Let $G$ be a finite group and let $A$ be any $G$ module. Then it is well known that $H^n(G,A)$ is a subgroup of $\oplus_p H^n(G_p, A)$, where $G_p$ denotes a sylow $p$ subgroup of $G$. This is achieved as a corollary to the fact that restriction composed with corestriction is multiplication by the index of the subgroup.
My Question: In the above direct sum, do we need to consider all sylow $p$ subgroups of $G$ for a fixed prime $p$?
I believe it is enough to consider a sylow $p$ subgroup for each fixed prime $p$. Hope somebody can clarify this for me.
The fact that
$$\operatorname{cor}\circ\operatorname{res}:H^n(G,A)\to H^n(G,A)$$
is multiplication by $(G:H)$ shows that the $p$-primary part of $H^n(G,A)$ embeds into $H^n(G_p,A)$, where $G_p$ is any Sylow $p$-subgroup. Since $H^n(G,A)$ is the direct sum of its $p$-primary parts, this shows that $H^n(G,A)$ is a subgroup of
$$\bigoplus_{p||G|}H^n(G_p,A)$$
where we index over primes dividing $|G|$, not all possible Sylow $p$-subgroups. In other words, we only need to take one Sylow $p$-subgroup for each prime dividing $p$ as a constituent in the direct sum.