Carmichael chains

A Carmichael number is a chain if when we remove repeatedly its largest factor, we still obtain a Carmicheal number.

$174470590282430768272287350512321$ is a Carmichael chain of length 10 (it has 10 factors).

$174470590282430768272287350512321=7.13.31.61.181.541.2161.187921.3570481.7140961$
$24432368456070656074481761=7.13.31.61.181.541.2161.187921.3570481$
$6842878720281848881=7.13.31.61.181.541.2161.187921$
$36413592521761=7.13.31.61.181.541.2161$
$16850343601=7.13.31.61.181.541$
$31146661=7.13.31.61.181$
$172081=7.13.31.61$
$2821=7.13.31$
All these numbers are Carmichael numbers.

Of course, we stop at 3 factors, as a Carmichael number can't have less than 3 factors.

Find the next Carmichael chain of length 10. We assume that no factor is greater than ${10}^7$.

[My timing: 60 sec]