# Every 1-completed subgroup is contained in a maximal subgroup

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## Statement

Every 1-completed subgroup is contained in a maximal subgroup.

In symbols, suppose $G$ is a group and $H$ is a subgroup of $G$ such that there exists $x \in G, x \notin H$ such that $\langle H, x \rangle = G$. Then, $H$ is contained in a maximal subgroup.

Note that the statement is obviously true for finite groups; the interesting case is that where $G$ is an infinite group (so that it is not necessarily true that every proper subgroup is contained in a maximal subgroup).