  We introduce two minimality properties of subgroups in topological groups. A
subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if
there is no strictly coarser Hausdorff group topology on $G$ which induces on
$H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal
subgroup is a key subgroup while the converse is not true. Every locally
compact co-compact subgroup is a key subgroup (but not always co-minimal). Any
relatively minimal subgroup is a co-key subgroup (but not vice versa).
Extending some results concerning the generalized Heisenberg groups, we prove
that the center ("corner" subgroup) of the upper unitriangular group
$\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is
a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of
$\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the
restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to
\mathcal{T}_{\downarrow}(H), \ \sigma \mapsto \sigma|_H$$ and show that it is
an isomorphism of sup-semilattices for every central co-minimal subgroup $H$,
where $\mathcal{T}_{\downarrow}(G)$ is the semilattice of coarser Hausdorff
group topologies on $G$.
