Which cycles are possible by repeated summation of the cubes of the digits of a number?

Question

Here :

Digital root with squared digits

the possible cycles of repeated summation of the squares of the digits of a number are mentioned. What about cubes ?

$1$ , $153$ , $370$ , $371$ and $407$ are fixpoints

$(136/244)$ , $(919/1459)$ , $(160/217/352)$ and $(55/250/133)$ are cycles.

What is the complete set of the possible cycles ?