Given the complete k-uniform hypergraph with vertex set U, a one-factor of U is defined as partition of U into k-subsets, and a one-facorization of U is a collection of one-factors, such that every k-subset is contained in precisely one of the one-factors. We say that, given vertex sets U and V and one-factorizations S and T of the complete k-uniform hypergraph on U and V (respectively), T contains S as a subsystem if U is contained in V and T restricted to U contains S. A necessary condition for the existence of such a subsystem is |V| ≥ 2|U|; in this talk we examine sufficient conditions for the existence of such a V and T given a fixed U and S, both for general k and the special case of k = 3 and k = 4.