(Joint work with P.-E. Caprace and C. Reid) The collection of topologically simple totally disconnected locally compact (t.d.l.c.) groups which are compactly generated and non-discrete forms a rich and compelling class of locally compact groups, denoted by S; members include the simple algebraic groups over non-archimedean local fields and the almost automorphism groups. In recent years, a general theory for these groups, which considers the interaction between the geometry and the topology, has emerged. In this talk, we study this interaction by considering locally compact groups H which embed densely into some group G in S. We show the structure of such groups H is very restrictive as soon as H is non-discrete. As applications, we show no group in S admits a compact open subgroup with an infinite solvable Sylow subgroup. We further place restrictions on the automorphism groups of groups in S.