for example the subgroup produced by is the set of elements all the way until u get your identity element, which in this group is 1. That set of elements is your subgroup produced by 2. Check that its a subgroup of in this case since your group is finite just check that <2> is closed under multiplication mod 19 (which is a subgroup test for finite groups). So since <2> generates the whole Group, the set of cosets is just <2> itself. For example, take now to find the cosets of the subgroup find an element in the group that isn't in <6>, take 2, now just multipily all elements of <6> with 2 under mod 19. Notice that there are 9 elements in <6> so there should be 18/9 = 2 cosets exactly such that their union produces the original group.

so 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

so the first coset is {6, 17, 7, 4, 5, 11, 9, 16, 1}

the second coset is 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

for the subgroup <6> in U(19)