If a manifold is contained in another manifold, is it a regular submanifold?

Question

I am trying to wrap my head around regular submanifolds. There is an easy proposition that states that if M is regular submanifold of N, then M is itself a manifold (with the subspace topology). Is however the reverse correct? If $M \subset N $ where N is a manifold and M manifold with the subspace topology, does that imply that it s a regular submanifold? If not, can you provide a counterexample?