For $H \subset G$, write $e_H$ for the basis element of $B(G)$ corresponding to the $G$-set $G/H$. Recall that for $K \subset G$, 
$$
e_H\cdot e_K = \sum\limits_{H g K \in H \backslash G / K} e_{H \cap g K g^{-1}}.
$$

Say that $K$ is subconjugate to $H$ if $K$ is conjugate to a subgroup of $H$. This gives rises to a partial order on the set of conjugacy classes of subgroups of $G$, and hence on the basis elements $e_H$. With respect to this partial ordering, $M_{e_H}$ is lower-triangular for each $H \subset G$. The eigenvalues are then the diagonal elements. The diagonal element corresponding to $e_{\{\textrm{id}_G\}}$ is $|G/H|$. The diagonal element corresponding to $e_K$ for $K \subset G$ is bounded above by the number of $(H, K)$-double cosets in $G$, which is bounded above by $|G/H|$.