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We study the group structure in terms of the number of Sylow $p$-subgroups, which is denoted by $n_p\left(G\right)$. The first part is on the group structure of finite group $G$ such that $n_p\left(G\right)=n_p(G/N)$, where $N$ is a normal subgroup of $G$. The second part is on the average Sylow number $\mathrm{asn}\left(G\right)$ and we prove that if $G$ is a finite nonsolvable group with $\mathrm{asn}\left(G\right)<39/4$ and $\mathrm{asn}\left(G\right)\ne 29/4$, then $G/F\left(G\right)\cong A_5$, where $F\left(G\right)$ is the Fitting subgroup of $G$. In the third part, we study the nonsolvable group with small sum of Sylow numbers.