Subgroup structure of Mathieu group:M10

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This article gives specific information, namely, subgroup structure, about a particular group, namely: Mathieu group:M10.
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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 793
number of conjugacy classes of subgroups 25
number of automorphism classes of subgroups 24
isomorphism classes of Sylow subgroups and the corresponding Sylow numbers 2-Sylow: semidihedral group:SD16, Sylow number is 45
3-Sylow: elementary abelian group:E9, Sylow number is 10
5-Sylow: cyclic group:Z5, Sylow number is 36
Hall subgroups No Hall subgroups other than the trivial subgroup, whole group, and Sylow subgroups
maximal subgroups maximal subgroups have order 16, 20, 72, 360
normal subgroups the only proper nontrivial normal subgroup is of index two and is isomorphic to alternating group:A6 (order 360)