Subgroup structure of Mathieu group:M11

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This article gives specific information, namely, subgroup structure, about a particular group, namely: Mathieu group:M11.
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This article describes the subgroup structure of Mathieu group:M11.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

Quick summary

Item Value
number of subgroups 8651
number of conjugacy classes of subgroups 39
isomorphism classes of Sylow subgroups and corresponding Sylow numbers 2-Sylow: semidihedral group:SD16, Sylow number is 495
3-Sylow: elementary abelian group:E9, Sylow number is 55
5-Sylow: cyclic group:Z5, Sylow number is 396
11-Sylow: cyclic group:Z11, Sylow number is 144
Hall subgroups apart from the whole group, trivial subgroup, and Sylow subgroups, there exist the following Hall subgroups: \{ 2,3 \}-Hall (order 144), \{ 5, 11\}-Hall (order 55), \{2,3,5 \}-Hall (order 720)
maximal subgroups maximal subgroups of order 48, 120, 144, 660, 720
normal subgroups the group is simple non-abelian, so the only normal subgroups are the whole group and the trivial subgroup
subgroups that are simple non-abelian groups (apart from the whole group) alternating group:A5 (order 60), alternating group:A6 (order 360), projective special linear group:PSL(2,11) (order 660)