# Subgroup structure of Mathieu group:M9

## Contents

View subgroup structure of particular groups | View other specific information about Mathieu group:M9

## 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 -- -- --