Groups with all subgroups either subnormal or self-normalizing

Let G be a group in which every subgroup that is not self-normalizing is subnormal. It is shown that if G is locally finite then either G has a nilpotent subgroup of prime index or every subgroup of G is subnormal, while if G is not periodic then every subgroup of G is subnormal. If G is a periodic group with the given property then G need not be locally finite, but if G is also locally graded then it is locally finite.