Generalized Heineken--Mohamed type groups

We prove that a torsion group G with all subgroups subnormal is a nilpotent group or G=N(A1 \times \cdots \times An) is a product of a normal nilpotent subgroup N and pi-subgroups Ai, where Ai=A1(i) \cdots Ami(i) \lhd G, Aj(i) is a Heineken--Mohamed type group, and p1, \ldots, pn are pairwise distinct primes (n\geq 1; i=1, ... ,n; j=1, ... ,mi and mi are positive integers).