Let Gbe an infinite group. mdenotes the number of the conjugacy classes of non-subnormal subgroups of G.  and  denote the number of the finitely length and infinitely length conjugacy classes of non-subnormal subgroups of G, respectively.

Let Gbe a group with  Then it is proved that

(1) there exists no infinite group with

(2) if  then there exists some normal subgroup  such that  is a finite non-nilpotent inner-abelian group, where Nis a group with all subgroups subnormal;

(3) if Gis infinite locally finite, then Gis a Baer group, and

non-subnormal subgroup, conjugacy class, locally nilpotent, locally finite.