Let $G$ be a finite subgroup and $H$ a subgroup of index three in $G$, not necessarily normal. Put $n=|H|$. We choose representatives $a_1$ and $a_2$ such that $G$ is the disjoint union
$ G=H \cup (a_1H) \cup (a_2H) $
We can then define
$ \begin{array}{lcl} H_1 &=& H \cap (a_1 H a_1^{-1}) \cap (a_1 H a_2^{-1}) \\ H_2 &=& H \cap (a_1 H a_1^{-1}) \cap (a_2 H a_2^{-1}) \\ H_3 &=& H \cap (a_2 H a_1^{-1}) \cap (a_1 H a_2^{-1}) \\ H_4 &=& H \cap (a_2 H a_1^{-1}) \cap (a_2 H a_2^{-1}) \\ \end{array}{} $ Let us put $n_i=|H_i|$. Since $H$ is the disjoint union of the $H_i$, we have $n_1+n_2+n_3+n_4=|H|$. Are any further inequalities involving $n_1$, $n_2$, $n_3$ and $n_4$ known ?
Intuitively, the $n_i$ should be balanced, so that there should be a lower bound for $n_1+n_2,n_1+n_3$ etc that tends to $+\infty$ when $n\to +\infty$, but I don’t know how to show this.