When is it possible for a symmetric group (denoted Sym(m)) to have a maximal subgroup that is isomorphic to another symmetric group (say, Sym(n))? One easy case is when m = n+1; for example, Sym(13) has a maximal subgroup that is isomorphic to Sym(12). However, there are other interesting and surprising ways in which this can happen. This talk will describe three different infinite families of ordered pairs (n,m) such that Sym(m) has a maximal subgroup that is isomorphic to Sym(n), and will prove that no other ordered pairs exist. Only basic knowledge of group theory will be assumed; if you understand this abstract, you will likely understand the bulk of the lecture.