Group of order $1575$ problem

Read Tobias comment: the N/C (*) theorem tells us that

$$N_G(H)/C_G(H)=G/C_G(H)\cong T\le\;\text{Aut}\,(H)$$

But since $\;H\cong C_3\times C_3\;$ or $\;H\cong C_9\;$ , we have that $\;|\text{Aut}\,(H)|\in\{6\;,\;48\}\;$ , and since

$$\left|G/C_G(H)\right|\ \left|\ \frac{1575}9\right.=175=5^2\cdot7$$

so it must be that $\;G/C_G(H)=1\iff C_G(H)=G\iff H\le Z(G)\;$

(*) The N/C theorem tells us that we have a homomorphism

$$\phi:N_G(H)\to\;\text{Aut}\,(H)\;,\;\;\phi(x):=\varphi_x$$

with $\;\varphi_x(h):=h^x:=x^{-1}hx\;,\;\;\forall\,h\in H\;$

and then the first isomorphism theorem gives us

$$N_G(H)/\ker\phi\cong T\le\;\text{Aut}\,(H)$$

for some subgroup $\;T\;$ of the automorphism group of $\;H\;$