In group theory, a branch of abstract algebra, a **maximal subgroup** is a specific type of subgroup of a given group. A subgroup \( M \) of a group \( G \) is called a maximal subgroup if it is proper (meaning that it is not equal to \( G \)) and is not contained in any other proper subgroup of \( G \). In other words, there are no subgroups \( N \) such that \( M < N < G \).
