  Diaconis and Isaacs define a supercharacter theory for algebra groups over a
finite field by constructing certain unions of conjugacy classes called
superclasses and certain reducible characters called supercharacters. This work
investigates the properties of algebra subgroups $H\subset G$ which are unions
of some set of the superclasses of $G$; we call such subgroups supernormal.
After giving a few useful equivalent formulations of this definition, we show
that products of supernormal subgroups are supernormal and that all normal
pattern subgroups are supernormal. We then classify the set of supernormal
subgroups of $U_n(q)$, the group of unipotent upper triangular matrices over
the finite field $\FF_q$, and provide a formula for the number of such
subgroups when $q$ is prime. Following this, we give supercharacter analogues
for Clifford's theorem and Mackey's "method of little groups." Specifically, we
show that a supercharacter restricted to a supernormal subgroup decomposes as a
sum of supercharacters with the same degree and multiplicity. We then describe
how the supercharacters of an algebra group of the form $U_\fkn = U_\fkh
\ltimes U_\fka$, where $U_\fka$ is supernormal and $\fka^2=0$, are parametrized
by $U_\fkh$-orbits of the supercharacters of $U_\fka$ and the supercharacters
of the stabilizer subgroups of these orbits.
