To solve this, we first need to convert the mixed numbers to improper fractions:

6-2 2/7 = (7*6 + 2) - 2/7 = 44/7 - 2/7 = 42/7 - 2/7 = 40/7

1 5/44 = (44*1 + 5) / 44 = 49/44

Now we can perform the multiplication and subtraction:

(40/7 49/44) = (40 49) / (7 * 44) = 1960 / 308 = 490 / 77

Now, we can multiply by 2 17/19:

(490/77 2 17/19) = (490/77 38/19) = (490 38) / (77 19) = 18620 / 1463 = 140 / 11

Therefore, the final result is 140/11.