  We study the pro-$p$ group $G$ whose finite quotients give the prototypical
Sylow $p$-subgroup of the general linear groups over a finite field of prime
characteristic $p$. In this article, we extend the known results on the
subgroup structure of $G$. In particular, we give an explicit embedding of the
Nottingham group as a subgroup and show that it is selfnormalising. Holubowski
(\cite{holub1,holub0,holub2}) studies a free product $C_p*C_p$ as a (discrete)
subgroup of $G$ and we prove that its closure is selfnormalising of infinite
index in the subgroup of $2$-periodic elements of $G$. We also discuss change
of rings: field extensions and a variant for the $p$-adic integers, this latter
linking $G$ with some well known $p$-adic analytic groups. Finally, we
calculate the Hausdorff dimensions of some closed subgroups of $G$ and show
that the Hausdorff spectrum of $G$ is the whole interval $[0,1]$ which is
obtained by considering partition subgroups only.
