I am trying to understand the relation between the direct sum and direct product of all the groups from the set
$$\{S_{N_1} \wr D_{m_1}, \ldots, S_{N_i} \wr D_{m_i}, \ldots, S_{N_t} \wr D_{m_t}\}$$
where $i, t, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.
It looks like there is a subgroup in the direct product which is isomorphic to the direct sum. But I am not able to tell more than that.
Anyone any help?