For a natural number n, let G(n) denote the set of groups of order n. A common problem in a first course in algebra is to describe G(n) for some fixed, small n (e.g. n=15). In this talk we discuss some surprising results that show the connection between the number theoretic properties of n and the group theoretic properties of the elements of G(n) is stronger than one might expect. In particular, we definitively show that in the land of groups, not all numbers are created equal.