Monster Group - Subgroup Structure

Subgroup Structure

The Monster has at least 44 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A₁₂. The Monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.

44 of the classes of maximal subgroups of the monster are given by the following list, which is (as of 2012) believed to be complete except possibly for subgroups normalizing simple subgroups of the form L₂(13), U₃(4), U₃(8), and Suz(8) (Wilson 2010), (Norton & Wilson 2012). However tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups on the list below were incorrectly omitted in some previous lists.

2.B Centralizer of an involution

21+24.Co₁ Centralizer of an involution

3.Fi₂₄ Normalizer of a subgroup of order 3.

22.2E₆(22):S₃ Normalizer of a 4-group

210+16.O₁₀+(2)

22+11+22.(M₂₄ × S₃)

31+12.2Suz.2 Normalizer of a subgroup of order 3.

25+10+20.(S₃ × L₅(2))

S₃ × Th Normalizer of a subgroup of order 3.

23+6+12+18.(L₃(2) × 3S₆)

38.O₈−(3).2₃

(D₁₀ × HN).2 Normalizer of a subgroup of order 5.

(32:2 × O₈+(3)).S₄

32+5+10.(M₁₁ × 2S₄)

33+2+6+6:(L₃(3) × SD₁₆)

51+6:2J₂:4 Normalizer of a subgroup of order 5.

(7:3 × He):2 Normalizer of a subgroup of order 7.

(A₅ × A₁₂):2

53+3.(2 × L₃(5))

(A₆ × A₆ × A₆).(2 × S₄)

(A₅ × U₃(8):3₁):2

52+2+4:(S₃ × GL₂(5))

(L₃(2) × S₄(4):2).2

71+4:(3 × 2S₇) Normalizer of a subgroup of order 7.

(52: × U₃(5)).S₃

(L₂(11) × M₁₂):2 Contains the normalizer (11.5 × M₁₂):2 of a subgroup of order 11.

(A₇ × (A₅ × A₅):22):2

54:(3 × 2L₂(25)):2₂

72+1+2:GL₂(7)

M₁₁ × A₆.22

(S₅ × S₅ × S₅):S₃

(L₂(11) × L₂(11)):4

132:2L₂(13).4

(72:(3 × 2A₄) × L₂(7)).2

(13:6 × L₃(3)).2 Normalizer of a subgroup of order 13.

131+2:(3 × 4S₄) Normalizer of a subgroup of order 13.

L₂(71) (Holmes & Wilson 2008)

L₂(59) (Holmes & Wilson 2004)

112:(5 × 2A₅)

L₂(41) Norton & Wilson (2012) found a maximal subgroup of this form; due to a subtle error, some previous lists and papers stated that no such maximal subgroup existed.

L₂(29):2 (Holmes & Wilson 2002)

72:SL₂(7) This was accidentally omitted on some previous lists of 7-local subgroups.

L₂(19):2 (Holmes & Wilson 2008)

41:40 Normalizer of a subgroup of order 41.