Subgroup structure of alternating group:A4
From Groupprops
This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A4.
View subgroup structure of particular groups | View other specific information about alternating group:A4
The alternating group on 
is a group of order 12.
There is no subgroup of order 
. The alternating group of degree four is the group of smallest possible order (in this case 
) not having subgroups of all orders dividing the group order.
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 | 10 |
| Number of conjugacy classes of subgroups | 5 |
| Number of automorphism classes of subgroups | 5 |
Table classifying subgroups up to automorphism
| Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Total number of subgroups | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| trivial subgroup |  |
trivial group | 1 | 12 | 1 | 1 | 1 | alternating group:A4 | 1 | trivial |
| subgroup generated by double transposition in A4 |  ,  ,  |
cyclic group:Z2 | 2 | 6 | 1 | 3 | 3 | -- | 2 | |
| V4 in A4 |  |
Klein four-group | 4 | 3 | 1 | 1 | 1 | cyclic group:Z3 | 1 | 2-Sylow, minimal normal, maximal |
| A3 in A4 |  ,  ,  ,  |
cyclic group:Z3 | 3 | 4 | 1 | 4 | 4 | -- | -- | 3-Sylow, maximal |
| whole group | all elements | alternating group:A4 | 12 | 1 | 1 | 1 | 1 | trivial group | 1 | whole |
| Total (5 rows) | -- | -- | -- | -- | 5 | -- | 10 | -- | -- | -- |
Table classifying isomorphism types of subgroups
| Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
|---|---|---|---|---|---|---|---|
| trivial group | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| cyclic group:Z2 | 2 | 1 | 3 | 1 | 1 | 0 | 0 |
| cyclic group:Z3 | 3 | 1 | 4 | 1 | 1 | 0 | 0 |
| Klein four-group | 4 | 2 | 1 | 1 | 1 | 1 | 1 |
| alternating group:A4 | 12 | 3 | 1 | 1 | 1 | 1 | 1 |
| Total | -- | -- | 10 | 5 | 5 | 3 | 3 |
Table listing numbers of subgroups by group property
| Group property | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
|---|---|---|---|---|---|
| cyclic group | 8 | 3 | 3 | 1 | 1 |
| abelian group | 9 | 4 | 4 | 2 | 2 |
| nilpotent group | 9 | 4 | 4 | 2 | 2 |
| solvable group | 10 | 5 | 5 | 3 | 3 |
