There are at least $p^{2n^3/27+O(n^2)}$ groups of order $p^n$, and in 2006 those of order $p^7$ were classified in over 600 pages of work. Yet, with such a multitude of groups, a structure theory seems impossible. One approach is to decompose the $p$-groups via central and related products to reduce the study to indecomposable groups. Using rings and Jordan algebras, a theorem is proved on the uniqueness of these decompositions, asymptotic estimates are given which show there are roughly equal numbers of decomposable and indecomposable groups, and the indecomposable groups are categorized into classical families.