The set of side lengths is $S = \{7, 14, 21, 28, 35\}$. If $x \le y \le z$ are side lengths of the triangle, then $x + y > z$ holds by the Triangle Inequality. The triangles $(x,y,z)$ that can be formed from $S$ include: Equilateral: $(7,7,7); (14,14,14); (21,21,21); (28,28,28); (35,35,35)$ Isosceles; $(7,14,14); (7,21,21); (7,28,28); (7,35,35); (14,14,21); (14,21,21); (14,28,28); (14,35,35)$; $(21,21,28); (21,21,35); (21,28,28); (21,35,35); (28,28,35); (28,35,35)$ Scalene: $(14,21,28); (14,28,35); (21,28,35)$ for a total of $\fbox{22}$ triangles.