A factorion is a natural number that equals the sum of the factorials of its decimal digits. There are only four factorions in base 10 and they are 1, 2, 145 and 40585.

• 1     = 1!
• 2     = 2!
• 145   = 1! + 4! + 5!
• 40585 = 4! + 0! + 5! + 8! + 5!

Let us define a factorionid as a factorion-like number which is a member of any of the factorial-like functions. Let us also define a double factorion as a factorionid of the double factorials, a triple factorion as a factorionid of the triple factorials, and a subfactorion as a factorionid of the subfactorials.
Of these three factorionidic groups, which group has the most members? What is the largest factorionid of this group? And, of the three groups, what is the largest factorionid?