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)) (element of) 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.