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Let $G$ be a finite group. The main supergraph $𝒮\left(G\right)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o\left(x\right)\mid o\left(y\right)$ or $o\left(y\right)\mid o\left(x\right)$. In this paper, we will show that $G\cong Sz\left(q\right)$ if and only if $𝒮\left(G\right)\cong 𝒮\left(Sz\right(q\left)\right)$, where $q=2^{2m+1}\ge 8$.

We prove that if the average number of Sylow subgroups of a finite group is less than $\frac{41}{5}$ and not equal to $\frac{29}{4}$, then $G$ is solvable or $G/F\left(G\right)\cong A_5$. In particular, if the average number of Sylow subgroups of a finite group is $\frac{29}{4}$, then $G/N\cong A_5$, where $N$ is the largest normal solvable subgroup of $G$. This generalizes an earlier result by Moretó et al.