  Much can be learned about a finite group from its character table, but
sometimes that table can be difficult to compute. Supercharacter theories are
generalizations of character theory defined by P. Diaconis and I.M. Isaacs, in
which certain (possibly reducible) characters called supercharacters take the
place of the irreducible characters, and a certain coarser partition of the
group takes the place of the conjugacy classes.
  We present five new ways to construct new supercharacter theories out of
supercharacter theories already known to exist, including a direct product, a
lattice-theoretic join, two products over normal subgroups, and a duality for
supercharacter theories of abelian groups.
