How many non-isomorphic finite groups with exactly $ n $ are there conjugation classes?

All I could prove was that their number is finite for everyone $ n in mathbb {N} $:

Suppose $ G $ has $ n $ conjugation classes $ C_1, … C_n $and suppose $ forall i <n $ $ g_i in C_i $. Then $ | C_i | = (G: C (g_i)) $. From that we can conclude, that $ 1 = | G | ^ {- 1} ( sum_ {i = 1} ^ n | C_i |) = | G | ^ {- 1} ( sum_ {i = 1} ^ n (G: C (g_i))) = sum_ {i = 1} ^ n frac {1} {C (g_i)} $. And it is known that there are only many fine ways to partition $ 1 $ in sum of $ n $ Egyptian fractions