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The main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.

We prove that a nonperfect locally graded minimal non-metahamiltonian group G is a soluble group with derived length of at most 4. On the other hand, if G is perfect, then G/Φ(G) is isomorphic to A5 , where Φ(G) is the Frattini subgroup of G and A5 is the alternating group. Moreover, we show that under some conditions, if G is a p -group, then G is metabelian, where p is a prime integer.

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