The classical Hurwitz bound gives a linear upper bound for automorphism groups of algebraic curves in characteristic zero, whereas algebraic curves in positive characteristic can have much larger automorphism groups. In this article, we firstly investigate the solvable subgroups and $p$-subgroups of $\operatorname{Aut}(F/\mathbb{F}_q)$ where $p=\operatorname{char}(\mathbb{F}_q)$ and $F/\mathbb{F}_q$ is a function field of genus $g_F\ge2$ with full constant field $\mathbb{F}_q$.
