  We prove that the Herzog-Sch\"onheim Conjecture holds for any group $G$ of
order smaller than $1440$. In other words we show that in any non-trivial coset
partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq
n$ such that $[G:U_i]=[G:U_j]$.
  We also study interaction between the indices of subgroups having cosets with
pairwise trivial intersection and harmonic integers. We prove that if
$U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting
cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.
