Generalized Heineken--Mohamed type groups

We prove that a torsion group G with all subgroups subnormal is a nilpotent group or G=N(A_1 times cdots times A_n) is a product of a normal nilpotent subgroup N and p_i-subgroups A_i, where A_i=A_1^{(i)} cdots A_{m_i}^{(i)} lhd G, A_j^{(i)} is a Heineken--Mohamed type group, and p_1, ldots, p_n are pairwise distinct primes (ngeq 1; i=1, ... ,n; j=1, ... ,m_i and m_i are positive integers).