  Let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. Most of the
known $\mathbb{F}_{q^2}$-maximal curves arise as quotient curves of the
$\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_{q}$. After a seminal
paper by Garcia, Stichtenoth and Xing, many papers have provided genera of
quotients of $\mathcal{H}_q$, but their complete determination is a challenging
open problem. In this paper we determine completely the spectrum of genera of
quotients of $\mathcal{H}_q$ for any $q\equiv1\pmod4$.
