C-Minimal topological groups

We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group $G$ contains a compact open normal subgroup $N$, then $G$ is c-totally minimal if and only if $G/N$ is hereditarily non-topologizable. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering a question by Dikranjan and Megrelishvili we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact. We also prove that the group $A\times F$ is c-(totally) minimal for every (respectively, totally) minimal abelian group $A$ and every finite group $F.$