  For any finite Coxeter group $W$ of rank $n$ we show that the order complex
of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a
connected chamber subcomplex into a spherical building of type $A_{n-1}$. We
use this to give a new proof of the fact that the non-crossing partition
lattice in type $A_n$ is supersolvable for all $n$ and show that in case $B_n$,
this is only the case if $n<4$. We also obtain a lower bound on the radius of
the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the
radius is ${n \choose 2}$.
