Is this class of $p$-groups large?

Question

Call a $p$-group $G$ good if for each subgroups $H, H_1, H_2\subseteq G$ for which $H_1\subseteq H$, $H_2\subseteq H$, $|H_1| = |H_2| = |H|/p$, $H_1\not= H_2$, $H'\not=\{e\}$ holds we have that there exists a subgroup $H_3\subset H$ such that 1, 2, 3 holds

1. $|H_3| = |H|/p$
2. $H_3\cap H_1 = H_3\cap H_2 = H_1\cap H_2$
3. $c(H_3)\geq c(H) - 1$, where $c(\cdot)$ denotes the nilpotency class of a group.

Is it true that for each $p\not= 2$, $N\in\mathbb{N}$ there exists a good $p$-group $G$ with $c(G) \geq N$?