B.H.Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If X is a class of groups, a group G is said to have X-conjugate classes of subgroups if G/CoreG(NG(H)) ∈ X for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroups if and only if it is central-by-(polycyclic-by-finite).

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For soluble groups, the Fitting length is bounded by a function of the maximum order of the Fitting subgroups of 2-generator subgroups.