Subgroup structure of 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.

Table classifying subgroups up to automorphisms
For more information on each automorphism type, follow the link.

Table classifying isomorphism types of subgroups
Note that the first part of the GAP ID is the order.

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 $$p$$.

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.

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.

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.

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.

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 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.

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