  Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974
Herzog and Sch\"onheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a
partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot
be distinct. In this paper we show that if $\Cal A$ covers all the elements of
G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal
to G, then $M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$ is not less than the
smallest prime divisor of $n_1... n_k$, moreover $\min_{1\ls i\ls k}\log
n_i=O(M\log^2 M)$ where the O-constant is absolute.
