Subgroup structure of symmetric group:S5

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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S5.
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The symmetric group of degree five has many subgroups. We'll take the five letters as ${1, 2, 3, 4, 5}$ . The group has order 120.

Note that since $S_{5}$ is a complete group, every automorphism of it is inner, so the classification of subgroups upto conjugacy is the same as the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

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 automorphisms

Note that the only normal subgroups are the trivial subgroup, the whole group, and A5 in S5, so we do not waste a column on specifying whether the subgroup is normal and on the quotient group.

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	120	1	1	1	trivial
S2 in S5	${(), (1, 2)}$	cyclic group:Z2	2	60	1	10	10
subgroup generated by double transposition in S5	${(), (1, 2) (3, 4)}$	cyclic group:Z2	2	60	1	15	15
subgroup generated by pair of disjoint transpositions in S5	${(), (1, 2), (3, 4), (1, 2) (3, 4)}$	Klein four-group	4	30	1	15	15
subgroup generated by double transpositions on four elements in S5	${(), (1, 2) (3, 4), (1, 3) (2, 4), (1, 4) (2, 3)}$	Klein four-group	4	30	1	5	5
Z4 in S5	${(), (1, 2, 3, 4), (1, 3) (2, 4), (1, 4, 3, 2)}$	cyclic group:Z4	4	30	1	15	15
D8 in S5	$⟨ (1, 2, 3, 4), (1, 3) ⟩$	dihedral group:D8	8	15	1	15	15	2-Sylow
Z3 in S5	${(), (1, 2, 3), (1, 3, 2)}$	cyclic group:Z3	3	40	1	10	10	3-Sylow
Z6 in S5	$⟨ (1, 2, 3), (4, 5) ⟩$	cyclic group:Z6	6	20	1	10	10
S3 in S5	$⟨ (1, 2, 3), (1, 2) ⟩$	symmetric group:S3	6	20	1	10	10
twisted S3 in S5	$⟨ (1, 2, 3), (1, 2) (4, 5) ⟩$	symmetric group:S3	6	20	1	10	10
direct product of S3 and S2 in S5	$⟨ (1, 2, 3), (1, 2), (4, 5) ⟩$	direct product of S3 and Z2	12	10	1	10	10	3-Sylow normalizer
A4 in S5	$⟨ (1, 2) (3, 4), (1, 2, 3) ⟩$	alternating group:A4	12	10	1	5	5
S4 in S5	$⟨ (1, 2, 3, 4), (1, 2) ⟩$	symmetric group:S4	24	5	1	5	5	(2,3)-Hall
Z5 in S5	$⟨ (1, 2, 3, 4, 5) ⟩$	cyclic group:Z5	5	24	1	6	6	5-Sylow
D10 in S5	$⟨ (1, 2, 3, 4, 5), (2, 5) (3, 4) ⟩$	dihedral group:D10	10	12	1	6	6
GA(1,5) in S5	$⟨ (1, 2, 3, 4, 5), (2, 3, 5, 4) ⟩$	general affine group:GA(1,5)	20	6	1	6	6
A5 in S5	$⟨ (1, 2, 3, 4, 5), (1, 2, 3) ⟩$	alternating group:A5	60	2	1	1	1	only proper nontrivial normal subgroup, quotient is cyclic group:Z2
whole group	$⟨ (1, 2, 3, 4, 5), (1, 2) ⟩$	symmetric group:S5	120	1	1	1	1
Total (19 rows)	--	--	--	--	19	--	156	--

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	25	2	2	0	0
cyclic group:Z3	3	1	10	1	1	0	0
cyclic group:Z4	4	1	15	1	1	0	0
Klein four-group	4	2	20	2	2	0	0
cyclic group:Z5	5	1	6	1	1	0	0
symmetric group:S3	6	1	20	2	2	0	0
cyclic group:Z6	6	2	10	1	1	0	0
dihedral group:D8	8	3	15	1	1	0	0
dihedral group:D10	10	1	6	1	1	0	0
alternating group:A4	12	3	5	1	1	0	0
direct product of S3 and Z2	12	4	10	1	1	0	0
general affine group:GA(1,5)	20	3	6	1	1	0	0
symmetric group:S4	24	12	5	1	1	0	0
alternating group:A5	60	5	1	1	1	1	1
symmetric group:S5	120	34	1	1	1	1	1
Total	--	--	156	19	19	3	3

Table listing number of subgroups by order

Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $p^{r}$ is congruent to $1$ modulo $p$ .

Group order	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Automorphism classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
1	1	1	1	1	1
2	25	2	2	0	0
3	10	1	1	0	0
4	35	3	3	0	0
5	6	1	1	0	0
6	30	3	3	0	0
8	15	1	1	0	0
10	6	1	1	0	0
12	15	2	2	0	0
20	6	1	1	0	0
24	5	1	1	0	0
60	1	1	1	1	1
120	1	1	1	1	1
Total	156	19	19	3	3