Groups with soluble minimax conjugate classes of subgroups

A classical result of Neumann characterizes the groups in which each 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)) G X for each subgroup H of G. Here we study groups which have soluble minimax conjugate classes of subgroups, giving a description in terms of G/Z(G). We also characterize FC-groups which have soluble minimax conjugate classes of subgroups.