  This paper concerns the non-commutative analog of the Normal Subgroup Theorem
for certain groups. Inspired by Kalantar-Panagopoulos, we show that all
$\Gamma$-invariant subalgebras of $L\Gamma$ and $C^*_r(\Gamma)$ are ($\Gamma$-)
co-amenable. The groups we work with satisfy a singularity phenomenon described
in Bader-Boutonnet-Houdayer-Peterson. The setup of singularity allows us to
obtain a description of $\Gamma$-invariant intermediate von Neumann subalgebras
$L^{\infty}(X,\xi)\subset\mathcal{M}\subset L^{\infty}(X,\xi)\rtimes\Gamma$ in
terms of the normal subgroups of $\Gamma$.
