Transitive subgroup of symmetric group

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Definition

Let $S_{n}$ be the symmetric group on $n$ letters. Let it act naturally on $X = {1, 2, \dots, n}$ . A subgroup $G \leq S_{n}$ is said to be a transitive subgroup of the symmetric group on n letters if its group action on $X$ made from the restriction of the action of $S_{n}$ on $X$ is a transitive group action.

Examples

General facts

Any $G \leq S_{n}$ that contains an $n$ -cycle is certainly transitive - successive applications of that cycle can send any $x \in X = {1, 2, \dots, n}$ to any other element of $X$ under the natural group action.
A partial converse of the above exists. For $p$ prime, if $G \leq S_{p}$ is transitive then it must contain a $p$ -cycle.

Small symmetric groups

n=1, n=2, symmetric groups of order 1, 2

In these cases, the symmetric group only has itself as a transitive subgroup. The symmetric group is isomorphic to the trivial group and cyclic group:Z2 respectively.

n=3, symmetric group of order 6

The transitive subgroups of symmetric group:S3 are, up to automorphism classes of subgroups:

Automorphism class of subgroups	List of all subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Note
A3 in S3	${(), (1, 2, 3), (1, 3, 2)}$	cyclic group:Z3	3	2	1	1	1	cyclic group:Z2	3-Sylow
whole group	${(), (1, 2, 3), (1, 3, 2),$ $(1, 2), (1, 3), (2, 3)}$	symmetric group:S3	6	1	1	1	1	trivial group
Total (2 rows)	--	--	--	--	4	--	6	--	--

n=4, symmetric group of order 24

The transitive subgroups of symmetric group:S4 are, up to automorphism classes of subgroups:

Automorphism class of subgroups	Representative	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
Z4 in S4	$⟨ (1, 2, 3, 4) ⟩$	cyclic group:Z4	4	6	1	3	3	--	--
normal Klein four-subgroup of S4	${(), (1, 2) (3, 4),$ $(1, 3) (2, 4), (1, 4) (2, 3)}$	Klein four-group	4	6	1	1	1	symmetric group:S3	1	2-core
D8 in S4	$⟨ (1, 2, 3, 4), (1, 3) ⟩$	dihedral group:D8	8	3	1	3	3	--	--	2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8
A4 in S4	$⟨ (1, 2, 3), (1, 2) (3, 4) ⟩$	alternating group:A4	12	2	1	1	1	cyclic group:Z2	1
whole group	$⟨ (1, 2, 3, 4), (1, 2) ⟩$	symmetric group:S4	24	1	1	1	1	trivial group	0
Total (5 rows)	--	--	--	--	11	--	30	--	--	--

n=5, symmetric group of order 120

Automorphism class of subgroups	Representative subgroup	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (= 1 iff automorph-conjugate subgroup)	Size of each conjugacy class (= 1 iff normal subgroup)	Total number of subgroups (= 1 iff characteristic subgroup)
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 (5 rows)	--	--	--	--	19	--	156	--