Finite groups with each non-normal subgroup having maximal normal cores

In this paper, we study a finite group G such that vertical bar H : H-G vertical bar is a prime for each non-normal subgroup H of G. We prove that such a group must contain a big abelian subgroup. More specifically, if such a group G is not supersoluble, then there is an abelian subgroup A such that vertical bar G : A vertical bar vertical bar 2pq, and if G is supersoluble, then there is an abelian subgroup A such that vertical bar G : A vertical bar vertical bar 2(5) or vertical bar G : A vertical bar vertical bar p(2)q(2), where p and q are primes.