The number of reflection subgroups of the associated Weyl group. Let $\mathcal{R} \subseteq W$ be the set of reflections in the Weyl group $W$. A (possibly empty) subset $X \subseteq \mathcal{R}$ generates a subgroup of $W$ that is again a reflection group. This is the number of all pairwise different subgroups of $W$ obtained this way (including the trivial subgroup). If $\Phi^+$ is an associated set of positive roots, then this also is the number of subsets $Y \subseteq \Phi^+$ such that $Y$ is a simple system of some type (including the empty system for type $A_0$). Such a subset $Y$ is identified as simple system if for all $x \neq y \in Y$ we have $\langle x,y \rangle \leq 0$.