A theorem of A. Weil asserts that a topological group embeds as a (dense)
subgroup of a locally compact group if and only if it contains a non-empty
precompact open set; such groups are called locally precompact. Within the
class of locally precompact groups, the authors classify those groups with the
following topological properties:

Dieudonn\'e completeness; local realcompactness; realcompactness; hereditary
realcompactness; connectedness; local connectedness; zero-dimensionality. They
also prove that an abelian locally precompact group occurs as the
quasi-component of a topological group if and only if it is precompactly
generated, that is, it is generated algebraically by a precompact subset.