In this work, we give a description of the structure of the normal subgroups of a Sylow $p$-subgroup $W_n$ of $\mathrm{Sym}(p^n)$, showing that they contain a term from the lower central series with bounded index. To this end, we explicitly determine the terms of the upper and the lower central series of $W_n$. We provide a similar description of these series in the Lie algebra associated to $W_n$, giving a new proof of the equality of their terms in both the group and the algebra contexts. Finally, we calculate the growth of the normalizer chain starting from an elementary abelian regular subgroup of $W_n$.