# Subgroup structure of alternating group:A5

View subgroup structure of particular groups | View other specific information about alternating group:A5

This article discusses the subgroup structure of alternating group:A5, which is the alternating group on the set $\{ 1, 2,3,4,5\}$. The group has order 60.

## Family contexts

Family name Parameter values General discussion of subgroup structure of family
alternating group degree $n = 5$, i.e., the group $A_5$ subgroup structure of alternating groups
projective special linear group of degree two field:F5, i.e., it is the group $PSL(2,5)$ subgroup structure of projective special linear group of degree two over a finite field
projective special linear group of degree two field:F4, i.e., it is the group $PSL(2,4)$ subgroup structure of projective special linear group of degree two over a finite field

## 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 59
Compared with $A_n, n = 3,4,5,\dots$: 2, 10, 59, 501, 3786, 48337, ...
number of conjugacy classes of subgroups 9
Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 22, 40, 137, ...
number of automorphism classes of subgroups 9
Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 16, 37, 112, ...
isomorphism classes of Sylow subgroups and the corresponding fusion systems 2-Sylow: Klein four-group (order 4) as V4 in A5 (with its simple fusion system -- see simple fusion system for Klein four-group). Sylow number is 5.
3-Sylow: cyclic group:Z3 (order 3) as Z3 in A5. Sylow number is 10.
5-Sylow: cyclic group:Z5 (order 5) as Z5 in A5. Sylow number is 6.
Hall subgroups In addition to the whole group, trivial subgroup, and Sylow subgroups: $\{ 2,3 \}$-Hall subgroup of order 12 (A4 in A5). There is no $\{ 2,5 \}$-Hall subgroup or $\{3,5\}$-Hall subgroup.
maximal subgroups maximal subgroups have orders 6 (twisted S3 in A5), 10 (D10 in A5), 12 (A4 in A5)
normal subgroups only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple.

### Table classifying subgroups up to automorphisms

Note that A5 is simple, and hence no proper nontrivial subgroup is normal or subnormal.

Automorphism class of subgroups Representative subgroup (full list if small, generating set if large) Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes Size of each conjugacy class Total number of subgroups Note
trivial subgroup $()$ trivial group 1 60 1 1 1 trivial
subgroup generated by double transposition in A5 $\{ (), (1,2)(3,4) \}$ cyclic group:Z2 2 30 1 15 15
V4 in A5 $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ Klein four-group 4 15 1 5 5 2-Sylow
A3 in A5 $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 20 1 10 10 3-Sylow
twisted S3 in A5 $\langle (1,2,3), (1,2) (4,5)\rangle$ symmetric group:S3 6 10 1 10 10 maximal
A4 in A5 $\langle (1,2)(3,4), (1,2,3) \rangle$ alternating group:A4 12 5 1 5 5 2,3-Hall, maximal
Z5 in A5 $\langle (1,2,3,4,5) \rangle$ cyclic group:Z5 5 12 1 6 6 5-Sylow
D10 in A5 $\langle (1,2,3,4,5), (2,5)(3,4) \rangle$ dihedral group:D10 10 6 1 6 6 maximal
whole group $\langle (1,2,3,4,5), (1,2,3) \rangle$ alternating group:A5 60 1 1 1 1
Total -- -- -- -- 9 -- 59 --

### 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 15 1 1 0 0
cyclic group:Z3 3 1 10 1 1 0 0
Klein four-group 4 2 5 1 1 0 0
cyclic group:Z5 5 1 6 1 1 0 0
symmetric group:S3 6 1 10 1 1 0 0
dihedral group:D10 10 1 6 1 1 0 0
alternating group:A4 12 3 5 1 1 0 0
alternating group:A5 60 5 1 1 1 1 1
Total -- -- 59 9 9 2 2