Radical subgroups of locally compact, totally disconnected groups

We observe a correspondence between collections of closed subgroups and normal subgroups in locally compact, totally disconnected groups. This correspondence is first applied to prove structure theorems for two classes of locally compact, totally disconnected, and second countable groups: the class of such groups with an open, solvable subgroup and the class of such groups with a compact, open, and pro-nilpotent subgroup. As a second application, we give new proofs and generalizations of results of G. Willis and Y. Barnea, M. Ershov, and T. Weigel on locally compact, totally disconnected groups which are topologically simple and compactly generated,