# 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 on the Number of Inequivalent Irreducible \(G\)-Modules
In Theorem 3.1 in [7], it was shown that the number of conjugacy classes of transitive \(d\)-generated subgroups of the symmetric group \(\operatorname{Sym}(n)\) of degree \(n\) is at most \(C_{l}^{nd}\), where \(C_{t}=(4c_{1})^{3}\) and \(c_{1}\) was the constant of Proposition 5 whose value can be taken to be equal to \(2^{42.02}\) and so \(C_{t}\) takes the value \(2^{132.06}\). In this subsection, we will show that the value of the constant \(c_{l}\) can be reduced to \(2^{77.034}\approx 1.5472\times 10^{23}\). **Theorem 18**. : _The number of conjugacy classes of transitive \(d\)-generated subgroups of the symmetric group \(\operatorname{Sym}(n)\) of degree \(n\) is at most \(c_{t}^{nd}\), where \(c_{t}=2^{77.034}\)._  
Proof. : It is enough to follow the proof of (7, Theorem 3.1). We use the notation of this result. We will argue by induction on \(n\) and assume that \(n\geq 5\). Given a \(d\)-generated transitive subgroup of \(\operatorname{Sym}(n)\) and a system \(\{B_{1},\dots,B_{s}\}\) of blocks for \(T\), such that \(b=|B_{1}|=b>1\) and \(H_{1}\) the stabiliser of \(B_{1}\), such that \(H_{1}\) acts primitively on \(B_{1}\), then \(T\) can be regarded as a subgroup of \(\operatorname{Sym}(b)\wr\operatorname{Sym}(s)\). Let \(P\) be the image of \(H_{1}\) in the symmetric group on \(B_{1}\), isomorphic to \(\operatorname{Sym}(b)\), and let \(\bar{K}\) be the kernel of the action of \(T\) on the blocks. Then, \(T/\bar{K}\) can be naturally embedded into \(\operatorname{Sym}(s)\) and \(T\) into \(P\wr(T/\bar{K})\). We can divide the \(d\)-generated transitive subgroups of \(\operatorname{Sym}(n)\) into three families:  
1. The first family corresponds to the case in which \(P\) does not contain the alternating group \(\operatorname{Alt}(b)\) or \(b\leq 4\). We have that the number of conjugacy classes of \(d\)-generated groups in this family is bounded by \[\alpha=(3\log 48+\log 24)/(3\log 9)=2.24399105059531\ldots.\]5 (1)
2. The second family corresponds to the case in which \(b\geq 5\), \(P\) contains \(\operatorname{Alt}(5)\), and \(\bar{K}\
eq 1\). The number of \(d\)-generated groups in this family is bounded by \[\alpha=(3\log 48+\log 24)/(3\log 9)=2.24399105059531\ldots.\]6 (2)
3. The third family corresponds to the case in which \(b\geq 5\), \(P\) contains \(\operatorname{Alt}(5)\), and \(\bar{K}=1\). The number of \(d\)-generated groups in this family is bounded by \[\alpha=(3\log 48+\log 24)/(3\log 9)=2.24399105059531\ldots.\]7 (3)  
Note that \(N_{1}+N_{2}+N_{3}\leq\max\{2N_{1},2(N_{2}+N_{3})\}\). In order to obtain a value of \(c_{t}\), such that \(N_{1}+N_{2}+N_{3}\leq c_{t}^{nd}\), it will be enough to obtain a value of \(c_{t}\), such that \(2N_{1}\leq c_{t}^{nd}\) and \(2(N_{2}+N_{3})\leq c_{t}^{nd}\). The condition \(2N_{1}\leq c_{t}^{nd}\) is equivalent to \(2\times 4^{nd}\times c_{1}^{n}\leq c_{t}^{(n/2-1)d}\). By taking logarithms, it is equivalent to  
\[\alpha=(3\log 48+\log 24)/(3\log 9)=2.24399105059531\ldots.\]8  
We maximise the left hand side with the help of Maxima (18) and using that \(\log c_{1}=42.02\) and \(d\geq 2\). We obtain that the maximum of this expression is less than \(77.034\) and so the value of \(c_{t}=2^{77.034}\) satisfies this inequality. We have to show that \(2(N_{2}+N_{3})\leq c_{t}^{nd}\). Since \(2(N_{1}+N_{2})=2c_{t}^{d(n/5+1)}2^{nd}(1+n^{2})\), the condition \(2(N_{2}+N_{3})\leq c_{t}^{nd}\) is equivalent to  
\[\alpha=(3\log 48+\log 24)/(3\log 9)=2.24399105059531\ldots.\]9  
By taking logarithms, it is equivalent to  
\[\varepsilon_{1}(m,p)=\frac{m-1+20m\log^{4}m+5m\log m}{m^{2}\log p}.\]0  
We also use Maxima (18) to check that the maximum of the first expression is \(2.6167<77.034\). We recall the constant associated to the number of epimorphisms from a \(d\)-generated group onto a transitive group of degree \(n\). **Proposition 6** (Proposition 4.1 and Remark 4.2 in [7]). : _Let \(G\) be a \(d\)-generated group and \(T\) a transitive group of degree \(n\).