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:A5, which is the alternating group on the set $\{1,2,3,4,5\}$. The group has order 360.

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

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.

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	$⟨(1,2,3,4)(5,6),(1,3)(5,6)⟩$	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	$⟨(1,2,3),(4,5,6)⟩$	elementary abelian group:E9	9	40	1	10	10	3-Sylow
diagonal S3 in A6	$⟨(1,2,3)(4,5,6),(1,2)(4,5)⟩$	symmetric group:S3	6	60	1	60	60
twisted S3 in A6	$⟨(1,2,3),(1,2)(4,5)⟩$	[symmetric group:S3]]	6	60	1	60	60
A4 in A6	$⟨(1,2)(3,4),(1,2,3)⟩$	alternating group:A4	12	30	1	15	15
twisted A4 in A6	$⟨(1,2,3)(4,5,6),(1,4)(2,5),(1,4)(3,6)⟩$	alternating group:A4	12	30	1	15	15
standard twisted S4 in A6	$⟨(1,2,3,4),(1,2)(5,6)⟩$	symmetric group:S4	24	15	1	15	15	maximal
exceptional twisted S4 in A6	$⟨(3,4)(5,6),(1,2)(5,6),(1,3,5)(2,4,6),(3,5)(4,6)⟩$	symmetric group:S4	24	15	1	15	15	maximal
generalized dihedral group for E9 in A6	$⟨(1,2,3),(4,5,6),(1,2)(4,5)⟩$	generalized dihedral group for E9	18	20	1	10	10
?	$⟨(4,5,6),(1,2,3),(2,3)(5,6),(1,4)(2,5,3,6)⟩$	?	36	10	1	10	10	maximal
Z5 in A6	$⟨(1,2,3,4,5)⟩$	cyclic group:Z5	5	72	1	36	36	5-Sylow
D10 in A6	$⟨(1,2,3,4,5),(2,5)(3,4)⟩$	dihedral group:D10	10	36	1	36	36
A5 in A6	$⟨(1,2,3,4,5),(1,2,3)⟩$	alternating group:A5	60	6	1	6	6	maximal
twisted A5 in A6	$⟨(1,2,3,4,5),(1,4)(5,6)⟩$	alternating group:A5	60	6	1	6	6	maximal
whole group	$⟨(1,2,3,4,5),(1,2,3),(1,2)(5,6)⟩$	alternating group:A6	360	1	1	1	1
Total	--	--	--	--	22	--	501	--