## Does S4 contain a subgroup of order 6?

Quick summary. maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4). There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4.

## What is the only subgroup of order 12 in S4?

The subgroup in S4 that I know has order 12 is the subgroup of all even permutations, otherwise known as the alternating group A4.

Is S4 a subgroup of S5?

. The group has order 120. 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….Quick summary.

Item Value
Hall subgroups -Hall subgroup: S4 in S5 (order 24) No -Hall subgroup or -Hall subgroup

### What are the subgroups of D4?

(a) The proper normal subgroups of D4 = {e, r, r2,r3, s, rs, r2s, r3s} are {e, r, r2,r3}, {e, r2, s, r2s}, {e, r2, rs, r3s}, and {e, r2}.

### What is the order of S4?

(a) The possible cycle types of elements in S4 are: identity, 2-cycle, 3-cycle, 4- cycle, a product of two 2-cycles. These have orders 1, 2, 3, 4, 2 respectively, so the possible orders of elements in S4 are 1, 2, 3, 4.

What is the order of S6?

The possible orders for elements in S6 are: 6, 5, 4, 3, 2, 1.

#### How many subgroups of order 4 are there?

A subgroup of order 4 is a subgroup of a Sylow 2-subgroup, so either cyclic ⟨ (i, j, k, l) ⟩ or one of the two kinds of Klein 4-subgroups ⟨ (i, j), (k, l) ⟩ (3 subgroups), or the true K4 ⟨ (i, j) (k, l), (i, k) (j, l) ⟩ (normal). seven subgroups of order 4, three conjugacy classes

#### How many normal subgroups are there in S4?

There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4.

How to sort subgroups by the magnitude of the Order?

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.

## How many subgroups of order 6 have a normal Sylow 3?

A subgroup of order 6 must have a normal Sylow 3-subgroup, so must live inside the normalizer (inside S4) of a Sylow 3-subgroup. The Sylow 3-subgroups are just the various alternating groups of degree 3, and their normalizers are various symmetric groups of degree 3, so are exactly the 4 subgroups of order 6.