It is impossible to fully represent a non-associative algebra using group theory due to the associativity requirement for the group operation. This precludes a full group theoretical cover of Cayley-Dickson algebras beginning with the generally non-associative Octonion Algebra doubling count. However, if we “forget” the sign attached to the result of the product of any two basis elements, every Cayley-Dickson algebra of doubling count n: CD(n) is modelled by the Exclusive Or group X(n), up to ignored sign of fixed nature or orientation choice. This paper explores the bijective correspondence between every CD(n) algebra and all of its subalgebras, and every X(n) group and all of its subgroups. A recursion relationship for determining the number of subalgebras of order 2 exponent m for any CD(n), which is equivalent to the number of subgroups of order 2 exponent m for the corresponding group X(n), is presented.