Using Result of Nontrivial Center show that Every prime power order group contain subgroups of all prime power order

Question

Let G be group with Order $p^n$ , p is prime number Then G has subgroup of order $p^i$ where $0\le i \le n$.
I wanted to show this using tool of G contain Non trivial center .
My attempt : I can assume |$Z(G)$|=$p^k$ where k . Center of group is abelian and Finite . So I can have subgroup of order $p^t$ where $0\le t\le k$..
As $Z(G)$ is normal therefore I had factor group which is of order $p^{n-k}$.Denote that Factor group as $G_1$ Now similar to above argument There Exist some Nontrivial center of $G_1$.....[I this argument will not work further because upto k we had already achieved So ].
By Class equation $$|G|=|Z(G)|+\sum [G:N_G(x)]$$
$\sum [G:N_G(x)]$=$p^n-p^k$...Here Onward I had no idea how to Obtain.Any help will be appreciated.