Subgroup structure of symmetric group:S3

TAKE A QUIZ ON THIS TOPIC and test the quality of your understanding of it

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

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.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable 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
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

Quick summary

Item	Value
Number of subgroups	6 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups	4 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups	4 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3-Sylow: cyclic group:Z3, Sylow number is 1, fusion system is non-inner fusion system for cyclic group:Z3
Hall subgroups	Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number
maximal subgroups	maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3).
normal subgroups	There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

Table classifying subgroups up to automorphisms

For more information on each automorphism type, follow the link.

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
trivial subgroup	$\{ () \}$	trivial group	1	6	1	1	1	symmetric group:S3	trivial
S2 in S3	$\{ (), (1,2) \}, \{ (), (2,3) \}, \{ (), (1,3) \}$	cyclic group:Z2	2	3	1	3	3	--	2-Sylow
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 (4 rows)	--	--	--	--	4	--	6	--	--

Table classifying isomorphism types of subgroups

Note that the first part of the GAP ID is the order.

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

Subgroup structure viewed as symmetric group

Classification based on partition given by orbit sizes

For any subgroup of $S_3$, the natural action on $\{ 1,2,3 \}$ induces a partition of the set $\{ 1,2,3 \}$ into orbits, which in turn induces an unordered integer partition of the number 3. Below, we classify this information for the subgroups.

Conjugacy class of subgroups	Size of conjugacy class	Induced partition of 3	Direct product of transitive subgroups on each orbit?	Illustration with representative
trivial subgroup	1	1 + 1 + 1	Yes	Under the action of the trivial subgroup, the orbits are singleton subsets.
S2 in S3	3	2 + 1	Yes	Under the action of $\{ (), (1,2) \}$, the orbits are $\{ 1,2 \}$ and $\{ 3 \}$.
A3 in S3	1	3	Yes	The action is a transitive group action, so $\{ 1,2,3 \}$ is a single orbit. Each of the elements sends the point $1$ to a different element of $\{ 1,2,3 \}$, covering everything.
whole group	1	3	Yes	The action is a transitive group action, so $\{ 1,2,3 \}$ is a single orbit.

Defining functions

Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function	What it means	Value as subgroup	Value as group	Order	Associated quotient-defining function	Value as group	Order (= index of subgroup)
center	elements that commute with every group element	trivial subgroup	trivial group	1	inner automorphism group	symmetric group:S3	6
derived subgroup	subgroup generated by all commutators	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	abelianization	cyclic group:Z2	2
Frattini subgroup	intersection of all maximal subgroups	trivial subgroup	trivial group	1	Frattini quotient	symmetric group:S3	6
Jacobson radical	intersection of all maximal normal subgroups	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2
socle	join of all minimal normal subgroups	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2
Fitting subgroup	join of all nilpotent normal subgroups	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2
solvable radical	join of all solvable normal subgroups	whole group	symmetric group:S3	6	?	trivial group	1
Brauer core	largest odd-order normal subgroup	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2
3-Sylow core or 3-core	normal core of any 3-Sylow subgroup	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2
3-Sylow closure	normal closure of any 3-Sylow subgroup	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	?	cyclic group:Z2	2

Subgroup series-defining functions

Series-defining function	Type	Zeroth member	First member	Second member	Third member	Stable member
upper central series	ascending	trivial	center: trivial	second center: trivial	trivial	trivial
lower central series	descending	--	whole group	derived subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	$\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	$\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3
derived series	descending	whole group	derived subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	second derived subgroup: trivial	trivial	trivial
Frattini series	descending	whole group	Frattini subgroup: trivial	trivial	trivial	trivial
Fitting series	ascending	trivial	Fitting subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	whole group	whole group	whole group
socle series	ascending	trivial	socle: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	whole group	whole group	whole group

Conjugacy class-defining functions

Conjugacy class-defining function	What it means	Value as subgroup	Value as group	Order	Index of subgroup	Number of subgroups
2-Sylow subgroup	subgroup whose order is a power of 2, index relatively prime to 2. Sylow subgroups exist, and Sylow implies order-conjugate	S2 in S3: $\{ (), (1,2) \}, \{ (), (2,3) \},$, $\{ (), (1,3) \}$	cyclic group:Z2	2	3	3
3-Sylow subgroup	subgroup whose order is a power of 3, index relatively prime to 3. Sylow subgroups exist, and Sylow implies order-conjugate	A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	2	1

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.

Subgroup operations

Intersection

For all pairs of subgroups, either one is contained in the other, or the intersection is trivial. This makes the intersection table easy to construct.

Subgroup/subgroup	trivial	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group
trivial	trivial	trivial	trivial	trivial	trivial	trivial
$\{ (), (1,2) \}$	trivial	$\{ (), (1,2) \}$	trivial	trivial	trivial	$\{ (), (1,2) \}$
$\{ (), (2,3) \}$	trivial	trivial	$\{ (), (2,3) \}$	trivial	trivial	$\{ (), (2,3) \}$
$\{ (), (1,3) \}$	trivial	trivial	trivial	$\{ (), (1,3) \}$	trivial	$\{ (), (1,3) \}$
$\{ (), (1,2,3), (1,3,2) \}$	trivial	trivial	trivial	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$
whole group	trivial	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group

Join of subgroups

For all pairs of subgroups, either one is contained in the other, or the join is the whole group. This makes the table for joins of subgroups easy to construct.

Subgroup/subgroup	trivial	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group
trivial	trivial	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group
$\{ (), (1,2) \}$	$\{ (), (1,2) \}$	$\{ (), (1,2) \}$	whole group	whole group	whole group	whole group
$\{ (), (2,3) \}$	$\{ (), (2,3) \}$	whole group	$\{ (), (2,3) \}$	whole group	whole group	whole group
$\{ (), (1,3) \}$	$\{ (), (1,3) \}$	whole group	whole group	$\{ (), (1,3)\}$	whole group	whole group
$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group	whole group	whole group	$\{ (), (1,2,3), (1,3,2) \}$	whole group
whole group	whole group	whole group	whole group	whole group	whole group	whole group

Commutators of pairs of subgroups

For any pair of subgroups, their commutator is trivial if one of them is trivial or they are both equal and proper, and is $\{ (), (1,2,3), (1,3,2) \}$ otherwise.

Subgroup/subgroup	trivial	$\{ (), (1,2) \}$	$\{ (), (2,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,2,3), (1,3,2) \}$	whole group
trivial	trivial	trivial	trivial	trivial	trivial	trivial
$\{ (), (1,2) \}$	trivial	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (1,3) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (2,3) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (1,2,3), (1,3,2) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	trivial	$\{ (), (1,2,3), (1,3,2) \}$
whole group	trivial	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$

Unary operations on subgroups

Subgroup	Normalizer	Centralizer	Normal closure	Normal core
trivial	whole group	whole group	trivial	trivial
$\{ (), (1,2) \}$	$\{ (), (1,2) \}$	$\{ (), (1,2) \}$	whole group	trivial
$\{ (), (2,3) \}$	$\{ (), (2,3) \}$	$\{ (), (2,3) \}$	whole group	trivial
$\{ (), (1,3) \}$	$\{ (), (1,3) \}$	$\{ (), (1,3) \}$	whole group	trivial
$\{ (), (1,2,3), (1,3,2) \}$	whole group	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$
whole group	whole group	trivial	whole group	whole group

Subgroup series

The max-length of the group is 2 (it cannot be more, based on the prime factorization of 6) and there are four subgroup series of this maximal length, one series for each proper nontrivial subgroup.

Series	Intermediate subgroup	Normal series?	Subnormal series?	Chief series?	Composition series?
$\{ () \} \le \{ (), (1,2) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$	$\{ (), (1,2) \}$ -- S2 in S3	No	No	No	No
$\{ () \} \le \{ (), (2,3) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$	$\{ (), (2,3) \}$ -- S2 in S3	No	No	No	No
$\{ () \} \le \{ (), (1,3) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$	$\{ (), (1,3) \}$ -- S2 in S3	No	No	No	No
$\{ () \} \le \{ (), (1,2,3), (1,3,2) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$	$\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3	Yes	Yes	Yes	Yes

In particular, there is a unique composition series which is also a unique chief series for the group.