Let $\mathbb{N}=A \cup B$ be a partition of the natural
numbers. We prove that if $A$ is sufficiently close to being a
subsemigroup, then there exists a basic sequence $Q$ where the set of
numbers that are $Q$-normal of all orders in $A$ and not $Q$-normal of
any orders in $B$ is non empty. Furthermore, these sets will have full
Hausdorff dimension under some additional conditions. This stands in
sharp contrast to the fact that if a real number is normal of order $k$
for the $b$-ary expansion, then it is also normal of all orders less
than $k$.