Finite groups in which all maximal subgroups are $\pi$-closed. I

Finite simple nonabelian groups $G$ that are not $\pi$-closed for some set of primes $\pi$ but have $\pi$-closed maximal subgroups (property $(*)$ for $(G,\pi)$) are studied. We give a list $\mathcal{L}$ of finite simple groups that contains any group $G$ with the above property (for some $\pi$). It is proved that $2\not\in\pi$ for any pair $(G,\pi)$ with property $(*)$ (Theorem 1). In addition, we specify for any sporadic simple group $G$ from $\mathcal{L}$ all sets of primes $\pi$ such that the pair $(G,\pi)$ has property $(*)$ (Theorem 2). The proof uses the author's results on the control of prime spectra of finite simple groups.