# 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
### Determination of the Constants  
We are now in a position to complete the proof of Theorem 6. Proof of Theorem 6.: The arguments of the proof of Corollary 9.1 in [7] show that the constant \(c_{p}\) appearing there is essentially the same as the constant \(C_{6}\) of Corollary 2. The constant \(c\) such that the number of maximal subgroups of index \(n\) of \(G\) is bounded by \(n^{cd}\max\{1,\operatorname{rk}_{n}(G)\}\) can be taken as \(c=C_{6}+1=375.06\). This is the constant that appears in Theorem 3. We can present the arguments of Sections 6 and 7 of [7] in a different way to improve the bound for the number of irreducible \(G\)-modules of order \(n\). **Theorem 19**. : _Let \(G\) be a \(d\)-generated group. The number of irreducible non-equivalent \(G\)-modules of size \(n\) is at most \(n^{c_{6}d+k_{6}}\max\{1,\operatorname{rk}_{n}(G)\}\), where \(c_{6}=2c_{4}+c_{5}+\log c_{t}+4=183.034\) and \(k_{6}=k_{5}+2=74\)._  
Proof. : As in Propositions 6.1 and 7.1 in [7], let \(T\) be an irreducible \(d\)-generated subgroup of \(\operatorname{GL}_{\mathbb{F}_{p}}(V)\) and let \(H\) be a subgroup of \(T\), such that the representation of \(T\) is induced from a primitive representation of \(H\). Let \(W\) be a primitive \(H\)-module such that \(V=\mathbb{F}_{p}T\oplus_{\mathbb{F}_{p}H}W\). Let \(P\) be the image of \(H\) in \(\operatorname{GL}_{\mathbb{F}_{p}}(W)\) and \(b=\dim_{\mathbb{F}_{p}}W\). Set \(\tilde{K}=H_{T}\). Then, \(T/\tilde{K}\) is a transitive group of degree \(s=m/b\), where \(m=\dim_{\mathbb{F}_{p}}V\), and \(T\) is a subgroup of \(P\wr(T/\tilde{K})\). We divide the \(d\)-generated irreducible subgroups of \(\operatorname{GL}_{\mathbb{F}_{p}}(V)\) into two families, as in Proposition 6.1 in [7]. In the first family, \(|P|\leq|W|^{c_{4}}\). Note that  
\[|T|\leq|P|^{s}|T/\tilde{K}|\leq|W|^{c_{4}s}|T|/|\tilde{K}|\]  
and so \(|\tilde{K}|\leq|W|^{sc_{4}}=|V|^{c_{4}}\), that is, we are in Case 1 in the proof of Proposition 7.1. The argument in the proof of Proposition 6.1 in [7], with the replacement of \(|W|^{c_{5}\operatorname{d}(H)}\) by \(|W|^{c_{5}\operatorname{d}(H)+k_{5}}\) and \(|V|^{c_{5}d}\) by \(|V|^{c_{5}d+k_{5}}\) shows that the number of conjugacy classes of \(d\)-generated irreducible subgroups in the first family is bounded by  
\[n|V|^{c_{5}d+k_{5}}c_{t}^{sd}|V|^{c_{4}d}\leq|V|^{(c_{5}+c_{4}+\log c_{t})d+k_ {5}+1}.\]  
Now, we obtain a bound for \(|\operatorname{Epi}(G,T)|\). The argument in Case 1 of Proposition 7.1 in [7] shows that  
\[|\operatorname{Epi}(G,T)|\leq|T|\max\{1,\tau_{|V|}(G)\}|V|^{d(c_{4}+4)}.\]  
By Lemma 7.2 in [7], we conclude that the number of irreducible \(\mathbb{F}_{p}G\)-modules of size \(n\) obtained starting from a primitive linear group \(P\) with \(|P|\leq|W|^{c_{4}}\) is bounded by  
\[|V|^{(c_{5}+2c_{4}+\log c_{t}+c_{4})d+k_{5}+2}\max\{1,\tau_{|V|}(G)\}.\]  
In the second family, \(|P|>|W|^{c_{4}}\). This corresponds to Case 2 in Proposition 7.1 in [7]. The number of choices of conjugacy classes of primitive \(d\)-generated groups in this family, according to the proof of Proposition 8, family 1, is bounded by \(|W|^{9d+9}\). As in the proof of Proposition 6.1 in [7], we can consider two subfamilies. In the subfamily 2.1, there are at most  
\[|V|^{3d+4}c_{t}^{dm}\]  
conjugacy classes of \(d\)-generated irreducible subgroups, while in the subfamily 2.2, there are at most  
\[|V|^{3d+5}c_{t}^{dm}\]  
conjugacy classes of \(d\)-generated irreducible subgroups. In Case 2 of Proposition 7.1 in [7], we see that  
\[|\operatorname{Epi}(G,T)|\leq\max\{1,\tau_{|V|}(G)\}|T||V|^{4d}.\]  
By Lemma 7.2 in [7], we conclude that the number of irreducible \(\mathbb{F}_{p}G\)-modules of size \(n\) obtained starting from a primitive linear group \(P\) with \(|P|\leq|W|^{c_{4}}\) is bounded by  
\[(|V|^{3d+4}+|V|^{3d+5})c_{t}^{dm}\max\{1,\tau_{|V|}(G)\}|V|^{4d}|V|\] \[\leq|V|^{7d+7}c_{t}^{dm}\max\{1,\tau_{|V|}(G)\}\leq|V|^{(7+\log c _{t})d+7}\max\{1,\tau_{|V|}(G)\},\]  
because \(c_{t}^{dm}=2^{md\log c_{t}}\leq|V|^{d\log c_{t}}\).