Given a discrete group G, we consider the set L(G) of all subgroups of G endowed with topology arising from the standard embedding of L(G) into the Cantor cube {0, 1} G. The cellularity c(X) is the supremum of cardinalities of disjoint families of open subsets of a topological space X. It has been shown in a joint paper of the speaker and I. Protasov that the cellularity c(L(G)) is countable for every infinite abelian group G; and, for any infinite cardinal τ , there exist a non-abelian group G with c(L(G)) = τ . In the second part of our talk we will review a complete description of the homomorphism type of L(G), when G is a countable abelian group. It has been done in a recent paper by Y. de Cornulier, L. Guyot and W. Pitsch.