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 Group, $|V|$ 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 $G=SL_4(2)$ 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 $|V|\mid |G|$?**

Any explanation, references suggestion and examples are appreciated.