Assume that we have two subgroups $G_1,G_2$ of $Sp(4,\mathbb{Z})$ that are conjugate in $GL(4,\mathbb{Q})$ $\big($or in $SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follows that their indices are equal: $[Sp(4,\mathbb{Z}):G_1]=[Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!