# Subgroup

In group theory, a branch of mathematics, given a group *G* under a binary operation ∗, a subset *H* of *G* is called a **subgroup** of *G* if *H* also forms a group under the operation ∗. More precisely, *H* is a subgroup of *G* if the restriction of ∗ to *H* × *H* is a group operation on *H*. This is often denoted *H* ≤ *G*, read as "*H* is a subgroup of *G*".

If *H* is a subgroup of *G*, then *G* is sometimes called an **overgroup** of *H*.

The same definitions apply more generally when *G* is an arbitrary semigroup, but this article will only deal with subgroups of groups.

where |*G*| and |*H*| denote the orders of *G* and *H*, respectively. In particular, the order of every subgroup of *G* (and the order of every element of *G*) must be a divisor of |*G*|.^{[6]}^{[7]}

If *aH* = *Ha* for every *a* in *G*, then *H* is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if *p* is the lowest prime dividing the order of a finite group *G,* then any subgroup of index *p* (if such exists) is normal.

Let S_{4} be the symmetric group on 4 elements.
Below are all the subgroups of S_{4}, listed according to the number of elements, in decreasing order.

The whole group S_{4} is a subgroup of S_{4}, of order 24. Its Cayley table is