A note on finite groups with the indice of some maximal subgroups being primes

The Theorem 12 in [A note on p-nilpotence and solvability of nite groups, J. Algebra 321 (2009) 1555-1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indices. In this note we show that this result can be applied to prove that the nite groups in which every non-nilpotent maximal subgroup has prime index are solvable.