We study the structure of the category of graded, connected, commutative and cocommutative Hopf algebras of finite type over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct sum of explicitly given indecomposables. This gives rise to a similar classification of not necessarily $p$-torsion objects that are either free as commutative algebras or cofree as cocommutative coalgebras. We also completely classify those objects that are indecomposable modulo $p$.