Bob: I hear you won the lottery.
Alice: So I did!
Bob: What six numbers did you win it with?
Alice: Can't remember. All I recall is that they were all different, and none greater than 28.
Bob: Anything else?
Alice: Yes. I checked that no other six different numbers, none greater than 28, have the same common factor graph as mine.
(The common factor graph of a set of integers is the graph whose vertices are the integers, two of which are joined by an edge if they have a common factor greater than 1.)
What six numbers did Alice win the lottery with?