  It is important to classify covering subgroups of the fundamental group of a
topological space using their topological properties in the topologized
fundamental group. In this paper, we introduce and study some topologies on the
fundamental group and use them to classify coverings, semicoverings, and
generalized coverings of a topological space. To do this, we use the concept of
subgroup topology on a group and discuss their properties. In particular, we
explore which of these topologies make the fundamental group a topological
group. Moreover, we provide some examples of topological spaces to compare
topologies of fundamental groups.
