Given a pair of matrices (A, B), we can associate to each nice basis of the controllability subspace of (A, B) a set of integers to be called the indices of the basis. Each set provides partial information about the similarity class of A. The relationship between several sets of indices as well as the constraints that they impose on the invariant factors of A are investigated. Special attention is paid to the controllability and Hermite indices of a given pair (A, B).