  In this paper we explore minimum odd and minimum even depth sub algebra pairs
in the context of double cross products of finite dimensional Hopf algebras. We
start by defining factorization algebras and outline how subring depth in this
context relates with the module depth of the regular left module representation
of the given subalgebra. Next we study minimum odd depth for double cross
product Hopf subalgebras and determine their value in terms of their related
module depth, we conclude that minimum odd depth of Drinfeld double Hopf
subalgebras is 3. Finaly we produce a necessary and sufficient condition for
depth 2 in double cross product Hopf subalgebra extensions. This sufficient
condition is then used to prove results regarding minimum depth 2 in Drinfeld
double Hopf subalgebras, particularly in the case of finite Group Hopf
algebras. Lastly we provide formulas for the centralizer of a normal Hopf
subalgebra in a double cross product scenario.
