On the number of group topologies on countable groups

If a countable group $G$ admits a non-discrete Hausdorff group topology, then the lattice of all group topologies of the group $G$ admits:
– continuum $c$ of non-discrete metrizable group topologies such that $\sup\{\tau_1,\tau_2\}$ is the discrete topology for any two of these topologies;
– two to the power of continuum of coatoms in the lattice of all group topologies.