We present a contribution to the structure theory of locally compact groups. The emphasis is put on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a complete description of groups all of whose proper quotients are compact, of characteristically simple groups and of groups admitting a subnormal series with all subquotients compact, or compactly generated Abelian, or compactly generated and topologically simple. Two appendices introduce results and examples around the concept of quasi-product.