Hadwiger numbers of self-complementary graphs

The Hadwiger number of a graph $G$, denoted by $h(G)$, is the order of the largest complete minor of $G$. A graph is said to be self-complementary if it is isomorphic to its complement. We prove that for all $n\equiv 0,1 (\text{mod 4})$ and for all $\lfloor \dfrac{n+1}{2} \rfloor \le h \le \lfloor \dfrac{3n}{5}\rfloor$, there exists a self-complementary graph $G$ with $n$ vertices whose Hadwiger number is $h$.