  Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its
group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds
for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of
groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic,
abelian, cyclic. We refine these results by finding similar bounds for groups
of odd order that are of these types. We also add more types of solvable groups
to that long list by establishing the optimal bounds for, among others, groups
of order $p^m q^n$. Moreover, we show that Zomorrodian's bound for $p$-groups
$G$ with $p\geq 5$, namely $|G|\leq \frac{2p}{p-3}(g-1)$, actually holds for
any group $G$ for which $p\geq 5$ is the smallest prime divisor of $|G|$.
