[bigpooch]

Using the "lingo" of the post by TC that was in white:

54 = 8^3 - 6 x 63 - 2 x 26 - 4 x 7

(Okay, so it's not THAT easy to see this geometrically!)

Also, looking at everything (mod 7),

53 is congruent to 39 and 20 = 27-7, so 39 = 20+20-1 is
possible
(if a cube can be dissected into x1 subcubes or x2 subcubes
it obviously can also be dissected into x1+x2-1 subcubes)

EDIT: much easier is 53 = 27+26

52 is congruent to 38 = 64-26
51 = 6^3 - 5 x 26 - 5 x 7 (geometrically ok)
50 is congruent to 1
49 = 6^3 - 4 x 26 - 9 x 7 (geometrically ok)
48 is congruent to 27

47 is not possible