I have been thinking about a composition series for $D_{14}\times D_{10}$ (where $D_{2n}$ is the dihedral group with $2n$ elements). Is the following a correct composition series for $D_{10}\times D_{14}$: $D_{14}\times D_{10}\vartriangleright \langle\sigma_1\rangle\times D_{10}\vartriangleright\langle\sigma_1\rangle\times\langle\sigma_2\rangle\vartriangleright\{id_1\}\times\{id_2\}?$ I also have to verify what the factors are, and in this case, I think the factors are $\begin{align}&(D_{14}\times D_{10})/(\sigma_1\times D_{10})\cong\mathbb{Z}_2\\& (\langle\sigma_1\rangle\times D_{10})/(\langle\sigma_1\rangle\times\langle\sigma_2\rangle)\cong\mathbb{Z}_2\\&\langle\sigma_1\rangle\times\langle\sigma_2\rangle\cong\mathbb{Z}_{35}.\end{align}$ I'm really not sure I understand the material very well, so if possible, please try to elaborate as much as you can. Thanks!
Please tell me if this is correct, and if it isn't, what I should change (and if maybe it isn't so clear, why it should be changed).