Some explicit bounds in groups with a finite number of non-normal subgroups

The number of conjugacy classes of non-normal subgroups is an invariant of a group G denoted by v(G). In this paper explicit upper bounds for the order of the commutator subgroup [G'] and for the index of the centre in G, [G : Z(G)] are given for a group G with only a finite number of non-normal subgroups. The bounds provided are functions of v(G) and the primes appearing as orders of elements of G.