In this paper we obtain significant bounds for the number of maximal subgroups a given index finite group. These results allow us to give new random generators needed generate $d$-generated group with high probability.

We initiate a study of maximal subgroups and von Neumann subalgebras which have the Haagerup property. determine inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ obtain several explicit instances where yield subalgebras. Our techniques are on one hand based group-theoretic considerations, other certain results intermediate algebras, in particular these allowing us to deduce that all algebras for ...