We examined what determines the designability of two-letter codes (H and P) lattice proteins from three points of view. First, whether the native structure is searched within all possible structures or within maximally compact structures. Second, whether the structure of the used lattice is bipartite or not. Third, the effect of the length of the chain, namely, the number of monomers on the chain. We found that the bipartiteness of the lattice structure is not a main factor that determines the designability. Our results suggest that highly designable structures will be found when the length of the chain is sufficiently long to make the hydrophobic core consisting of a large enough number of monomers.