how to check if a subgroup is maximal?

Question

Is there any general strategy to check whether a subgroup is maximal or not ? For example, in case of rings, we know that an ideal $I$ of a ring $R$ is maximal if and only if $R/I$ is a field. Is there something like this for groups ?

Answer 1

There is an equivalent condition, that a subgroup $H$ of index $n>1$ in a group $G$ is maximal if and only if the image of the permutation action of $G$ on the cosets of $H$ in $G$ is a primitive subgroup of $S_n$. This test is often used in computer calculations, because there are fast tests for primitivity.