On groups satisfying the double chain condition on nonascendant subgroups

If θ is a subgroup property, a group G is said to satisfy the double chain condition on θ-subgroups if it admits no infinite double chain · · · X−n · · · X−1 X0 X1 · · · Xn · · · consisting of θ-subgroups. Here we want to describe the structure of locally finite and locally nilpotent groups satisfying the double chain condition on non-ascendant subgroups in term of chain conditions and of ascendant subgroups.