Two more additions. The first digit of the last group is the cycle number, 0 being the beginning of the month and 9 being the last. (Or the last digit of the middle group -- it's been a while since I dealt with this. My card is 67 in that position so I can't tell.)

A few Amex cards start with 34 instead of 37. I think they assigned a few in that range to keep it reserved for future use.