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

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 $1 \leq i, j \leq n$suchthat$[G:U_i]=[G:U_j]$. 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 $n\le 4$ then $[G:U_1]$,…,$[G:U_n]$ are harmonic integers.