\begin{definition}[Definition:Maximal Subgroup]
Let $G$ be a group.
Let $M \le G$ be a proper subgroup of $G$.
Then $M$ is a '''maximal subgroup of $G$''' {{iff}}:
: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 $M$.
\end{definition}