Maximal Subgroups of Finite Groups

What ingredients are necessary to describe all maximal subgroups of the general finite group G? This paper is concerned with providing such an analysis. A good first reduction is to take into account the first isomorphism theorem, which tells us that the maximal subgroups containing a given normal subgroup N of G correspond, under the natural projection, to the maximal subgroups of the quotient group GIN. Let n = nG denote the collection of maximal subgroups of G, and let n * be the subset of those MEn with KerG(M) = 1, where KerG(M) denotes the largest normal subgroup of G contained in M. Then the first isomorphism theorem allows us to identify n with the disjoint union UN <lG nS/N' Actually, what we really want to parameterize are the conjugacy classes of maximal subgroups, but this too works well: If W = WG denotes the set of G-conjugacy classes of elements of n, and W* is defined similarly, then we have