Subgroup structure of alternating group:A6

This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A6.
View subgroup structure of particular groups | View other specific information about alternating group:A6

This article discusses the subgroup structure of alternating group:A6, which is the alternating group on the set $\{1,2,3,4,5,6\}$. The group has order 360.

Family contexts

Family name	Parameter values	General discussion of subgroup structure of family
alternating group	degree $n=6$, i.e., the group $A_{6}$	subgroup structure of alternating groups
projective special linear group of degree two	field:F9, i.e., it is the group $PSL(2,9)$	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	501 Compared with $A_{n},n=3,4,5,\dots$: 2, 10, 59, 501, 3786, 48337, ...
number of conjugacy classes of subgroups	22 Compared with $A_{n},n=3,4,5,\dots$: 2, 5, 9, 22, 40, 137, ...
number of automorphism classes of subgroups	16 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: dihedral group:D8 (order 8) as D8 in A6 (with its simple fusion system -- see simple fusion system for dihedral group:D8). Sylow number is 45. 3-Sylow: elementary abelian group:E9 (order 9) as E9 in A6. Sylow number is 10. 5-Sylow: cyclic group:Z5 (order 5) as Z5 in A6. Sylow number is 36.
Hall subgroups	no Hall subgroups other than the Sylow subgroups, whole group, and trivial subgroup. In particular, there is no $\{2,3\}$-Hall subgroup, $\{2,5\}$-Hall subgroup, and $\{3,5\}$-Hall subgroup.
maximal subgroups	maximal subgroups have orders 24, 36, and 60.
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 permutation automorphisms

Note that alternating groups are simple (with an exception for degree 1,2,4), so in particular $A_{6}$ is simple. Hence no proper nontrivial subgroup is normal or subnormal.

The below lists subgroups up to automorphisms arising from permutations, i.e., automorphisms arising from conjugation in symmetric group:S6. This is not the same as the classification up to automorphisms because of the presence of other automorphisms, a phenomenon unique to degree six.

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Permutation 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	360	1	1	1	trivial
subgroup generated by double transposition in A6	$\{(),(1,2)(3,4)\}$	cyclic group:Z2	2	180	1	45	45
V4 with two fixed points in A6	$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$	Klein four-group	4	90	1	15	15
V4 without fixed points in A6	$\{(),(1,2)(5,6),(1,2)(3,4),(3,4)(5,6)\}$	Klein four-group	4	90	1	15	15
Z4 in A6	$\{(),(1,2,3,4)(5,6),(1,3)(2,4),(1,4,3,2)(5,6)\}$	cyclic group:Z4	4	90	1	45	45
D8 in A6	$\langle (1,2,3,4)(5,6),(1,3)(5,6)\rangle$	dihedral group:D8	8	45	1	45	45	2-Sylow
A3 in A6	$\{(),(1,2,3),(1,3,2)\}$	cyclic group:Z3	3	120	1	20	20
diagonal A3 in A6	$\{(),(1,2,3)(4,5,6),(1,3,2)(4,6,5)\}$	cyclic group:Z3	3	120	1	20	20
E9 in A6	$\langle (1,2,3),(4,5,6)\rangle$	elementary abelian group:E9	9	40	1	10	10	3-Sylow; also maximal among $\{3,5\}$-subgroups
diagonal S3 in A6	$\langle (1,2,3)(4,5,6),(1,2)(4,5)\rangle$	symmetric group:S3	6	60	1	60	60
twisted S3 in A6	$\langle (1,2,3),(1,2)(4,5)\rangle$	symmetric group:S3	6	60	1	60	60
A4 in A6	$\langle (1,2)(3,4),(1,2,3)\rangle$	alternating group:A4	12	30	1	15	15
twisted A4 in A6	$\langle (1,2,3)(4,5,6),(1,4)(2,5),(1,4)(3,6)\rangle$	alternating group:A4	12	30	1	15	15
standard twisted S4 in A6	$\langle (1,2,3,4)(5,6),(1,2)(5,6)\rangle$	symmetric group:S4	24	15	1	15	15	maximal; also maximal among $\{2,3\}$-subgroups
exceptional twisted S4 in A6	$\langle (3,4)(5,6),(1,2)(5,6),(1,3,5)(2,4,6),(3,5)(4,6)\rangle$	symmetric group:S4	24	15	1	15	15	maximal; also maximal among $\{2,3\}$-subgroups
generalized dihedral group for E9 in A6	$\langle (1,2,3),(4,5,6),(1,2)(4,5)\rangle$	generalized dihedral group for E9	18	20	1	10	10
?	$\langle (4,5,6),(1,2,3),(2,3)(5,6),(1,4)(2,5,3,6)\rangle$	SmallGroup(36,9)	36	10	1	10	10	maximal; also maximal among $\{2,3\}$-subgroups
Z5 in A6	$\langle (1,2,3,4,5)\rangle$	cyclic group:Z5	5	72	1	36	36	5-Sylow, also maximal among $\{3,5\}$-subgroups
D10 in A6	$\langle (1,2,3,4,5),(2,5)(3,4)\rangle$	dihedral group:D10	10	36	1	36	36	maximal among $\{2,5\}$-subgroups
A5 in A6	$\langle (1,2,3,4,5),(1,2,3)\rangle$	alternating group:A5	60	6	1	6	6	maximal
twisted A5 in A6	$\langle (1,2,3,4,5),(1,4)(5,6)\rangle$	alternating group:A5	60	6	1	6	6	maximal
whole group	$\langle (1,2,3,4,5),(1,2,3),(1,2)(5,6)\rangle$	alternating group:A6	360	1	1	1	1
Total	--	--	--	--	22	--	501	--

Table classifying subgroups up to automorphisms

Permutation 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	360	1	1	1	trivial
subgroup generated by double transposition in A6	$\{(),(1,2)(3,4)\}$	cyclic group:Z2	2	180	1	45	45
V4 with two fixed points in A6, V4 without fixed points in A6	$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$ $\{(),(1,2)(5,6),(1,2)(3,4),(3,4)(5,6)\}$	Klein four-group	4	90	2	15	30
Z4 in A6	$\{(),(1,2,3,4)(5,6),(1,3)(2,4),(1,4,3,2)(5,6)\}$	cyclic group:Z4	4	90	1	45	45
D8 in A6	$\langle (1,2,3,4)(5,6),(1,3)(5,6)\rangle$	dihedral group:D8	8	45	1	45	45	2-Sylow
A3 in A6 diagonal A3 in A6	$\{(),(1,2,3),(1,3,2)\}$ $\{(),(1,2,3)(4,5,6),(1,3,2)(4,6,5)\}$	cyclic group:Z3	3	120	2	20	40
E9 in A6	$\langle (1,2,3),(4,5,6)\rangle$	elementary abelian group:E9	9	40	1	10	10	3-Sylow
diagonal S3 in A6 twisted S3 in A6	$\langle (1,2,3)(4,5,6),(1,2)(4,5)\rangle$ $\langle (1,2,3),(1,2)(4,5)\rangle$	symmetric group:S3	6	60	2	60	120
A4 in A6 twisted A4 in A6	$\langle (1,2)(3,4),(1,2,3)\rangle$ $\langle (1,2,3)(4,5,6),(1,4)(2,5),(1,4)(3,6)\rangle$	alternating group:A4	12	30	2	15	30
standard twisted S4 in A6, exceptional twisted S4 in A6	$\langle (1,2,3,4)(5,6),(1,2)(5,6)\rangle$, $\langle (3,4)(5,6),(1,2)(5,6),(1,3,5)(2,4,6),(3,5)(4,6)\rangle$	symmetric group:S4	24	15	2	15	30	maximal
generalized dihedral group for E9 in A6	$\langle (1,2,3),(4,5,6),(1,2)(4,5)\rangle$	generalized dihedral group for E9	18	20	1	10	10
?	$\langle (4,5,6),(1,2,3),(2,3)(5,6),(1,4)(2,5,3,6)\rangle$	?	36	10	1	10	10	maximal
Z5 in A6	$\langle (1,2,3,4,5)\rangle$	cyclic group:Z5	5	72	1	36	36	5-Sylow
D10 in A6	$\langle (1,2,3,4,5),(2,5)(3,4)\rangle$	dihedral group:D10	10	36	1	36	36
A5 in A6 twisted A5 in A6	$\langle (1,2,3,4,5),(1,2,3)\rangle$ $\langle (1,2,3,4,5),(1,4)(5,6)\rangle$	alternating group:A5	60	6	2	6	12	maximal
whole group	$\langle (1,2,3,4,5),(1,2,3),(1,2)(5,6)\rangle$	alternating group:A6	360	1	1	1	1
Total	--	--	--	--	22	--	501	--

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	45	1	1	0	0
cyclic group:Z3	3	1	40	2	1	0	0
cyclic group:Z4	4	1	45	1	1	0	0
Klein four-group	4	2	30	2	1	0	0
cyclic group:Z5	5	1	36	1	1	0	0
symmetric group:S3	6	1	120	2	1	0	0
dihedral group:D8	8	3	45	1	1	0	0
elementary abelian group:E9	9	2	10	1	1	0	0
dihedral group:D10	10	1	36	1	1	0	0
alternating group:A4	12	3	30	2	1	0	0
generalized dihedral group for E9	18	4	10	1	1	0	0
symmetric group:S4	24	12	30	2	1	0	0
SmallGroup(36,9)	36	9	10	1	1	0	0
alternating group:A5	60	5	12	2	1	0	0
alternating group:A6	360	118	1	1	1	1	1
Total	--	--	501	22	16	2	2