# 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 $$[G:H]. Is right that $$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 ?