Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Groups, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=\mathrm{SL}_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|$ divides $|G|$?

Any explanation, references suggestion and examples are appreciated.