# Bounds on the Number of Maximal Subgroups of Finite Groups: Applications
## 3 Upper Bounds for the Number of Maximal Subgroups of a Given Index in a Group  
### Bounds for the Number of Socles of Primitive Groups of Type 2  
In Lemma 2.3 in [17], it is shown that the symmetric group \(\operatorname{Sym}(n)\) has at most \(\operatorname{O}(n)\) isomorphism classes of non-abelian simple subgroups. We will adapt the proof in this paper in order to give precise values to the constants associated with this bound. The number \(g(n)\) of isomorphism classes of non-abelian simple subgroups of the symmetric group \(\operatorname{Sym}(n)\) for \(n\geq 5\) satisfies the inequalities \(g(n)\leq g_{1}(n)\leq g_{2}(n)\), where  
\[g_{1}(n)=\frac{9}{2}n+22+\frac{7}{2}(\log n)(\sqrt{n}-1)-\frac{7}{2}\sqrt{n}+ 7\log n+7(\ln\log n)\log n\]  
and  
\[g_{2}(n)=4.8869n+1088.\]  
Proof. : The number of subgroups of alternating type or sporadic type is at most \((n-4)+26\). By all results mentioned in the proof of Theorem 3.1, the minimum degree of a permutation representation of a simple group \(T=X_{k}(q)\) of Lie type of rank \(k\) over \(\mathbb{F}_{q}\) is \(1(T)\geq q^{k}\), with the only exception of \(\operatorname{PSL}_{2}(9)\cong\operatorname{Alt}(6)\), already considered. Since \(1(T)\leq n\), we obtain that \(k\leq\log n\). For each \(k\), the number of possibilities for odd \(q\) is at most \(\lfloor(n^{1/k}-1)/2\rfloor\) (namely \(3^{k},5^{k},7^{k},\ldots,\lfloor n^{1/k}\rfloor^{k}\)) and for even \(q\) it is at most \(\lfloor\log n^{1/k}\rfloor\) (namely \(2^{k},4^{k},8^{k},\ldots,\)\((2^{\lfloor\log n^{1/k}\rfloor})^{k}\)). Once \(q\) and \(k\) are given, there are at most \(7\) possibilities for the simple group \(T\) (up to isomorphism). Recall that the harmonic sum \(H_{n}=\sum_{j=1}^{n}\frac{1}{j}\) satisfies the inequality \(H_{n}\leq\ln n+1\), as we can check by using the integral test. Hence, the number \(g(n)\) of non-abelian simple subgroups \(T\leq\operatorname{Sym}(n)\) is bounded by  
\[g(n) \leq n+22+7\sum_{k=1}^{\lfloor\log n\rfloor}\left(\frac{n^{1/k}-1 }{2}+\frac{1}{k}\log n\right)\] \[\leq n+22+\frac{7n}{2}+\frac{7}{2}\sum_{k=2}^{\lfloor\log n \rfloor}n^{1/k}-\frac{7}{2}\lfloor\log n\rfloor+7(1+\ln\lfloor\log n\rfloor )\log n\] \[\leq\frac{9}{2}n+22+\frac{7}{2}\sum_{k=2}^{\lfloor\log n\rfloor} n^{1/k}-\frac{7}{2}\lfloor\log n\rfloor+7(1+\ln\lfloor\log n\rfloor )\log n\] \[\leq\frac{9}{2}n+22+\frac{7}{2}(\lfloor\log n\rfloor-1)\sqrt{n}- \frac{7}{2}\lfloor\log n\rfloor+7(1+\ln\lfloor\log n\rfloor)\log n\] \[=\frac{9}{2}n+22+\frac{7}{2}\lfloor\log n\rfloor(\sqrt{n}-1)- \frac{7}{2}\sqrt{n}+7\log n+7(\ln\lfloor\log n\rfloor)\log n\] \[\leq\frac{9}{2}n+22+\frac{7}{2}(\log n)(\sqrt{n}-1)-\frac{7}{2} \sqrt{n}+7\log n+7(\ln\log n)\log n\] \[=g_{1}(n).\]  
Since the second derivative of the function \(g_{1}\) is negative, the function \(g_{1}\) is concave. Therefore, the graph of \(g_{1}\) lies below its tangent on any point, for example, the tangent on \(n=4096\), that is, \(g_{1}(n)\leq 4.8869n+1088\) as we can compute with Maxima [18]. It is also shown in Lemma 2.3 in [17] that the number of almost simple subgroups of \(\operatorname{Sym}(n)\) up to isomorphism is at most \(\operatorname{O}(n\log^{6}n)\). We specify the bound in that lemma by adapting its arguments and show that the exponent \(6\) can be reduced to \(3\) by using Theorem 3.1. We need the following bound for the number of subgroups of the outer automorphism group of a non-abelian simple group. [Theorem B in [10]] The number of subgroups of the outer automorphism group of a non-abelian simple group \(S\) is bounded by \(\log^{3}1(S)\). **Lemma 5**. : _The number of isomorphism classes of almost simple subgroups of \(\operatorname{Sym}(n)\) is at most \(g_{1}(n)\log^{3}n\leq g_{2}(n)\log^{3}n\), where \(g_{1}(n)\) and \(g_{2}(n)\) are the functions defined in Lemma 3._  
Proof. : Let \(H\leq\operatorname{Sym}(n)\) be an almost simple subgroup and let \(T=\operatorname{Soc}(H)\). There are \(g(n)\) possibilities for \(T\).