I have a some confusion regarding the accepted answer solution here: Structure of groups of order $pq$, where $p,q$ are distinct primes.
Consider a group $G$ with order $pq$ where $p$ and $q$ are distinct primes. Let $P$ be a subgroup of $G$ with order $p$ and $Q$ be a subgroup with order $q$ (from Sylow's theorems we know these exist cf. this). The assumption $q \pmod p \neq 1$ implies $P$ is normal in $G$ and $p \pmod q \neq 1$ implies that $Q$ is normal in $G$. We need at least one of these assumptions to write $G$ as a semi-direct product of $P$ and $Q$. The accepted solution to the other questions does that.
But what happens if neither of these assumptions hold true, and neither $P$ nor $Q$ is normal in $G$? Then we can no longer write $G\cong C_p\rtimes C_q$ or $G\cong C_q\rtimes C_p$ (because for writing a group as a semi-direct product of its subgroups, at least one of the subgroups must be normal cf. Wikipedia). How would the group structure look in that case and how to write a mathematical description of it? The accepted answer does not deal that case and I'm not sure if I'm missing something.
Perhaps at least one of the subgroups $P$ or $Q$ simply has to be normal? But why?
