  Let $k$ be any field and let $G$ be a connected reductive algebraic
$k$-group. Associated to $G$ is an invariant first studied by Satake and Tits
that is called the index of $G$ (a Dynkin diagram along with some additional
combinatorial information). Tits showed that the $k$-isogeny class of $G$ is
uniquely determined by its index and the $k$-isogeny class of its anisotropic
kernel $G_a$. For the cases where $G$ is absolutely simple, Satake and Tits
classified all possibilities for the index of $G$. Let $H$ be a connected
reductive $k$-subgroup of maximal rank in $G$. We introduce an invariant of the
$G(k)$-conjugacy class of $H$ in $G$ called the embedding of indices of $H$ in
$G$. This consists of the index of $H$ and the index of $G$ along with an
embedding map that satisfies certain compatibility conditions. We introduce an
equivalence relation called index-conjugacy on the set of $k$-subgroups of $G$,
and observe that the $G(k)$-conjugacy class of $H$ in $G$ is determined by its
index-conjugacy class and the $G(k)$-conjugacy class of $H_a$ in $G$. We show
that the index-conjugacy class of $H$ in $G$ is uniquely determined by its
embedding of indices. For the cases where $G$ is absolutely simple of
exceptional type and $H$ is maximal connected in $G$, we classify all
possibilities for the embedding of indices of $H$ in $G$. Finally, we establish
some existence results. In particular, we consider which embeddings of indices
exist when $k$ has cohomological dimension $1$ (resp. $k=R$, $k$ is $p$-adic).
