I've been trying to google this question, but to no avail. The question sounds elementary but I hope it's suitable for the experts at MO!

Let $B_n$ be the braid group on $n$-strands and $P_n$ be the pure braid group on $n$-strands. We have a well know decomposition of $P_n$ as $F_n \rtimes P_{n-1}$. In this light, my question is simply:

**Are there known decompositions of $B_n$ in terms of direct/semidirect products of $B_{k}$ for $1\leq k < n$ and (possibly) other subgroups of $B_n$**

I'd even be interested for special cases of $n$

Thanks!