Is right that $gHg^{-1}subset_{neq} G$, for every $H$, proper subgroup of $G$?

Let $G$ group and H, proper subgroup of G with $[G:H]<infty$. Is right that $bigcup_{gin G}gHg^{-1}$ is proper subset of G ? If G is finite group we can proof this claim, but is this right for infity groups ?