I want to select four points lie on the sphere `(x-1)^2 + (y-3)^2 + (z-5)^2 = 15^2` so that its coordinates are integer numbers. I tried

 ClearAll[a, b, r, c];
 a = 1;
 b = 3;
 c = 5;
 r = 15; ss = 
 Subsets[{x, y, z} /. 
 Solve[{(x - a)^2 + (y - b)^2 + (z - c)^2 == r^2}, {x, y, z}, 
 Integers], {4}];
 list = Select[
 ss, ( EuclideanDistance[#[[1]], #[[2]]] == 
 EuclideanDistance[#[[1]], #[[3]]] == 
 EuclideanDistance[#[[1]], #[[4]]] == 
 EuclideanDistance[#[[2]], #[[4]]] == 
 EuclideanDistance[#[[2]], #[[3]]] == 
 EuclideanDistance[#[[3]], #[[4]]] && 
 Det[{#[[1]] - #[[2]], #[[1]] - #[[3]], #[[1]] - #[[4]]}] != 
 0 &) ]

About ten minutes, I can not get the result? How can I get the result?