Subgroup structure of Mathieu group:M9
This article gives specific information, namely, subgroup structure, about a particular group, namely: Mathieu group:M9.
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 | -- | -- | -- |