A "prime number theorem" for simple groups

Question

Our professor recently introduced us to simple groups. Since all groups of prime order are simple, I became interested in simple groups of composite order. Suppose I represent the number of simple groups less than $n$ by the function $f(n)$. An immediate conclusion is that eventually $\pi(n)<f(n)$. I am interested in the difference $f(n)-\pi(n)$. This will give me a way to estimate the number of composite simple groups less than $n$. I wanted to know if such an estimate exists or if there have been attempts at the same. For example, if this difference becomes constant or bounded then it tells us that composite simple groups are somehow "controlled" by the prime simple groups lying in their vicinity. On the other hand, if the difference is unbounded, I would like to find out if there is a way to find the rate at which the difference increases.

Answer 1

You should read this, the Classification of the Finite Simple Groups is one of the hallmarks of group theory. All of the simple groups of order less than $1000$ are of the form $PSL(2,q)$, where $q \in \{5, 7, 8, 9, 11\}$, the corresponding group orders are $60, 168, 504, 360$ and $660$. Every non-abelian group is either alternating $A_n$, for $n \geq 5$, a member of an infinite family (see above) parametrized by prime-powers and integers, or one of $26$ so-called sporadic simple groups. The largest one is called the Monster and has order $2^{46} · 3^{20} · 5^9 · 7^6 · 11^2 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8×10^{53}.$