  In this paper, we study finite dimensional non-cosemisimple Hopf algebras
through its underlying coalgebra. We decompose such a Hopf algebra as a direct
sum of `blocks'. Blocks are closely related to each other by `rules' and form
the `block system'. Through the block system, we are able to give a lower bound
for $\dim H$, where $H$ is a non-cosemisimple Hopf algebras with no nontrivial
skew-primitives, and a series of results about non-cosemisimple Hopf algebras
of dimension $12p, 15p, 16p, 20p\ \text{and}\ 21p$ with $p$ being a prime
number.
