Wikipedia states that the time complexity of the General Number Field Sieve (GNFS) is $$\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\log N)^{\frac{1}{3}}(\log \log N)^{\frac{2}{3}}\right),$$ where $N$ is the number to be factored.

The same site also mentions quasi-polynomial time is $$\mbox{QP} = \bigcup_{c \in \mathbb{N}} \mbox{DTIME} \left(2^{\log^c n}\right),$$ where $n$ is the length of the input, in other words, $n=\log N$.

Now my question is whether GNFS is in quasi-polynomial time. It appears not, because of the existence of the $(\log N)^{1/3}$ exponent.

How about sub-exponential? Wikipedia says $$\text{SUBEXP}=\bigcap_{\varepsilon>0} \text{DTIME}\left(2^{n^\varepsilon}\right).$$ I think the answer is still negative because $\varepsilon=1/3$ in this case and it can’t be smaller as in the intersection.

So the running time is exponential. Am I right?