Groups with weak maximality condition for nonnilpotent subgroups

A group G satisfies the weak maximality condition for nonnilpotent subgroups [or, briefly, the Wmax-(nonnil) condition if G does not have infinite increasing chains {Hn | n ∈ ℕ} of nonnilpotent subgroups such that the indices |Hn+1: Hn| are infinite for each n ∈ ℕ. We study the structure of hypercentral groups satisfying the weak maximality condition for nonnilpotent subgroups.