Maximal subgroup

Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finite partially ordered set under inclusion. There are, however, infinite abelian groups that contain no maximal subgroups, for example the Prüfer group. ==Maximal normal subgroup==

Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N < G and there is no normal subgroup K of G such that N < K < G. We have the following theorem: :Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple. ==Hasse diagrams==

These Hasse diagrams show the lattices of subgroups of the symmetric group S4, the dihedral group D4, and C23, the third direct power of the cyclic group C2. The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram. ==References==