of all the polynomials over $\mathbb{q}_p$, those of degree divisible by $p$ are most complicated. similarly, the $p$-local subgroup structure of $\operatorname{gl}_n(\mathbb{c})$ is more complicated if $p$ divides $n$. the talk will be an example based introduction to these ideas, which fall under the ``wild aspect'' of the local langlands correspondence. along the way we will define $\mathbb{q}_p$, some reductive groups of exceptional type, and exhibit wild $p$-structures in both of these examples.