Let $G$ be a $p$-group. Let $H$ be any subgroup of $G$. How to prove that there exists subgroups of $G$ such that $$H = H_0 \lt H_1 \lt H_2\lt \cdots \lt H_n=G$$ such that $|H_{i+1}/H_i|=p$?
I have proved the theorem when H is normal subgroup of G, & when H is trivial subgroup.
Here H is any given fixed subgroup to start with (not necessarily normal)