Is it possible for a group $\mathcal{G}$ to have normal subgroups $\mathcal{H}_1$ and $\mathcal{H}_2$, such that $\mathcal{H}_1$ and $\mathcal{H}_2$ have the same order and are both the largest, but their elements are all different except $e$?
If this is true, would the decomposition of a group in terms of its largest normal subgroup still be unique?
What about $\;\{1\}\times C_p\;,\;\;C_p\times\{1\}\;$ in $\;C_p\times C_p\;$ ,with $\;C_p:=$ the (one and unique up to isomorphism) cyclic group of order a prime $\;p\;$