I am a bit confused about why a finitely generated group has infinite Prufer rank?
A group is said to have finite Prufer rank $r$ if every finitely generated subgroup can be generated by $r$ elements and $r$ is the least such integer. Otherwise it has infinite Prufer rank.
If the subgroups we are considering is finitely generated, surely we can find a max over all these subgroups and find $r$? I am not sure what I am missing here...