Subgroup structure of Mathieu group:M9

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Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable 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
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

Quick summary

Item Value
Number of subgroups 68
Number of conjugacy classes of subgroups 14
Number of automorphism classes of subgroups 10
Sylow subgroups, corresponding Sylow numbers, and fusion systems 2-Sylow: quaternion group, Sylow number 9, fusion system is inner because it has a normal complement (i.e., the group is 2-nilpotent)
3-Sylow: elementary abelian group:E9, Sylow number 1, fusion system is quaternion group fusion system for elementary abelian group:E9
Hall subgroups The only primes dividing the order of the group are 2 and 3. Since there are only two distinct prime divisors, the only Hall subgroups are the whole group, the trivial subgroup, and the Sylow subgroups.
maximal subgroups order 8 (Q8 in M9), order 36

Table classifying subgroups up to automorphisms

Automorphism class of subgroups List of subgroups Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes (=1 iff automorph-conjugate subgroup) Size of each conjugacy class (=1 iff normal subgroup) Total number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth (if subnormal) Nilpotency class (if nilpotent) Note
trivial subgroup trivial group 1 72 1 1 1 Mathieu group:M9 1 0 trivial
cyclic group:Z2 2 36 1 9 9 -- -- 1
cyclic group:Z4 4 18 3 9 27 -- -- 1
Q8 in M9 quaternion group 8 9 1 9 9 -- -- 2 2-Sylow
cyclic group:Z3 3 24 1 4 4 -- 2 1
3-Sylow subgroup of Mathieu group:M9 elementary abelian group:E9 9 8 1 1 1 quaternion group 1 1 3-Sylow
symmetric group:S3 6 12 1 12 12 -- -- --
generalized dihedral group for E9 18 4 1 1 1 Klein four-group 1 --
SmallGroup(36,9) 36 2 3 1 3 cyclic group:Z2 1 --
whole group Mathieu group:M9 72 1 1 1 1 trivial group 0 --
Total (10 rows) -- -- -- -- 14 -- 68 -- -- --