Let $N\subset M$ be a be factors acting on a Hilbert space $H$. Denote by $\mathrm{Cyc}(A)\subset H$ the set of cyclic vectors for $A = M,N$.

I am interested in the equality case of the inclusion $Cyc(N)\subseteq Cyc(M)$?

If, e.g., $M= M_2(\mathbb C) \otimes N$, then it is easy to find vectors vectors which are cyclic for $M$ but not for $N$ (any product vector with respect to the induced splitting $H = \mathbb C^2\otimes H_0$ works). I was not able to find counterexamples for cases where the relative commutant is trivial, so I want to ask:

Does $\mathrm{Cyc}(M)=\mathrm{Cyc}(N)$ hold if the relative commutant is trivial: $N'\cap M=\mathbb C$?

Any help is much appreciated