In 2006, Bacher introduced a family of Riordan group automorphisms parametrized by three complex numbers. Bacher's family is a subgroup of the group of automorphisms of the Riordan group and so is the subfamily parametrized only by two real numbers. Here, we study some of the algebraic properties of this subfamily and use the elements to point out isomorphisms between Riordan subgroups. In this context, we prove that the set of Riordan arrays whose row sum sequence is a sequence of partial sums, forms a Riordan subgroup. Moreover, we show that the well-known recursive matrices may be constructed from sequences of images of a Riordan array under automorphisms. Our construction also discloses a correspondence between the recursive matrices and a pair of well-defined Riordan arrays.