For a subgroup $H$ of a given group $G$, I say $H$ is "big" if it has nonempty intersection with each conjugacy class of $G$. I have known that, trivially, $G$ itself is "big". And if $H$ is a normal subgroup and it is "big", then $H=G$. I have also known that a finite group has no proper "big" subgroup. My question is "Is there an infinite group who has a proper 'big' subgroup?"
Yes, for example in Osin's infinite group with 2 conjugacy classes every proper subgroup is big. Of course if you do not care about the number of generators, you can consider the (much easier) infinitely generated group constructed by Higman-Neumann-Neumann where all non-identity elements are conjugate. There also every proper subgroup is big.