A Lower Bound on the Number of Maximal Subgroups in a Finite Group

For a finite group G, let m(G) denote the set of maximal subgroups of G and $$\pi (G)$$ denote the set of primes which divide |G|. In this paper, we prove a lower bound on |m(G)| when G is not nilpotent, that is, $$|m(G)| \ge | \pi (G)| + p$$, where $$p \in \pi (G)$$ is the smallest prime that divides |G| such that the Sylow p- subgroup of G is not normal in G.