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

The Herzog-Sch\"onheim Conjecture for small groups and harmonic subgroups