Let G be a locally compact group and let \({\mathcal {SUB}}\left( G\right) \) be the set of closed subgroups of G equipped with the Chabauty topology. In general, \({\mathcal {SUB}}\left( G\right) \) is not connected (e.g. if the connected component of the identity element of G is compact). For every closed subgroup H of G, the connected component \({\textbf{X}_{H}^{G}}\) of H in \({\mathcal {SUB}}\left( G\right) \) contains \(\left\{ g H g^{-1}\, \mid \, g\in G_0\right\} \) , where \(G_0\) denotes the identity component of G. This paper is an investigation of the class \(\mathfrak {X}\) of locally compact groups G such that \({\textbf{X}_{H}^{G}}=\left\{ g H g^{-1}\, \mid \, g\in G_0\right\} \) , for every \(H\in {\mathcal {SUB}}\left( G\right) \) . We establish some stability properties of the class \(\mathfrak {X}\) and we prove that the class \(\mathfrak {X}\) contains the class of locally elliptic groups (We recall that a locally compact group G is called locally elliptic if any compact subset of G is contained in a compact subgroup).