Can a group $G$ have a maximal subgroup and at the same time have a subgroup $H$ which is not contained in a maximal subgroup?
2026-03-25 13:51:03.1774446663
maximal and usual subgroups
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Let $H$ be a group with no maximal subgroups ($H = ({\mathbb Q},+)$ for example), $K = C_2$, and let $G = H \times K$. Then $G$ has the maximal subgroup $H$, but the subgroup $K$ of $G$ is not contained in a maximal subgroup.