroups of
  [22XG[122X.  The  following small [5XGAP[105X function can be used for that. It returns [9Xfalse[109X
  if  the table of marks of the group with the name [10Xname[110X is not available, and
  the  list  [10X[  name,  n,  super  ][110X otherwise, where [10Xn[110X is the maximal order of
  solvable  subgroups  of  [22XG[122X, and [10Xsuper[110X is a list of lists; for each conjugacy
  class  of solvable subgroups [22XS[122X of order [10Xn[110X, [10Xsuper[110X contains the list of orders
  of  representatives  [22XM[122X  of the classes of maximal subgroups of [22XG[122X such that [22XM[122X
  contains a conjugate of [22XS[122X.[133X
  
  [33X[0;0YNote  that  a  subgroup  in  the  [22Xi[122X-th  class of a table of marks contains a
  subgroup in the [22Xj[122X-th class if and only if the entry in the position [22X(i,j)[122X of
  the  table  of marks is nonzero. For tables of marks objects in [5XGAP[105X, this is
  the  case  if and only if [22Xj[122X is contained in the [22Xi[122X-th row of the list that is
  stored  as  the value of the attribute [10XSubsTom[110X of the table of marks object;
  for this test, one need not unpack the matrix of marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaximalSolvableSubgroupInfoFromTom:= function( name )[127X[104X
    [4X[25X>[125X [27X    local tom,          # table of marks for `name'[127X[104X
    [4X[25X>[125X [27X          n,            # maximal order of a solvable subgroup[127X[104X
    [4X[25X>[125X [27X          maxsubs,      # numbers of the classes of subgroups of order `n'[127X[104X
    [4X[25X>[125X [27X          orders,       # list of orders of the classes of subgroups[127X[104X
    [4X[25X>[125X [27X          i,            # loop over the classes of subgroups[127X[104X
    [4X[25X>[125X [27X          maxes,        # list of positions of the classes of max. subgroups[127X[104X
    [4X[25X>[125X [27X          subs,         # `SubsTom' value[127X[104X
    [4X[25X>[125X [27X          cont;         # list of list of positions of max. subgroups[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    tom:= TableOfMarks( name );[127X[104X
    [4X[25X>[125X [27X    if tom = fail then[127X[104X
    [4X[25X>[125X [27X      return false;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    n:= 1;[127X[104X
    [4X[25X>[125X [27X    maxsubs:= [];[127X[104X
    [4X[25X>[125X [27X    orders:= OrdersTom( tom );[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( orders ) ] do[127X[104X
    [4X[25X>[125X [27X      if IsSolvableTom( tom, i ) then[127X[104X
    [4X[25X>[125X [27X        if orders[i] = n then[127X[104X
    [4X[25X>[125X [27X          Add( maxsubs, i );[127X[104X
    [4X[25X>[125X [27X        elif orders[i] > n then[127X[104X
    [4X[25X>[125X [27X          n:= orders[i];[127X[104X
    [4X[25X>[125X [27X          maxsubs:= [ i ];[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    maxes:= MaximalSubgroupsTom( tom )[1];[127X[104X
    [4X[25X>[125X [27X    subs:= SubsTom( tom );[127X[104X
    [4X[25X>[125X [27X    cont:= List( maxsubs, j -> Filtered( maxes, i -> j in subs[i] ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return [ name, n, List( cont, l -> orders{ l } ) ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.2-2 [33X[0;0YUse Information from the Character Table Library[133X[101X
  
  [33X[0;0YThe [5XGAP[105X Character Table Library contains the character tables of all maximal
  subgroups  of  sporadic  simple  groups,  except for the Monster group. This
  information can be used as follows.[133X
  
  [33X[0;0YWe  start,  for a sporadic simple group [22XG[122X, with a known solvable subgroup of
  order [22Xn[122X, say, in [22XG[122X. In order to show that [22XG[122X contains no solvable subgroup of
  larger  order,  it suffices to show that no maximal subgroup of [22XG[122X contains a
  larger solvable subgroup.[133X
  
  [33X[0;0YThe  point  is that usually the orders of the maximal subgroups of [22XG[122X are not
  much  larger  than  [22Xn[122X,  and  that  a  maximal subgroup [22XM[122X contains a solvable
  subgroup  of  order  [22Xn[122X only if the factor group of [22XM[122X by its largest solvable
  normal  subgroup [22XN[122X contains a solvable subgroup of order [22Xn/|N|[122X. This reduces
  the question to relatively small groups.[133X
  
  [33X[0;0YWhat  we  can check [13Xautomatically[113X from the character table of [22XM/N[122X is whether
  [22XM/N[122X  can  contain  subgroups  (solvable  or not) of indices between five and
  [22X|M|/n[122X,  by computing possible permutation characters of these degrees. (Note
  that  a  solvable  subgroup  of a nonsolvable group has index at least five.
  This  lower  bound could be improved for example by considering the smallest
  degree of a nontrivial character, but this is not an issue here.)[133X
  
  [33X[0;0YThen  we  are  left with a –hopefully short– list of maximal subgroups of [22XG[122X,
  together  with  upper  bounds on the indices of possible solvable subgroups;
  excluding these possibilities then yields that the initially chosen solvable
  subgroup of [22XG[122X is indeed the largest one.[133X
  
  [33X[0;0YThe  following  [5XGAP[105X function can be used to compute this information for the
  character  table [10XtblM[110X of [22XM[122X and a given order [10Xminorder[110X. It returns [9Xfalse[109X if [22XM[122X
  cannot  contain  a solvable subgroup of order at least [10Xminorder[110X, otherwise a
  list  [10X[  tblM,  m,  k  ][110X where [10Xm[110X is the maximal index of a subgroup that has
  order  at  least [10Xminorder[110X, and [10Xk[110X is the minimal index of a possible subgroup
  of  [22XM[122X (a proper subgroup if [22XM[122X is nonsolvable), according to the [5XGAP[105X function
  [2XPermChars[102X ([14XReference: PermChars[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSolvableSubgroupInfoFromCharacterTable:= function( tblM, minorder )[127X[104X
    [4X[25X>[125X [27X    local maxindex,  # index of subgroups of order `minorder'[127X[104X
    [4X[25X>[125X [27X          N,         # class positions describing a solvable normal subgroup[127X[104X
    [4X[25X>[125X [27X          fact,      # character table of the factor by `N'[127X[104X
    [4X[25X>[125X [27X          classes,   # class sizes in `fact'[127X[104X
    [4X[25X>[125X [27X          nsg,       # list of class positions of normal subgroups[127X[104X
    [4X[25X>[125X [27X          i;         # loop over the possible indices[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    maxindex:= Int( Size( tblM ) / minorder );[127X[104X
    [4X[25X>[125X [27X    if   maxindex = 0 then[127X[104X
    [4X[25X>[125X [27X      return false;[127X[104X
    [4X[25X>[125X [27X    elif IsSolvableCharacterTable( tblM ) then[127X[104X
    [4X[25X>[125X [27X      return [ tblM, maxindex, 1 ];[127X[104X
    [4X[25X>[125X [27X    elif maxindex < 5 then[127X[104X
    [4X[25X>[125X [27X      return false;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    N:= [ 1 ];[127X[104X
    [4X[25X>[125X [27X    fact:= tblM;[127X[104X
    [4X[25X>[125X [27X    repeat[127X[104X
    [4X[25X>[125X [27X      fact:= fact / N;[127X[104X
    [4X[25X>[125X [27X      classes:= SizesConjugacyClasses( fact );[127X[104X
    [4X[25X>[125X [27X      nsg:= Difference( ClassPositionsOfNormalSubgroups( fact ), [ [ 1 ] ] );[127X[104X
    [4X[25X>[125X [27X      N:= First( nsg, x -> IsPrimePowerInt( Sum( classes{ x } ) ) );[127X[104X
    [4X[25X>[125X [27X    until N = fail;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    for i in Filtered( DivisorsInt( Size( fact ) ),[127X[104X
    [4X[25X>[125X [27X                       d -> 5 <= d and d <= maxindex ) do[127X[104X
    [4X[25X>[125X [27X      if Length( PermChars( fact, rec( torso:= [ i ] ) ) ) > 0 then[127X[104X
    [4X[25X>[125X [27X        return [ tblM, maxindex, i ];[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return false;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YCases where the Table of Marks is available in [5XGAP[105X[101X[1X[133X[101X
  
  [33X[0;0YFor  twelve sporadic simple groups, the [5XGAP[105X library of Tables of Marks knows
  the tables of marks, so we can use [10XMaximalSolvableSubgroupInfoFromTom[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsolvinfo:= Filtered( List([127X[104X
    [4X[25X>[125X [27X        AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X                                IsDuplicateTable, false ),[127X[104X
    [4X[25X>[125X [27X        MaximalSolvableSubgroupInfoFromTom ), x -> x <> false );;[127X[104X
    [4X[25Xgap>[125X [27Xfor entry in solvinfo do[127X[104X
    [4X[25X>[125X [27X     MaxSolv.( entry[1] ):= entry[2];[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xfor entry in solvinfo do                                 [127X[104X
    [4X[25X>[125X [27X     Print( String( entry[1], 5 ), String( entry[2], 7 ),[127X[104X
    [4X[25X>[125X [27X            String( entry[3], 28 ), "\n" );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X  Co3  69984     [ [ 3849120, 699840 ] ][128X[104X
    [4X[28X   HS   2000      [ [ 252000, 252000 ] ][128X[104X
    [4X[28X   He  13824  [ [ 138240 ], [ 138240 ] ][128X[104X
    [4X[28X   J1    168                 [ [ 168 ] ][128X[104X
    [4X[28X   J2   1152                [ [ 1152 ] ][128X[104X
    [4X[28X   J3   1944                [ [ 1944 ] ][128X[104X
    [4X[28X  M11    144                 [ [ 144 ] ][128X[104X
    [4X[28X  M12    432        [ [ 432 ], [ 432 ] ][128X[104X
    [4X[28X  M22    576                [ [ 5760 ] ][128X[104X
    [4X[28X  M23   1152         [ [ 40320, 5760 ] ][128X[104X
    [4X[28X  M24  13824              [ [ 138240 ] ][128X[104X
    [4X[28X  McL  11664      [ [ 3265920, 58320 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  for [22XJ_1[122X, [22XJ_2[122X, [22XJ_3[122X, [22XM_11[122X, and [22XM_12[122X, the subgroup [22XS[122X is maximal.
  For [22XM_12[122X and [22XHe[122X, there are two classes of subgroups [22XS[122X. For the other groups,
  the  class  of  subgroups  [22XS[122X  is unique, and there are one or two classes of
  maximal  subgroups  of  [22XG[122X  that  contain  [22XS[122X.  From the shown orders of these
  maximal  subgroups,  their structures can be read off from the [5XAtlas[105X, on the
  pages listed in Table II.[133X
  
  [33X[0;0YSimilarly,  the  [5XAtlas[105X  tells  us about the extensions of the subgroups [22XS[122X in
  Aut[22X(G)[122X. In particular,[133X
  
  [30X    [33X[0;6Ythe  order  [22X2000[122X subgroups of [22XHS[122X are contained in maximal subgroups of
        the type [22XU_3(5).2[122X (two classes) which do not extend to [22XHS.2[122X, but there
        are  novelties of the type [22X5^1+2_+:[2^5][122X and of the order [22X4000[122X, so the
        solvable subgroups of maximal order in [22XHS[122X do in fact extend to [22XHS.2[122X.[133X
  
  [30X    [33X[0;6Ythe  order [22X13824[122X subgroups of [22XHe[122X are contained in maximal subgroups of
        the type [22X2^6:3S_6[122X (two classes) which do not extend to [22XHe.2[122X, but there
        are  novelties of the type [22X2^4+4.(S_3 × S_3).2[122X and of the order [22X18432[122X.
        (So  the  solvable subgroups [22XS[122X of maximal order in [22XHe[122X do not extend to
        [22XHe.2[122X but there are larger solvable subgroups in [22XHe.2[122X.)[133X
  
        [33X[0;6YWe  inspect  the maximal subgroups of [22XHe.2[122X in order to show that these
        are  in  fact the solvable subgroups of maximal order (see [CCN+85, p.
        104]): Any other solvable subgroup of order at least [22Xn[122X in [22XHe.2[122X must be
        contained in a subgroup of one of the types [22XS_4(4).4[122X (of index at most
        [22X212[122X),  [22X2^2.L_3(4).D_12[122X  (of index at most [22X52[122X), or [22X2^1+6_+.L_3(2).2[122X (of
        index at most [22X2[122X). By [CCN+85, pp. 44, 23, 3], this is not the case.[133X
  
  [30X    [33X[0;6Ythe  maximal  subgroups  of  order  [22X1152[122X in [22XJ_2[122X extend to subgroups of
        order [22X2304[122X in [22XJ_2.2[122X.[133X
  
  [30X    [33X[0;6Ythe  maximal subgroups of order [22X1944[122X in [22XJ_3[122X extend to subgroups of the
        type [22X3^2.3^1+2_+:8.2[122X and of order [22X3888[122X in [22XJ_3.2[122X. (The structure stated
        in [CCN+85, p. 82] is not correct, see [BN95].)[133X
  
  [30X    [33X[0;6Ythe maximal subgroups of order [22X432[122X in [22XM_12[122X (two classes) do [13Xnot[113X extend
        in [22XM_12.2[122X, and we see from the table of marks of [22XM_12.2[122X that there are
        no  larger  solvable  subgroups  in  this  group,  i. e., the solvable
        subgroups of maximal order in [22XM_12.2[122X lie in [22XM_12[122X.[133X
  
  [30X    [33X[0;6Ythe  order [22X576[122X subgroups of [22XM_22[122X are contained in maximal subgroups of
        the  type  [22X2^4:A_6[122X  which  extend  to subgroups of the type [22X2^4:S_6[122X in
        [22XM_22.2[122X,  so the solvable subgroups of maximal order in [22XM_22.2[122X have the
        type  [22X2^4:3^2:D_8[122X  and  the order [22X1152[122X. In fact the structure is [22XS_4 ≀
        S_2[122X.[133X
  
  [30X    [33X[0;6Ythe order [22X11664[122X subgroups of [22XMcL[122X are contained in maximal subgroups of
        the type [22X3^1+4_+:2S_5[122X which extend to subgroups of the type [22X3^1+4:4S_5[122X
        in [22XMcL.2[122X, so the solvable subgroups of maximal order in [22XMcL.2[122X have the
        type [22X3^1+4:4S_4[122X and the order [22X23328[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "HS.2" ):= 2 * MaxSolv.( "HS" );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 2^(4+4) * ( 6 * 6 ) * 2;  MaxSolv.( "He.2" ):= n;;[127X[104X
    [4X[28X18432[128X[104X
    [4X[25Xgap>[125X [27XList( [ Size( CharacterTable( "S4(4).4" ) ),[127X[104X
    [4X[25X>[125X [27X           Factorial( 5 )^2 * 2,[127X[104X
    [4X[25X>[125X [27X           Size( CharacterTable( "2^2.L3(4).D12" ) ),[127X[104X
    [4X[25X>[125X [27X           2^7 * Size( CharacterTable( "L3(2)" ) ) * 2,[127X[104X
    [4X[25X>[125X [27X           7^2 * 2 * Size( CharacterTable( "L2(7)" ) ) * 2,[127X[104X
    [4X[25X>[125X [27X           3 * Factorial( 7 ) * 2 ], i -> Int( i / n ) );[127X[104X
    [4X[28X[ 212, 1, 52, 2, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "J2.2" ):= 2 * MaxSolv.( "J2" );;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "J3.2" ):= 2 * MaxSolv.( "J3" );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= MaximalSolvableSubgroupInfoFromTom( "M12.2" );[127X[104X
    [4X[28X[ "M12.2", 432, [ [ 95040 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "M12.2" ):= info[2];;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "M22.2" ):= 2 * MaxSolv.( "M22" );;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "McL.2" ):= 2 * MaxSolv.( "McL" );;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YCases where the Table of Marks is not available in [5XGAP[105X[101X[1X[133X[101X
  
  [33X[0;0Y}  We  use  the  [5XGAP[105X  function  [10XSolvableSubgroupInfoFromCharacterTable[110X,  and
  individual  arguments.  In several cases, information about smaller sporadic
  simple groups is needed, so we deal with the groups in increasing order.[133X
  
  
  [1X6.4-1 [33X[0;0Y[22XG = Ru[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XRu[122X  contains  exactly  two  conjugacy  classes  of nonisomorphic
  solvable subgroups of order [22Xn = 49152[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Ru" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 49152;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "2^3+8:L3(2)" ), 7, 7 ], [128X[104X
    [4X[28X  [ CharacterTable( "2.2^4+6:S5" ), 5, 5 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups of the structure [22X2.2^4+6:S_5[122X in [22XRu[122X contain one class
  of  solvable  subgroups  of  order  [22Xn[122X  and  with  the structure [22X2.2^4+6:S_4[122X,
  see [CCN+85, p. 126, p. 2].[133X
  
  [33X[0;0YThe  maximal  subgroups  of  the  structure  [22X2^3+8:L_3(2)[122X  in [22XRu[122X contain two
  classes  of  solvable subgroups of order [22Xn[122X and with the structure [22X2^3+8:S_4[122X,
  see [CCN+85,  p. 126, p. 3]. These groups are the stabilizers of vectors and
  two-dimensional subspaces, respectively, in the three-dimensional submodule;
  note  that  each  [22X2^3+8:L_3(2)[122X type subgroup [22XH[122X of [22XRu[122X is the normalizer of an
  elementary  abelian group of order eight all of whose involutions are in the
  [22XRu[122X-class  [10X2A[110X and are conjugate in [22XH[122X. Since the [22X2.2^4+6:S_5[122X type subgroups of
  [22XRu[122X  are  the  normalizers of [10X2A[110X-elements in [22XRu[122X, the groups in one of the two
  classes  in  question  coincide  with  the largest solvable subgroups in the
  [22X2.2^4+6:S_5[122X  type subgroups. The groups in the other class do not centralize
  a  [10X2A[110X-element  in  [22XRu[122X  and are therefore not isomorphic with the [22X2.2^4+6:S_4[122X
  type groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Ru" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27Xs:= info[1][1];;[127X[104X
    [4X[25Xgap>[125X [27Xcls:= SizesConjugacyClasses( s );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),[127X[104X
    [4X[25X>[125X [27X                   x -> Sum( cls{ x } ) = 2^3 );[127X[104X
    [4X[28X[ [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcls{ nsg[1] };[127X[104X
    [4X[28X[ 1, 7 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
  [4X[32X[104X
  
  
  [1X6.4-2 [33X[0;0Y[22XG = Suz[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XSuz[122X  contains  a unique conjugacy class of solvable subgroups of
  order [22Xn = 139968[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Suz" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 139968;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "G2(4)" ), 1797, 416 ], [128X[104X
    [4X[28X  [ CharacterTable( "3_2.U4(3).2_3'" ), 140, 72 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^5:M11" ), 13, 11 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^4+6:3a6" ), 7, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^2+4:2(2^2xa4)2" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  [22XS[122X  of the structure [22X3^2+4:2(A_4 × 2^2).2[122X in [22XSuz[122X are
  solvable and have order [22Xn[122X, see [CCN+85, p. 131].[133X
  
  [33X[0;0YIn  order  to  show  that  [22XSuz[122X contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in  [22XG_2(4)[122X  of  index at most [22X1797[122X (see [CCN+85, p. 97]), in [22XU_4(3).2_3^'[122X of
  index  at  most  [22X140[122X  (see [CCN+85,  p.  52]),  in  [22XM_11[122X of index at most [22X13[122X
  (see [CCN+85, p. 18]), and in [22XA_6[122X of index at most [22X7[122X (see [CCN+85, p. 4]).[133X
  
  [33X[0;0YThe  group  [22XS[122X  extends to a group of the structure [22X3^2+4:2(S_4 × D_8)[122X in the
  automorphism group [22XSuz.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Suz" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Suz.2" ):= 2 * n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-3 [33X[0;0Y[22XG = ON[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XON[122X  contains  a  unique conjugacy class of solvable subgroups of
  order [22X25920[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "ON" );;                                            [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 25920;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "L3(7).2" ), 144, 114 ], [128X[104X
    [4X[28X  [ CharacterTable( "ONM2" ), 144, 114 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^4:2^(1+4)D10" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal subgroups [22XS[122X of the structure [22X3^4:2^1+4_-D_10[122X in [22XON[122X are solvable
  and have order [22Xn[122X, see [CCN+85, pp. 132].[133X
  
  [33X[0;0YIn  order  to  show  that  [22XON[122X  contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in [22XL_3(7).2[122X of index at most [22X144[122X (see [CCN+85, p. 50]); note that the groups
  in the second class of maximal subgroups of [22XON[122X are isomorphic with [22XL_3(7).2[122X.[133X
  
  [33X[0;0YThe  group  [22XS[122X  extends  to  a group of order [22X|S.2|[122X in the automorphism group
  [22XON.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "ON" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "ON.2" ):= 2 * n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-4 [33X[0;0Y[22XG = Co_2[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XCo_2[122X  contains a unique conjugacy class of solvable subgroups of
  order [22X2359296[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Co2" );;                                           [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 2359296;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "U6(2).2" ), 7796, 672 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^10:m22:2" ), 385, 22 ], [128X[104X
    [4X[28X  [ CharacterTable( "McL" ), 380, 275 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^1+8:s6f2" ), 315, 28 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^1+4+6.a8" ), 17, 8 ], [128X[104X
    [4X[28X  [ CharacterTable( "U4(3).D8" ), 11, 8 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^(4+10)(S5xS3)" ), 5, 5 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  of  the structure [22X2^4+10(S_5 × S_3)[122X in [22XCo_2[122X contain
  solvable  subgroups  [22XS[122X  of order [22Xn[122X and with the structure [22X2^4+10(S_4 × S_3)[122X,
  see [CCN+85, p. 154].[133X
  
  [33X[0;0YThe  subgroups  [22XS[122X  are  contained  also in the maximal subgroups of the type
  [22X2^1+8_+:S_6(2)[122X; note that the [22X2^1+8_+:S_6(2)[122X type subgroups are described as
  normalizers of elements in the [22XCo_2[122X-class [10X2A[110X, and [22XS[122X normalizes an elementary
  abelian  group  of  order  [22X16[122X  containing  an [22XS[122X-class of length five that is
  contained in the [22XCo_2[122X-class [10X2A[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= info[7][1];[127X[104X
    [4X[28XCharacterTable( "2^(4+10)(S5xS3)" )[128X[104X
    [4X[25Xgap>[125X [27Xcls:= SizesConjugacyClasses( s );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),[127X[104X
    [4X[25X>[125X [27X                   x -> Sum( cls{ x } ) = 2^4 );[127X[104X
    [4X[28X[ [ 1, 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcls{ nsg[1] };[127X[104X
    [4X[28X[ 1, 5, 10 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  stabilizers  of these involutions in [22X2^4+10(S_5 × S_3)[122X have index five,
  they  are solvable, and they are contained in [22X2^1+8_+:S_6(2)[122X type subgroups,
  so they are [22XCo_2[122X-conjugates of [22XS[122X. (The corresponding subgroups of [22XS_6(2)[122X are
  maximal and have the type [22X2.[2^6]:(S_3 × S_3)[122X.)[133X
  
  [33X[0;0YIn order to show that [22XG[122X contains no other solvable subgroups of order larger
  than  or  equal  to  [22X|S|[122X,  we  check that there are no solvable subgroups in
  [22XU_6(2)[122X  of  index at most [22X7796[122X (see [CCN+85, p. 115]), in [22XM_22.2[122X of index at
  most  [22X385[122X  (see [CCN+85, p. 39] or Section [14X6.3[114X), in [22XMcL[122X of index at most [22X380[122X
  (see [CCN+85,  p.  100]  or  Section [14X6.3[114X),  in  [22XA_8[122X  of  index  at  most  [22X17[122X
  (see [CCN+85,  p.  20]), and in [22XU_4(3).D_8[122X of index at most [22X11[122X (see [CCN+85,
  p. 52]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Co2" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-5 [33X[0;0Y[22XG = Fi_22[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XFi_22[122X contains a unique conjugacy class of solvable subgroups of
  order [22X5038848[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Fi22" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 5038848;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "2.U6(2)" ), 3650, 672 ], [128X[104X
    [4X[28X  [ CharacterTable( "O7(3)" ), 910, 351 ], [128X[104X
    [4X[28X  [ CharacterTable( "Fi22M3" ), 910, 351 ], [128X[104X
    [4X[28X  [ CharacterTable( "O8+(2).3.2" ), 207, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^10:m22" ), 90, 22 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^(1+6):2^(3+4):3^2:2" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  [22XS[122X  of  the structure [22X3^1+6:2^3+4:3^2:2[122X in [22XFi_22[122X are
  solvable and have order [22Xn[122X, see [CCN+85, p. 163].[133X
  
  [33X[0;0YIn  order  to  show that [22XFi_22[122X contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in  [22XU_6(2)[122X  of index at most [22X3650[122X (see [CCN+85, p. 115]), in [22XO_7(3)[122X of index
  at  most  [22X910[122X  (see [CCN+85,  p. 109]), in [22XO_8^+(2).S_3[122X of index at most [22X207[122X
  (see [CCN+85,  p.  85]),  and in [22XM_22.2[122X of index at most [22X90[122X (see [CCN+85, p.
  39]  or  Section [14X6.3[114X);  note  that  the groups in the third class of maximal
  subgroups of [22XFi_22[122X are isomorphic with [22XO_7(3)[122X.[133X
  
  [33X[0;0YThe  group  [22XS[122X  extends  to  a group of order [22X|S.2|[122X in the automorphism group
  [22XFi_22.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Fi22" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Fi22.2" ):= 2 * n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-6 [33X[0;0Y[22XG = HN[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XHN[122X  contains  a  unique conjugacy class of solvable subgroups of
  order [22X2000000[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "HN" );; [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;                               [127X[104X
    [4X[25Xgap>[125X [27Xn:= 2000000;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "A12" ), 119, 12 ], [128X[104X
    [4X[28X  [ CharacterTable( "5^(1+4):2^(1+4).5.4" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal subgroups [22XS[122X of the structure [22X5^1+4:2^1+4.5.4[122X in [22XHN[122X are solvable
  and have order [22Xn[122X, see [CCN+85, p. 166].[133X
  
  [33X[0;0YIn  order  to  show  that  [22XHN[122X  contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in [22XA_12[122X of index at most [22X119[122X (see [CCN+85, p. 91]).[133X
  
  [33X[0;0YThe  group  [22XS[122X  extends  to  a group of order [22X|S.2|[122X in the automorphism group
  [22XHN.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "HN" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "HN.2" ):= 2 * n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-7 [33X[0;0Y[22XG = Ly[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XLy[122X  contains  a  unique conjugacy class of solvable subgroups of
  order [22X900000[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Ly" );;                                            [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 900000;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "G2(5)" ), 6510, 3906 ], [128X[104X
    [4X[28X  [ CharacterTable( "3.McL.2" ), 5987, 275 ], [128X[104X
    [4X[28X  [ CharacterTable( "5^3.psl(3,5)" ), 51, 31 ], [128X[104X
    [4X[28X  [ CharacterTable( "2.A11" ), 44, 11 ], [128X[104X
    [4X[28X  [ CharacterTable( "5^(1+4):4S6" ), 10, 6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  of the structure [22X5^(1+4):4S6[122X in [22XLy[122X contain solvable
  subgroups  [22XS[122X of order [22Xn[122X and with the structure [22X5^1+4:4.3^2.D_8[122X, see [CCN+85,
  p. 174].[133X
  
  [33X[0;0YIn  order  to  show  that  [22XLy[122X  contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in [22XG_2(5)[122X of index at most [22X6510[122X (see [CCN+85, p. 114]), in [22XMcL.2[122X of index at
  most  [22X5987[122X (see [CCN+85, p. 100] or Section [14X6.3[114X), in [22XL_3(5)[122X of index at most
  [22X51[122X  (see [CCN+85,  p. 38]), and in [22XA_11[122X of index at most [22X44[122X (see [CCN+85, p.
  75]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Ly" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-8 [33X[0;0Y[22XG = Th[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XTh[122X  contains  exactly  two  conjugacy  classes  of nonisomorphic
  solvable subgroups of order [22Xn = 944784[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Th" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 944784;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "2^5.psl(5,2)" ), 338, 31 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^1+8.a9" ), 98, 9 ], [128X[104X
    [4X[28X  [ CharacterTable( "U3(8).6" ), 35, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "ThN3B" ), 1, 1 ], [128X[104X
    [4X[28X  [ CharacterTable( "ThM7" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups [22XS[122X of the structures [22X[3^9].2S_4[122X and [22X3^2.[3^7].2S_4[122X in
  [22XTh[122X are solvable and have order [22Xn[122X, see [CCN+85, p. 177].[133X
  
  [33X[0;0YIn  order  to  show  that  [22XTh[122X  contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in  [22XL_5(2)[122X  of  index  at most [22X338[122X (see [CCN+85, p. 70]), in [22XA_9[122X of index at
  most  [22X98[122X  (see [CCN+85,  p.  37]),  and  in  [22XU_3(8).6[122X  of  index  at most [22X35[122X
  (see [CCN+85, p. 66]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Th" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-9 [33X[0;0Y[22XG = Fi_23[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XFi_23[122X contains a unique conjugacy class of solvable subgroups of
  order [22Xn = 3265173504[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Fi23" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 3265173504;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "2.Fi22" ), 39545, 3510 ], [128X[104X
    [4X[28X  [ CharacterTable( "O8+(3).3.2" ), 9100, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4" ), 1, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  [22XS[122X  of the structure [22X3^1+8_+.2^1+6_-.3^1+2_+.2S_4[122X in
  [22XFi_23[122X are solvable and have order [22Xn[122X, see [CCN+85, p. 177].[133X
  
  [33X[0;0YIn  order  to  show that [22XFi_23[122X contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in  [22XFi_22[122X  of index at most [22X39545[122X (see Section [14X6.4-5[114X) and in [22XO_8^+(3).S_3[122X of
  index at most [22X9100[122X (see [CCN+85, p. 140]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Fi23" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-10 [33X[0;0Y[22XG = Co_1[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XCo_1[122X  contains a unique conjugacy class of solvable subgroups of
  order [22Xn = 84934656[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Co1" );;                                           [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 84934656;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "Co2" ), 498093, 2300 ], [128X[104X
    [4X[28X  [ CharacterTable( "3.Suz.2" ), 31672, 1782 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^11:M24" ), 5903, 24 ], [128X[104X
    [4X[28X  [ CharacterTable( "Co3" ), 5837, 276 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^(1+8)+.O8+(2)" ), 1050, 120 ], [128X[104X
    [4X[28X  [ CharacterTable( "U6(2).3.2" ), 649, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^(2+12):(A8xS3)" ), 23, 8 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^(4+12).(S3x3S6)" ), 10, 6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups of the structure [22X2^4+12.(S_3 × 3S_6)[122X in [22XCo_1[122X contain
  solvable  subgroups  [22XS[122X  of  order  [22Xn[122X  and  with  the structure [22X2^4+12.(S_3 ×
  3^1+2_+:D_8)[122X, see [CCN+85, p. 183].[133X
  
  [33X[0;0YIn  order  to  show  that [22XCo_1[122X contains no other solvable subgroups of order
  larger  than  or equal to [22X|S|[122X, we check that there are no solvable subgroups
  in  [22XCo_2[122X  of  index at most [22X498093[122X (see Section [14X6.4-4[114X), in [22XSuz.2[122X of index at
  most  [22X31672[122X  (see  Section [14X6.4-2[114X),  in  [22XM_24[122X  of  index  at  most  [22X5903[122X (see
  Section [14X6.3[114X),  in  [22XCo_3[122X  of  index  at  most  [22X5837[122X  (see [CCN+85, p. 134] or
  Section [14X6.3[114X),  in  [22XO_8^+(2)[122X of index at most [22X1050[122X (see [CCN+85, p. 185]), in
  [22XU_6(2).S_3[122X  of index at most [22X649[122X (see [CCN+85, p. 115]), and in [22XA_8[122X of index
  at most [22X23[122X (see [CCN+85, p. 22]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Co1" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-11 [33X[0;0Y[22XG = J_4[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XJ_4[122X  contains  a unique conjugacy class of solvable subgroups of
  order [22X28311552[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J4" );; [127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 28311552;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );[127X[104X
    [4X[28X[ [ CharacterTable( "mx1j4" ), 17710, 24 ], [128X[104X
    [4X[28X  [ CharacterTable( "c2aj4" ), 770, 22 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^10:L5(2)" ), 361, 31 ], [128X[104X
    [4X[28X  [ CharacterTable( "J4M4" ), 23, 5 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups  of  the structure [22X2^11:M_24[122X in [22XJ_4[122X contain solvable
  subgroups  [22XS[122X  of  order  [22Xn[122X  and with the structure [22X2^11:2^6:3^1+2_+:D_8[122X, see
  Section [14X6.3[114X and [CCN+85, p. 190].[133X
  
  [33X[0;0Y(The  subgroups  in  the first four classes of maximal subgroups of [22XJ_4[122X have
  the  structures  [22X2^11:M_24[122X, [22X2^1+12_+.3M_22:2[122X, [22X2^10:L_5(2)[122X, and [22X2^3+12.(S_5 ×
  L_3(2))[122X, in this order.)[133X
  
  [33X[0;0YThe  subgroups  [22XS[122X  are  contained  also in the maximal subgroups of the type
  [22X2^1+12_+.3M_22:2[122X;  note that these subgroups are described as normalizers of
  elements  in  the [22XJ_4[122X-class [10X2A[110X, and [22XS[122X normalizes an elementary abelian group
  of  order [22X2^11[122X containing an [22XS[122X-class of length [22X1771[122X that is contained in the
  [22XJ_4[122X-class [10X2A[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= info[1][1];[127X[104X
    [4X[28XCharacterTable( "mx1j4" )[128X[104X
    [4X[25Xgap>[125X [27Xcls:= SizesConjugacyClasses( s );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= Filtered( ClassPositionsOfNormalSubgroups( s ),[127X[104X
    [4X[25X>[125X [27X                   x -> Sum( cls{ x } ) = 2^11 );[127X[104X
    [4X[28X[ [ 1, 2, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcls{ nsg[1] };[127X[104X
    [4X[28X[ 1, 276, 1771 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 3, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe stabilizers of these involutions in [22X2^11:M_24[122X have index [22X1771[122X, they have
  the  structure  [22X2^11:2^6:3.S_6[122X,  and  they are contained in [22X2^1+12_+.3M_22:2[122X
  type  subgroups;  so  also  [22XS[122X,  which  has  index  [22X10[122X  in [22X2^11:2^6:3.S_6[122X, is
  contained in [22X2^1+12_+.3M_22:2[122X. (The corresponding subgroups of [22XM_22:2[122X are of
  course the solvable groups of maximal order described in Section [14X6.3[114X.)[133X
  
  [33X[0;0YIn order to show that [22XG[122X contains no other solvable subgroups of order larger
  than  or  equal  to  [22X|S|[122X,  we  check that there are no solvable subgroups in
  [22XL_5(2)[122X  of  index  at  most [22X361[122X (see [CCN+85, p. 70]) and in [22XS_5 × L_3(2)[122X of
  index at most [22X23[122X (see [CCN+85, pp. 2, 3]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "J4" ):= n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-12 [33X[0;0Y[22XG = Fi_24^'[122X[101X[1X[133X[101X
  
  [33X[0;0YThe group [22XFi_24^'[122X contains a unique conjugacy class of solvable subgroups of
  order [22X29386561536[122X, and no larger solvable subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Fi24'" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= 29386561536;;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= List( mx, x -> SolvableSubgroupInfoFromCharacterTable( x, n ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= Filtered( info, IsList );                                        [127X[104X
    [4X[28X[ [ CharacterTable( "Fi23" ), 139161244, 31671 ], [128X[104X
    [4X[28X  [ CharacterTable( "2.Fi22.2" ), 8787, 3510 ], [128X[104X
    [4X[28X  [ CharacterTable( "(3xO8+(3):3):2" ), 3033, 6 ], [128X[104X
    [4X[28X  [ CharacterTable( "O10-(2)" ), 851, 495 ], [128X[104X
    [4X[28X  [ CharacterTable( "3^(1+10):U5(2):2" ), 165, 165 ], [128X[104X
    [4X[28X  [ CharacterTable( "2^2.U6(2).3.2" ), 7, 6 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  maximal  subgroups of the structure [22X3^1+10_+:U5(2):2[122X in [22XFi_24^'[122X contain
  solvable    subgroups    [22XS[122X    of    order   [22Xn[122X   and   with   the   structure
  [22X3^1+10_+:2^1+6_-:3^1+2_+:2S_4[122X, see [CCN+85, p. 73, p. 207].[133X
  
  [33X[0;0YIn order to show that [22XG[122X contains no other solvable subgroups of order larger
  than or equal to [22X|S|[122X, we check that there are no solvable subgroups in [22XFi_23[122X
  of order at least [22Xn[122X (see Section [14X6.4-9[114X), in [22XFi_22.2[122X of order at least [22Xn[122X (see
  Section [14X6.4-5[114X),  in  [22XO_8^+(3).S_3[122X  of  index  at  most [22X3033[122X (see [CCN+85, p.
  140]),  in  [22XO_10^-(2)[122X  of  index  at most [22X851[122X (see [CCN+85, p. 147]), and in
  [22XU_6(2).S_3[122X of index at most [22X7[122X (see [CCN+85, p. 115]).[133X
  
  [33X[0;0YThe  group  [22XS[122X  extends  to  a group of order [22X|S.2|[122X in the automorphism group
  [22XFi_24[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Fi24'" ):= n;;[127X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "Fi24'.2" ):= 2 * n;;[127X[104X
  [4X[32X[104X
  
  
  [1X6.4-13 [33X[0;0Y[22XG = B[122X[101X[1X[133X[101X
  
  [33X[0;0YThe group [22XB[122X contains a unique conjugacy class of solvable subgroups of order
  [22Xn = 29686813949952[122X, and no larger solvable subgroups.[133X
  
  [33X[0;0YThe  maximal subgroups of the structure [22X2^2+10+20(M_22:2 × S_3)[122X in [22XB[122X contain
  solvable subgroups [22XS[122X of order [22Xn[122X and with the structure [22X2^2+10+20(2^4:3^2:D_8
  × S_3)[122X, see [CCN+85, p. 217] and Section [14X6.3[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:= 29686813949952;;[127X[104X
    [4X[25Xgap>[125X [27Xn = 2^(2+10+20) * 2^4 * 3^2 * 8 * 6;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn = 2^(2+10+20) * MaxSolv.( "M22.2" ) * 6;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBy [Wil99,  Table  1],  the only maximal subgroups of [22XB[122X of order bigger than
  [22X|S|[122X have the following structures.[133X
  
        [22X2.^2E_6(2).2[122X    [22X2^1+22.Co_2[122X            [22XFi_23[122X                     [22X2^9+16S_8(2)[122X        
        [22XTh[122X              [22X(2^2 × F_4(2)):2[122X       [22X2^2+10+20(M_22:2 × S_3)[122X   [22X2^5+5+10+10L_5(2)[122X   
        [22XS_3 × Fi_22:2[122X   [22X2^[35](S_5 × L_3(2))[122X   [22XHN:2[122X                      [22XO_8^+(3):S_4[122X        
  
  [33X[0;0Y(The  character tables of the maximal subgroups of [22XB[122X are meanwhile available
  in [5XGAP[105X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( b ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27XFiltered( mx, x -> Size( x ) >= n );[127X[104X
    [4X[28X[ CharacterTable( "2.2E6(2).2" ), CharacterTable( "2^(1+22).Co2" ), [128X[104X
    [4X[28X  CharacterTable( "Fi23" ), CharacterTable( "2^(9+16).S8(2)" ), [128X[104X
    [4X[28X  CharacterTable( "Th" ), CharacterTable( "(2^2xF4(2)):2" ), [128X[104X
    [4X[28X  CharacterTable( "2^(2+10+20).(M22.2xS3)" ), [128X[104X
    [4X[28X  CharacterTable( "[2^30].L5(2)" ), CharacterTable( "S3xFi22.2" ), [128X[104X
    [4X[28X  CharacterTable( "[2^35].(S5xL3(2))" ), CharacterTable( "HN.2" ), [128X[104X
    [4X[28X  CharacterTable( "O8+(3).S4" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  subgroups  [22X2^1+22.Co_2[122X,  [22XFi_23[122X,  [22XTh[122X,  [22XS_3 × Fi_22:2[122X, and [22XHN:2[122X, the
  solvable  subgroups of maximal order are known from the previous sections or
  can be derived from known values, and are smaller than [22Xn[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( [ 2^(1+22) * MaxSolv.( "Co2" ),[127X[104X
    [4X[25X>[125X [27X           MaxSolv.( "Fi23" ),[127X[104X
    [4X[25X>[125X [27X           MaxSolv.( "Th" ),[127X[104X
    [4X[25X>[125X [27X           6 * MaxSolv.( "Fi22.2" ),[127X[104X
    [4X[25X>[125X [27X           MaxSolv.( "HN.2" ) ], i -> Int( i / n ) );[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  one of the remaining maximal groups [22XU[122X from the above list has a solvable
  subgroup of order at least [22Xn[122X then the index of this subgroup in [22XU[122X is bounded
  as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( [ Size( CharacterTable( "2.2E6(2).2" ) ),[127X[104X
    [4X[25X>[125X [27X           2^(9+16) * Size( CharacterTable( "S8(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^3 * Size( CharacterTable( "F4(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^(2+10+20) * Size( CharacterTable( "M22.2" ) ) * 6,[127X[104X
    [4X[25X>[125X [27X           2^30 * Size( CharacterTable( "L5(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^35 * Factorial(5) * Size( CharacterTable( "L3(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           Size( CharacterTable( "O8+(3)" ) ) * 24 ],[127X[104X
    [4X[25X>[125X [27X         i -> Int( i / n ) );[127X[104X
    [4X[28X[ 10311982931, 53550, 892, 770, 361, 23, 4 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group [22XO_8^+(3):S_4[122X is nonsolvable, and its order is less than [22X5 n[122X, thus
  its solvable subgroups have orders less than [22Xn[122X.[133X
  
  [33X[0;0YThe  largest  solvable  subgroup  of  [22XS_5  ×  L_3(2)[122X  has index [22X35[122X, thus the
  solvable subgroups of [22X2^[35](S_5 × L_3(2))[122X have orders less than [22Xn[122X.[133X
  
  [33X[0;0YThe  groups  of  type [22X2^5+5+10+10L_5(2)[122X cannot contain solvable subgroups of
  order  at least [22Xn[122X because [22XL_5(2)[122X has no solvable subgroup of index up to [22X361[122X
  –such a subgroup would be contained in [22X2^4:L_4(2)[122X, of index at most [22X⌊ 361/31
  ⌋  = 11[122X (see [CCN+85, p. 70]), and [22XL_4(2) ≅ A_8[122X does not have such subgroups
  (see [CCN+85, p. 22]).[133X
  
  [33X[0;0YThe  largest  proper  subgroup  of  [22XF_4(2)[122X  has index [22X69615[122X (see [CCN+85, p.
  170]),  which  excludes  solvable  subgroups  of  order at least [22Xn[122X in [22X(2^2 ×
  F_4(2)):2[122X.[133X
  
  [33X[0;0YRuling  out the group [22X2.^2E_6(2).2[122X is more involved. We consider the list of
  maximal  subgroups  of  [22X^2E_6(2)[122X  in [CCN+85,  p.  191]  (which is complete,
  see [BN95]),  and  compute  the  maximal  index of a group of order [22Xn/4[122X; the
  possible subgroups of [22X^2E_6(2)[122X to consider are the following[133X
  
        [22X2^1+20:U_6(2)[122X   [22X2^8+16:O_8^-(2)[122X   [22XF_4(2)[122X                         [22X2^2.2^9.2^18:(L_3(4) × S_3)[122X   
        [22XFi_22[122X           [22XO_10^-(2)[122X         [22X2^3.2^12.2^15:(S_5 × L_3(2))[122X                                 
  
  [33X[0;0Y(The order of [22XS_3 × U_6(2)[122X is already smaller than [22Xn/4[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( [ 2^(1+20) * Size( CharacterTable( "U6(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^(8+16) * Size( CharacterTable( "O8-(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           Size( CharacterTable( "F4(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^(2+9+18) * Size( CharacterTable( "L3(4)" ) ) * 6,[127X[104X
    [4X[25X>[125X [27X           Size( CharacterTable( "Fi22" ) ),[127X[104X
    [4X[25X>[125X [27X           Size( CharacterTable( "O10-(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           2^(3+12+15) * 120 * Size( CharacterTable( "L3(2)" ) ),[127X[104X
    [4X[25X>[125X [27X           6 * Size( CharacterTable( "U6(2)" ) ) ],[127X[104X
    [4X[25X>[125X [27X         i -> Int( i / ( n / 4 ) ) );[127X[104X
    [4X[28X[ 2598, 446, 446, 8, 8, 3, 2, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  indices  of the solvable groups of maximal orders in the groups [22XU_6(2)[122X,
  [22XO_8^-(2)[122X, [22XF_4(2)[122X, [22XL_3(4)[122X, and [22XFi_22[122X are larger than the bounds we get for [22Xn[122X,
  see [CCN+85, pp. 115, 89, 170, 23, 163].[133X
  
  [33X[0;0YIt  remains  to  consider  the subgroups of the type [22X2^9+16S_8(2)[122X. The group
  [22XS_8(2)[122X contains maximal subgroups of the type [22X2^3+8:(S_3 × S_6)[122X and of index
  [22X5355[122X  (see [CCN+85,  p.  123]), which contain solvable subgroups [22XS'[122X of index
  [22X10[122X. This yields solvable subgroups of order [22X2^9+16+3+8 ⋅ 6 ⋅ 72 = n[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2^(9+16+3+8) * 6 * 72 = n;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere are no other solvable subgroups of larger or equal order in [22XS_8(2)[122X: We
  would  need  solvable  subgroups  of index at most [22X446[122X in [22XO_8^-(2):2[122X, [22X393[122X in
  [22XO_8^+(2):2[122X,  [22X210[122X  in [22XS_6(2)[122X, or [22X23[122X in [22XA_8[122X, which is not the case by [CCN+85,
  pp. 89, 85, 46, 22].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xindex:= Int( 2^(9+16) * Size( CharacterTable( "S8(2)" ) ) / n );[127X[104X
    [4X[28X53550[128X[104X
    [4X[25Xgap>[125X [27XList( [ 120, 136, 255, 2295 ], i -> Int( index / i ) );[127X[104X
    [4X[28X[ 446, 393, 210, 23 ][128X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "B" ):= n;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  [22X2^9+16S_8(2)[122X type subgroups of [22XB[122X yield solvable subgroups [22XS'[122X of the
  type [22X2^9+16.2^3+8:(S_3 × 3^2:D_8)[122X, and of order [22Xn[122X.[133X
  
  [33X[0;0YWe  want  to show that [22XS'[122X is a [22XB[122X-conjugate of [22XS[122X. For that, we first show the
  following:[133X
  
  [33X[0;0YLemma:[133X
  
  [33X[0;0YThe  group  [22XB[122X  contains  exactly  two conjugacy classes of Klein four groups
  whose involutions lie in the class [10X2B[110X. (We will call these Klein four groups
  [10X2B[110X-pure.)  Their  normalizers  in  [22XB[122X  have  the orders [22X22858846741463040[122X and
  [22X292229574819840[122X, respectively.[133X
  
  [33X[0;0Y[13XProof.[113X Let [22XV[122X be a [10X2B[110X-pure Klein four group in [22XB[122X, and set [22XN = N_B(V)[122X. Let [22Xx ∈
  V[122X  be  an  involution and set [22XH = C_B(x)[122X, then [22XH[122X is maximal in [22XB[122X and has the
  structure  [22X2^1+22.Co_2[122X. The index of [22XC = C_B(V) = C_H(V)[122X in [22XN[122X divides [22X6[122X, and
  [22XC[122X  stabilizes  the  central  involution  in [22XH[122X and another [10X2B[110X involution. The
  group [22XH[122X contains exactly four conjugacy classes of [10X2B[110X elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh:= mx[2];[127X[104X
    [4X[28XCharacterTable( "2^(1+22).Co2" )[128X[104X
    [4X[25Xgap>[125X [27Xpos:= Positions( GetFusionMap( h, b ), 3 );[127X[104X
    [4X[28X[ 2, 4, 11, 20 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe [22XB[122X-classes of [10X2B[110X-pure Klein four groups arise from those of these classes
  [22Xy^H  ⊂  H[122X  such  that  [22Xx  ≠ y[122X holds and [22Xx y[122X is a [10X2B[110X element. We compute this
  subset.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos:= Filtered( Difference( pos, [ 2 ] ), i -> ForAny( pos,[127X[104X
    [4X[25X>[125X [27X            j -> NrPolyhedralSubgroups( h, 2, i, j ).number <> 0 ) );[127X[104X
    [4X[28X[ 4, 11 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  two classes have lengths [22X93150[122X and [22X7286400[122X, thus the index of [22XC[122X in [22XH[122X is
  one of these numbers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( h ){ pos };[127X[104X
    [4X[28X[ 93150, 7286400 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we compute the number [22Xn_0[122X of [10X2B[110X-pure Klein four groups in [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnr:= NrPolyhedralSubgroups( b, 3, 3, 3 );[127X[104X
    [4X[28Xrec( number := 14399283809600746875, type := "V4" )[128X[104X
    [4X[25Xgap>[125X [27Xn0:= nr.number;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [22XB[122X-conjugacy  class of [22XV[122X has length [22X[B:N] = [B:H] ⋅ [H:C] / [N:C][122X, where
  [22X[N:C][122X divides [22X6[122X. We see that [22X[N:C] = 6[122X in both cases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= List( pos, i -> Size( b ) / SizesCentralizers( h )[i] / 6 );[127X[104X
    [4X[28X[ 181758140654146875, 14217525668946600000 ][128X[104X
    [4X[25Xgap>[125X [27XSum( cand ) = n0;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  orders  of  the  normalizers  of  the two classes of [10X2B[110X-pure Klein four
  groups are as claimed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( cand, x -> Size( b ) / x );[127X[104X
    [4X[28X[ 22858846741463040, 292229574819840 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  subgroup  [22XS[122X of order [22Xn[122X is contained in a maximal subgroup [22XM[122X of the type
  [22X2^2+10+20(M_22:2  ×  S_3)[122X  in  [22XB[122X. The group [22XM[122X is the normalizer of a [10X2B[110X-pure
  Klein  four  group in [22XB[122X, and the other class of normalizers of [10X2B[110X-pure Klein
  four  groups does not contain subgroups of order [22Xn[122X. Thus the conjugates of [22XS[122X
  are  uniquely determined by [22X|S|[122X and the property that they normalize [10X2B[110X-pure
  Klein four groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= mx[7];[127X[104X
    [4X[28XCharacterTable( "2^(2+10+20).(M22.2xS3)" )[128X[104X
    [4X[25Xgap>[125X [27XSize( m );[127X[104X
    [4X[28X22858846741463040[128X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfMinimalNormalSubgroups( m );[127X[104X
    [4X[28X[ [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( m ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, b ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27XList( cand, x -> Size( b ) / ( n * x ) );[127X[104X
    [4X[28X[ 770, 315/32 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  consider  the  subgroup  [22XS'[122X of order [22Xn[122X, which is contained in a maximal
  subgroup  of  the  type  [22X2^9+16S_8(2)[122X  in  [22XB[122X.  In  order to prove that [22XS'[122X is
  [22XB[122X-conjugate  to  [22XS[122X,  it is enough to show that [22XS'[122X normalizes a [10X2B[110X-pure Klein
  four group.[133X
  
  [33X[0;0YThe  unique  minimal  normal  subgroup  [22XV[122X of [22X2^9+16S_8(2)[122X has order [22X2^8[122X. Its
  involutions lie in the class [10X2B[110X of [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= mx[4];[127X[104X
    [4X[28XCharacterTable( "2^(9+16).S8(2)" )[128X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfMinimalNormalSubgroups( m );[127X[104X
    [4X[28X[ [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( m ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 255 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, b ){ nsg[1] };[127X[104X
    [4X[28X[ 1, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group  [22XV[122X  is  central  in  the  normal  subgroup  [22XW = 2^9+16[122X, since all
  nonidentity  elements  of  [22XV[122X  lie in one conjugacy class of odd length. As a
  module  for  [22XS_8(2)[122X, [22XV[122X is the unique irreducible eight-dimensional module in
  characteristic two.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterDegrees( CharacterTable( "S8(2)" ) mod 2 );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 8, 1 ], [ 16, 1 ], [ 26, 1 ], [ 48, 1 ], [ 128, 1 ], [128X[104X
    [4X[28X  [ 160, 1 ], [ 246, 1 ], [ 416, 1 ], [ 768, 1 ], [ 784, 1 ], [128X[104X
    [4X[28X  [ 2560, 1 ], [ 3936, 1 ], [ 4096, 1 ], [ 12544, 1 ], [ 65536, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YHence  we  are  done  if  the  restriction of the [22XS_8(2)[122X-action on [22XV[122X to [22XS'/W[122X
  leaves  a  two-dimensional  subspace  of  [22XV[122X  invariant. In fact we show that
  already  the  restriction of the [22XS_8(2)[122X-action on [22XV[122X to the maximal subgroups
  of the structure [22X2^3+8:(S_3 × S_6)[122X has a two-dimensional submodule.[133X
  
  [33X[0;0YThese maximal subgroups have index [22X5355[122X in [22XS_8(2)[122X. The primitive permutation
  representation    of   degree   [22X5355[122X   of   [22XS_8(2)[122X   and   the   irreducible
  eight-dimensional  matrix  representation  of [22XS_8(2)[122X over the field with two
  elements  are  available  via  the  [5XGAP[105X  package  [5XAtlasRep[105X, see [WPN+19]. We
  compute  generators  for  an  index [22X5355[122X subgroup in the matrix group via an
  isomorphism to the permutation group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpermg:= AtlasGroup( "S8(2)", NrMovedPoints, 5355 );[127X[104X
    [4X[28X<permutation group of size 47377612800 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xmatg:= AtlasGroup( "S8(2)", Dimension, 8 );[127X[104X
    [4X[28X<matrix group of size 47377612800 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xhom:= GroupHomomorphismByImagesNC( matg, permg,[127X[104X
    [4X[25X>[125X [27X             GeneratorsOfGroup( matg ), GeneratorsOfGroup( permg ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmax:= PreImages( hom, Stabilizer( permg, 1 ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThese  generators  define the action of the index [22X5355[122X subgroup of [22XS_8(2)[122X on
  the eight-dimensional module. We compute the dimensions of the factors of an
  ascending composition series of this module.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= GModuleByMats( GeneratorsOfGroup( max ), GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= MTX.CompositionFactors( m );;[127X[104X
    [4X[25Xgap>[125X [27XList( comp, r -> r.dimension );[127X[104X
    [4X[28X[ 2, 4, 2 ][128X[104X
  [4X[32X[104X
  
  
  [1X6.4-14 [33X[0;0Y[22XG = M[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group [22XM[122X contains exactly two conjugacy classes of solvable subgroups of
  order [22Xn = 2849934139195392[122X, and no larger solvable subgroups.[133X
  
  [33X[0;0YThe  maximal subgroups of the structure [22X2^1+24_+.Co_1[122X in the group [22XM[122X contain
  solvable  subgroups [22XS[122X of order [22Xn[122X and with the structure [22X2^1+24_+.2^4+12.(S_3
  × 3^1+2_+:D_8)[122X, see [CCN+85, p. 234] and Section [14X6.4-10[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:= 2^25 * MaxSolv.( "Co1" );[127X[104X
    [4X[28X2849934139195392[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   solvable   subgroups   of   maximal  order  in  groups  of  the  types
  [22X2^2+11+22.(M_24 × S_3)[122X and [22X2^[39].(L_3(2) × 3S_6)[122X have order [22Xn[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2^(2+11+22) * MaxSolv.( "M24" ) * 6 = n;    [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X2^39 * 24 * 3 * 72 = n;                 [127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  inspecting  the  other  maximal  subgroups of [22XM[122X, we use the description
  from [NW13]. Currently [22X44[122X classes of maximal subgroups are listed there, and
  any  possible  other  maximal  subgroup  of [22XG[122X has socle isomorphic to one of
  [22XL_2(13)[122X, [22XSz(8)[122X, [22XU_3(4)[122X, [22XU_3(8)[122X; so these maximal subgroups are isomorphic to
  subgroups  of the automorphism groups of these groups – the maximum of these
  group  orders  is  smaller  than  [22Xn[122X,  hence  we  may  ignore  these possible
  subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= [ "L2(13)", "Sz(8)", "U3(4)", "U3(8)" ];;[127X[104X
    [4X[25Xgap>[125X [27XList( cand, nam -> ExtensionInfoCharacterTable( [127X[104X
    [4X[25X>[125X [27XCharacterTable( nam ) ) );[127X[104X
    [4X[28X[ [ "2", "2" ], [ "2^2", "3" ], [ "", "4" ], [ "3", "(S3x3)" ] ][128X[104X
    [4X[25Xgap>[125X [27Xll:= List( cand, x -> Size( CharacterTable( x ) ) );[127X[104X
    [4X[28X[ 1092, 29120, 62400, 5515776 ][128X[104X
    [4X[25Xgap>[125X [27X18* ll[4];[127X[104X
    [4X[28X99283968[128X[104X
    [4X[25Xgap>[125X [27X2^39 * 24 * 3 * 72;[127X[104X
    [4X[28X2849934139195392[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus only the following maximal subgroups of [22XM[122X have order bigger than [22X|S|[122X.[133X
  
        [22X2.B[122X                 [22X2^1+24_+.Co_1[122X            [22X3.Fi_24[122X            [22X2^2.^2E_6(2):S_3[122X           
        [22X2^10+16.O_10^+(2)[122X   [22X2^2+11+22.(M_24 × S_3)[122X   [22X3^1+12_+.2Suz.2[122X    [22X2^5+10+20.(S_3 × L_5(2))[122X   
        [22XS_3 × Th[122X            [22X2^[39].(L_3(2) × 3S_6)[122X   [22X3^8.O_8^-(3).2_3[122X   [22X(D_10 × HN).2[122X              
  
  [33X[0;0YFor  the  subgroups  [22X2.B[122X,  [22X3.Fi_24[122X,  [22X3^1+12_+.2Suz.2[122X,  [22XS_3 × Th[122X, and [22X(D_10 ×
  HN).2[122X, the solvable subgroups of maximal order are smaller than [22Xn[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( [ 2 * MaxSolv.( "B" ),[127X[104X
    [4X[25X>[125X [27X           6 * MaxSolv.( "Fi24'" ),[127X[104X
    [4X[25X>[125X [27X           3^13 * 2 * MaxSolv.( "Suz" ) * 2,[127X[104X
    [4X[25X>[125X [27X           6 * MaxSolv.( "Th" ),[127X[104X
    [4X[25X>[125X [27X           10 * MaxSolv.( "HN" ) * 2 ], i -> Int( i / n ) );[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe subgroup [22X2^2.^2E_6(2):S_3[122X can be excluded by the fact that this group is
  only six times larger than the subgroup [22X2.^2E_6(2):2[122X of [22XB[122X, but [22Xn[122X is [22X96[122X times
  larger than the maximal solvable subgroup in [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn / MaxSolv.( "B" );[127X[104X
    [4X[28X96[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group  [22X3^8.O_8^-(3).2_3[122X  can  be  excluded  by the fact that a solvable
  subgroup  of  order  at  least  [22Xn[122X  would  imply  the existence of a solvable
  subgroup  of  index  at  most  [22X46[122X  in  [22XO_8^-(3).2_3[122X,  which  is not the case
  (see [CCN+85, p. 141]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInt( 3^8 * Size( CharacterTable( "O8-(3)" ) ) * 2 / n );[127X[104X
    [4X[28X46[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSimilarly,  the  existence  of  a  solvable  subgroup of order at least [22Xn[122X in
  [22X2^5+10+20.(S_3 × L_5(2))[122X would imply the existence of a solvable subgroup of
  index  at  most [22X723[122X in [22XL_5(2)[122X and in turn of a solvable subgroup of index at
  most [22X23[122X in [22XL_4(2)[122X, which is not the case (see [CCN+85, p. 70]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInt( 2^(10+16) * Size( CharacterTable( "O10+(2)" ) ) / n );    [127X[104X
    [4X[28X553350[128X[104X
    [4X[25Xgap>[125X [27XInt( 2^(5+10+20) * 6 * Size( CharacterTable( "L5(2)" ) ) / n );  [127X[104X
    [4X[28X723[128X[104X
    [4X[25Xgap>[125X [27XInt( 723 / 31 );[127X[104X
    [4X[28X23[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  remains  to  exclude the subgroup [22X2^10+16.O_10^+(2)[122X, which means to show
  that [22XO_10^+(2)[122X does not contain a solvable subgroup of index at most [22X553350[122X.
  If  such  a  subgroup  would  exist then it would be contained in one of the
  following  maximal  subgroups of [22XO_10^+(2)[122X (see [CCN+85, p. 146]): in [22XS_8(2)[122X
  (of  index  at  most  [22X1115[122X),  in  [22X2^8:O_8^+(2)[122X  (of  index at most [22X1050[122X), in
  [22X2^10:L_5(2)[122X  (of  index  at most [22X241[122X), in [22X(3 × O_8^-(2)):2[122X (of index at most
  [22X27[122X), in [22X(2^1+12_+:(S_3 × A_8)[122X (of index at most [22X23[122X), or in [22X2^3+12:(S_3 × S_3
  ×  L_3(2))[122X  (of index at most [22X4[122X). By [CCN+85, pp. 123, 85, 70, 89, 22], this
  is not the case.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xindex:= Int( 2^(10+16) * Size( CharacterTable( "O10+(2)" ) ) / n );    [127X[104X
    [4X[28X553350[128X[104X
    [4X[25Xgap>[125X [27XList( [ 496, 527, 2295, 19840, 23715, 118575 ], i -> Int( index / i ) );[127X[104X
    [4X[28X[ 1115, 1050, 241, 27, 23, 4 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  a  consequence,  we  have shown that the largest solvable subgroups of [22XM[122X
  have order [22Xn[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxSolv.( "M" ):= n;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to prove the statement about the conjugacy of subgroups of order [22Xn[122X
  in [22XM[122X, we first show the following.[133X
  
  [33X[0;0YLemma:[133X
  
  [33X[0;0YThe  group  [22XM[122X contains exactly three conjugacy classes of [10X2B[110X-pure Klein four
  groups.  Their  normalizers  in  [22XM[122X  have  the  orders  [22X50472333605150392320[122X,
  [22X259759622062080[122X, and [22X9567039651840[122X, respectively.[133X
  
  [33X[0;0Y[13XProof.[113X   The   idea  is  the  same  as  for  the  Baby  Monster  group,  see
  Section [14X6.4-13[114X.  Let  [22XV[122X  be  a  [10X2B[110X-pure  Klein  four group in [22XM[122X, and set [22XN =
  N_M(V)[122X.  Let [22Xx ∈ V[122X be an involution and set [22XH = C_M(x)[122X, then [22XH[122X is maximal in
  [22XM[122X and has the structure [22X2^1+24_+.Co_1[122X. The index of [22XC = C_M(V) = C_H(V)[122X in [22XN[122X
  divides  [22X6[122X,  and  [22XC[122X  stabilizes  the  central involution in [22XH[122X and another [10X2B[110X
  involution.[133X
  
  [33X[0;0YThe group [22XH[122X contains exactly five conjugacy classes of [10X2B[110X elements, three of
  them  consist  of elements that generate a [10X2B[110X-pure Klein four group together
  with [22Xx[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "2^1+24.Co1" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xpos:= Positions( GetFusionMap( h, m ), 3 );[127X[104X
    [4X[28X[ 2, 4, 7, 9, 16 ][128X[104X
    [4X[25Xgap>[125X [27Xpos:= Filtered( Difference( pos, [ 2 ] ), i -> ForAny( pos,[127X[104X
    [4X[25X>[125X [27X            j -> NrPolyhedralSubgroups( h, 2, i, j ).number <> 0 ) );[127X[104X
    [4X[28X[ 4, 9, 16 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  two classes have lengths [22X93150[122X and [22X7286400[122X, thus the index of [22XC[122X in [22XH[122X is
  one of these numbers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSizesConjugacyClasses( h ){ pos };[127X[104X
    [4X[28X[ 16584750, 3222483264000, 87495303168000 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we compute the number [22Xn_0[122X of [10X2B[110X-pure Klein four groups in [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnr:= NrPolyhedralSubgroups( m, 3, 3, 3 );[127X[104X
    [4X[28Xrec( number := 87569110066985387357550925521828244921875, [128X[104X
    [4X[28X  type := "V4" )[128X[104X
    [4X[25Xgap>[125X [27Xn0:= nr.number;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [22XM[122X-conjugacy  class of [22XV[122X has length [22X[M:N] = [M:H] ⋅ [H:C] / [N:C][122X, where
  [22X[N:C][122X divides [22X6[122X. We see that [22X[N:C] = 6[122X in both cases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= List( pos, i -> Size( m ) / SizesCentralizers( h )[i] / 6 );[127X[104X
    [4X[28X[ 16009115629875684006343550944921875, [128X[104X
    [4X[28X  3110635203347364905168577322802100000000, [128X[104X
    [4X[28X  84458458854522392576698341855475200000000 ][128X[104X
    [4X[25Xgap>[125X [27XSum( cand ) = n0;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  orders  of  the  normalizers of the three classes of [10X2B[110X-pure Klein four
  groups are as claimed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( cand, x -> Size( m ) / x );[127X[104X
    [4X[28X[ 50472333605150392320, 259759622062080, 9567039651840 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs we have seen above, the group [22XM[122X contains exactly the following (solvable)
  subgroups of order [22Xn[122X.[133X
  
  [31X1[131X   [33X[0;6YOne class in [22X2^1+24_+.Co_1[122X type subgroups,[133X
  
  [31X2[131X   [33X[0;6Yone class in [22X2^2+11+22.(M_24 × S_3)[122X type subgroups, and[133X
  
  [31X3[131X   [33X[0;6Ytwo classes in [22X2^[39].(L_3(2) × 3S_6)[122X type subgroups.[133X
  
  [33X[0;0YNote  that  [22X2^[39].(L_3(2)  ×  3S_6)[122X  contains  an elementary abelian normal
  subgroup  of order eight whose involutions lie in the class [10X2B[110X, see [CCN+85,
  p.  234].  As  a  module  for  the  group  [22XL_3(2)[122X,  this  normal subgroup is
  irreducible,  and  the  restriction  of the action to the two classes of [22XS_4[122X
  type  subgroups  fixes  a one- and a two-dimensional subspace, respectively.
  Hence we have one class of subgroups of order [22Xn[122X that centralize a [10X2B[110X element
  and  one  class  of subgroups of order [22Xn[122X that normalize a [10X2B[110X-pure Klein four
  group.  Clearly the subgroups in the first class coincide with the subgroups
  of  order  [22Xn[122X in [22X2^1+24_+.Co_1[122X type subgroups. By the above classification of
  [10X2B[110X-pure  Klein  four groups in [22XM[122X, the subgroups in the second class coincide
  with the subgroups of order [22Xn[122X in [22X2^2+11+22.(M_24 × S_3)[122X type subgroups.[133X
  
  [33X[0;0YIt  remains to show that the subgroups of order [22Xn[122X do [13Xnot[113X stabilize both a [10X2B[110X
  element  [13Xand[113X  a  [10X2B[110X-pure Klein four group. We do this by direct computations
  with  a  [22X2^2+11+22.(M_24  ×  S_3)[122X  type  group,  which  is available via the
  [5XAtlasRep[105X package, see [WPN+19].[133X
  
  [33X[0;0YFirst  we  fetch  the  group,  and  factor  out  the largest solvable normal
  subgroup, by suitable actions on blocks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "2^(2+11+22).(M24xS3)" );[127X[104X
    [4X[28X<permutation group of size 50472333605150392320 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( g );[127X[104X
    [4X[28X294912[128X[104X
    [4X[25Xgap>[125X [27Xbl:= Blocks( g, MovedPoints( g ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( bl );[127X[104X
    [4X[28X147456[128X[104X
    [4X[25Xgap>[125X [27Xhom1:= ActionHomomorphism( g, bl, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xact1:= Image( hom1 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g ) / Size( act1 );[127X[104X
    [4X[28X8192[128X[104X
    [4X[25Xgap>[125X [27Xbl2:= Blocks( act1, MovedPoints( act1 ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( bl2 );[127X[104X
    [4X[28X72[128X[104X
    [4X[25Xgap>[125X [27Xhom2:= ActionHomomorphism( act1, bl2, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xact2:= Image( hom2 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( act2 );[127X[104X
    [4X[28X1468938240[128X[104X
    [4X[25Xgap>[125X [27XSize( MathieuGroup( 24 ) ) * 6;[127X[104X
    [4X[28X1468938240[128X[104X
    [4X[25Xgap>[125X [27Xbl3:= AllBlocks( act2 );;[127X[104X
    [4X[25Xgap>[125X [27XList( bl3, Length );                                             [127X[104X
    [4X[28X[ 24, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xbl3:= Orbit( act2, bl3[2], OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xhom3:= ActionHomomorphism( act2, bl3, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xact3:= Image( hom3 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  an  isomorphism  from the factor group of type [22XM_24[122X to the
  group that belongs to [5XGAP[105X's table of marks. Then we use the information from
  the  table  of marks to compute a solvable subgroup of maximal order in [22XM_24[122X
  (which  is  [22X13824[122X), and take the preimage under the isomorphism. Finally, we
  take the preimage of this group in te original group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M24" );;[127X[104X
    [4X[25Xgap>[125X [27Xtomgroup:= UnderlyingGroup( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xiso:= IsomorphismGroups( act3, tomgroup );;[127X[104X
    [4X[25Xgap>[125X [27Xpos:= Positions( OrdersTom( tom ), 13824 );[127X[104X
    [4X[28X[ 1508 ][128X[104X
    [4X[25Xgap>[125X [27Xsub:= RepresentativeTom( tom, pos[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xpre:= PreImages( iso, sub );;[127X[104X
    [4X[25Xgap>[125X [27Xpre:= PreImages( hom3, pre );;[127X[104X
    [4X[25Xgap>[125X [27Xpre:= PreImages( hom2, pre );;[127X[104X
    [4X[25Xgap>[125X [27Xpre:= PreImages( hom1, pre );;[127X[104X
    [4X[25Xgap>[125X [27XSize( pre ) = n;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  subgroups  stabilizes  a  Klein  four group. It does not stabilize a [10X2B[110X
  element because its centre is trivial.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpciso:= IsomorphismPcGroup( pre );;[127X[104X
    [4X[25Xgap>[125X [27XSize( Centre( Image( pciso ) ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  
  [1X6.5 [33X[0;0YProof of the Corollary[133X[101X
  
  [33X[0;0YWith  the  computations  in  the  previous  sections,  we have collected the
  information that is needed to show the corollary stated in Section [14X6.1[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( Set( RecNames( MaxSolv ) ), [127X[104X
    [4X[25X>[125X [27X             x -> MaxSolv.( x )^2 >= Size( CharacterTable( x ) ) );[127X[104X
    [4X[28X[ "Fi23", "J2", "J2.2", "M11", "M12", "M22.2" ][128X[104X
  [4X[32X[104X
  
