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Contents 1 Tables for quick information 1.1 Table classifying subgroups up to automorphisms 1.2 Table classifying isomorphism types of subgroups 1.3 Table listing number of subgroups by order 1.4 Table listing numbers of subgroups by group property 1.5 Table listing numbers of subgroups by subgroup property 2 The three-element subgroup (type (3)) 2.1 Subgroup-defining functions yielding this subgroup 2.2 Subgroup properties satisfied by this subgroup 2.3 Subgroup properties not satisfied by this subgroup 3 The two-element subgroups (type (2)) 3.1 Subgroup properties satisfied by these subgroups 3.2 Subgroup properties not satisfied by these subgroups 4 Lattice of subgroups 4.1 The entire lattice 4.2 The sublattice of normal subgroups

This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S3.
View subgroup structure of particular groups | View other specific information about symmetric group:S3

The symmetric group on three letters has six subgroups. These are described below.

1. The identity element is the trivial subgroup (1)
2. There are three 2-element subgroups, generated by the transpositions. These are all conjugate subgroups, and each is isomorphic to the cyclic group of order two (3)
3. There is one 3-element subgroup, generated by a 3-cycle. This is a characteristic subgroup, and is isomorphic to the cyclic group of order three. This is, concretely, the alternating group on three letters (i.e., the group of even permutations on three letters). (1)
4. The whole group (1)

Since this group is a complete group, every automorphism of it is inner, and in particular, this means that the classification of subgroups upto conjugacy is the same as the classification up to automorphism. Types (1), (3) and (4) are normal, and in fact, characteristic subgroups. All the subgroups in type (2) are conjugate to each other.

Tables for quick information

Table classifying subgroups up to automorphisms

Automorphism class of subgroups	Isomorphism class	Number of conjugacy classes	Size of each conjugacy class	Isomorphism class of quotient (if exists)
trivial subgroup	trivial group	1	1	symmetric group:S3
S2 in S3	cyclic group:Z2	1	3	--
A3 in S3	cyclic group:Z3	1	1	cyclic group:Z2
whole group	symmetric group:S3	1	1	trivial group

Table classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	(1,1)	1	1	1	1
Cyclic group:Z2	(2,1)	3	1	0	0
Cyclic group:Z3	(3,1)	1	1	1	1
Symmetric group:S3	(6,1)	1	1	1	1
Total	--	6	4	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	Occurrences as normal subgroup	Occurrences as characteristic subgroup
1	1	1	1	1
2	3	1	0	0
3	1	1	1	1
6	1	1	1	1
Total	6	4	3	3

Table listing numbers of subgroups by group property

Group property	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Cyclic group	5	3	2	2
Abelian group	5	3	2	2
Nilpotent group	5	3	2	2
Solvable group	6	4	3	3

Table listing numbers of subgroups by subgroup property

Subgroup property	Occurences as subgroup	Conjugacy classes of occurrences as subgroup	Automorphism classes of occurrences as subgroup
Subgroup	6	4	4
Normal subgroup	3	3	3
Characteristic subgroup	3	3	3
Fully characteristic subgroup	3	3	3
Self-centralizing subgroup	5	3	3
Pronormal subgroup	6	4	4
Retract	3	3	3

The three-element subgroup (type (3))

In the language of permutations, this subgroup is {(),(123),(132)}.

Subgroup-defining functions yielding this subgroup

• The commutator subgroup
• The Jacobson radical: It is the intersection of all maximal normal subgroups. In fact, it is the unique maximal normal subgroup.
• The socle: It is the subgroup generated by all minimal normal subgroups. In fact, it is the unique minimal normal subgroup.
• The Brauer core: It is the subgroup generated by all normal subgroups of odd order.

Subgroup properties satisfied by this subgroup

On account of being the commutator subgroup, as well as on account of being the set of squares in the whole group, this subgroup is a verbal subgroup -- it is a subgroup generatedby words of a certain form. Thus, it satisfies the following properties:

• Fully characteristic subgroup: It is invariant under every endomorphism of the whole group. Further information: Verbal implies fully characteristic
• Image-closed fully characteristic subgroup: Its image under any surjective homomorphism is fully characteristic in the image. Further information: Verbal implies image-closed fully characteristic
• Image-closed characteristic subgroup: Further information: Verbal implies image-closed characteristic
• Characteristic subgroup

There are further properties it satisfies, including:

• Normal Sylow subgroup
• Order-unique subgroup
• Isomorph-free subgroup
• Minimal normal subgroup
• Simple normal subgroup
• Transitively normal subgroup
• Maximal normal subgroup
• Maximal subgroup
• Hereditarily normal subgroup
• Complemented normal subgroup
• Intermediately characteristic subgroup

Subgroup properties not satisfied by this subgroup

• Direct factor: This subgroup is not a direct factor of the whole group. It does have a complement, but the complement is not normal.
• Central factor

The two-element subgroups (type (2))

These subgroups are conjugate; in particular, they are automorphic. Thus, they satisfy the same subgroup properties.

Subgroup properties satisfied by these subgroups

• Core-free subgroup
• Contranormal subgroup
• Self-normalizing subgroup
• Abnormal subgroup
• Maximal subgroup
• Sylow subgroup
• Pronormal subgroup
• Retract

Subgroup properties not satisfied by these subgroups

• Normal subgroup
• Subnormal subgroup

Lattice of subgroups

The entire lattice

The lattice of subgroups of the symmetric group of degree three has the following interesting features:

• Every non-identity automorphism of the whole group acts nontrivially on the lattice. Note that since the symmetric group of degree three is a complete group, all the automorphisms are inner.
• In fact, the non-identity automorphisms give rise to all possible permutations of the three non-abelian subgroups of order two. More specifically, a permutation of the letters 1,2,3 gives rise to an inner automorphism that permutes the two-element subgroups fixing these elements the same way. For instance, the 3-cycle (1,2,3), acting by conjugation, sends the subgroup stabilizing 1 (namely {(),(2,3)}) to the subgroup stabilizing 2 (namely {(),(1,3)}).

The sublattice of normal subgroups

The lattice of normal subgroups, which is in this case also the lattice of characteristic subgroups, is a totally ordered sublattice comprising the trivial subgroup, the subgroup of order three, and the whole group. This sublattice is preserved by all automorphisms.