5. Given that $a$, $b$, and $c$ are all positive numbers. Then
$$
M=\min \left\{\frac{1}{a}, \frac{2}{b}, \frac{4}{c}, \sqrt[3]{a b c}\right\}
$$

the maximum value of $M$ is . $\qquad$
5. $\sqrt{3}$.

From the problem, we know $M \leqslant \frac{1}{a}, M \leqslant \frac{2}{b}, M \leqslant \frac{4}{c}$. Thus, $M^{3} \leqslant \frac{8}{a b c} \Rightarrow \sqrt[3]{a b c} \leqslant \frac{2}{M}$.
Combining $M \leqslant \sqrt[3]{a b c}$, we get
$$
M \leqslant \frac{2}{M} \Rightarrow M \leqslant \sqrt{2} \text {. }
$$

Therefore, $M_{\max }=\sqrt{2}$, which is achieved when $c=2 b=4 a=2 \sqrt{2}$.