# Definition:Maximal Subgroup

## Definition

Let $G$ be a group.

Let $M \le G$ be a proper subgroup of $G$.

Then $M$ is a maximal subgroup of $G$ if and only if:

For every subgroup $H$ of $G$, $M \subseteq H \subseteq G$ means $M = H$ or $H = G$.

That is, if there is no subgroup of $G$, except $M$ and $G$ itself, which contains $M$.