If L is not regular and is a proper subset of L1, does it follow that L1 is not regular?

If $L$ is not regular and $ L \subset L_1$, does it follow that $L_1$ is not regular also? Can you please provide an explanation? Thanks in advance.

@Vladislav's answer is probably more interesting, but observe that every language over an alphabet $\Sigma$ is a subset of $\Sigma^*$, which is certainly a regular language.

StackExchange Computer Science Q#123314, answer score: 18