The paper presents some results and a list of examples concerning the structure of subgroups \(\Gamma\) (with respect to addition) of a Hilbert space \(H\), especially their topological dimension \(ind\) and fundamental sets, i.e. closed and convex subsets \(K\) of \(H\) such that \(\pi(K)=H/\Gamma\) and \(\pi_{\mid\text{rint}(K)}\) is a homeomorphism onto a dense subset of \(H/\Gamma\), where \(\pi: H\to H/\Gamma\) is the natural projection and \(\text{rint}(K)\) is the radial interior of \(K\). It is shown in particular that if \(\Gamma\) is non-discrete weakly closed and contains no linear subspace of dimension \(>0\), then \(ind(\Gamma)=1\). For weakly sequentially closed \(\Gamma\) the existence of a fundamental set is proved.