We define a notion of depth for an inclusion of multimatrix algebras $B
\subseteq A$ based on a comparison of powers of the induction-restriction table
$M$ (and its transpose matrix). The depth of the semisimple subalgebra $B$ in
the semisimple algebra $A$ is the least positive integer $n \geq 2$ for which
$M^{n+1} \leq qM^{n-1}$ for some $q \in \Z_+$. We prove that a depth two
subalgebra is a normal subalgebra, and conversely. As a corollary, a depth $n$
subalgebra is a normal subalgebra of its $(n-2)$'nd iterated endomorphism
algebra in a tower above $B \subseteq A$. Odd and even depth may be expressed
in terms of graphical distance between $B$-simples in the inclusion diagram of
$B \subseteq A$. Applied to subgroups via complex group algebras, the depth of
a subgroup $H$ in a finite group $G$ is bounded above by $2n$ if the core is an
intersection of $n$ conjugates of $H$ in $G$; with upper bound $2n-1$ if the
core is trivial. We prove that the subgroup depth of symmetric groups

$S_n < S_{n+1}$ is $2n-1$. An appendix by S. Danz and B. K\"ulshammer
determines the subgroup depth of alternating groups $A_n < A_{n+1}$ to be
$2(n-\lceil\sqrt{n} \rceil)+1$.