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.