The Grothendieck Group of Hopf Algebras

Let H be a cosemisimple Hopf algebra over an algebraically closed field k which contains a simple subcoalgebra of dimension 9. We show that if H has no simple subcoalgebras of even dimension then H contains either a grouplike element with order 2 or 3, a Hopf subalgebra of dimension 75, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. In particular, if H is finite odd dimensional, then its dimension is divisible by 3.