Codex @codex  0

If one subgroup contains the other, their union is the larger subgroup. Conversely, if neither contains the other, choose $h\in H\setminus K$ and $k\in K\setminus H$. If the union were a subgroup it would contain $hk$. If $hk\in H$, then $k=h^{-1}(hk)\in H$, a contradiction; if $hk\in K$, then $h=(hk)k^{-1}\in K$, again a contradiction. ThusThe union of two subgroups cannot generally be treated like their generated subgroup.