1. For a manifold with no solvable fundamental group, take for example a finite unsolvable group $G$ and embed it into $SU(N)$ as a discrete subgroup. This embedding is obtained by realizing $G$ as a subgroup of the permutations $S(G)$ of the set $G$, then note $S(G)$ is isomorphic to a subgroup of $SU(n)$ via monomial matrices. The quotient $SU(n)/G$ carries a natural manifold structure with fundamental group $G$. Note these examples are even compact.

2. For manifolds with cyclic subgroups of finite index, consider products of real projective spaces $RP^n$ or lens spaces $S^{2n+1}/Z_p$ with manifolds with finite fundamental group.

To get an example with 1) and 2), take a product of a manifold with finite unsolvable fundamental group with a lens space.

1. To construct a manifold with no solvable fundamental group, take for example a finite unsolvable group $G$ and embed it into $SU(n)$ as a discrete subgroup. This embedding is obtained by realizing $G$ as a subgroup of the permutations $S(G)$ of the set $G$, then note $S(G)$ is isomorphic to a subgroup of $SU(n)$ via monomial matrices ($n=|G|$). The quotient $SU(n)/G$ carries a natural manifold structure with fundamental group $G$. Note these examples are compact. Taking product with $R$, one obtains non-compact examples.

2. For manifolds with cyclic subgroups of finite index, consider products of real projective spaces $RP^n$ or lens spaces $S^{2n+1}/Z_p$ with manifolds with finite fundamental group.

To combine 1) and 2), take products.