# Bounds on the Number of Maximal Subgroups of Finite Groups: Applications
## 2 Lower Bounds for the Number of Maximal Subgroups of a Given Index in a Group  
In this section we will obtain the inequality of Theorem 4. For this, we will need the results of [10] on the existence of conjugacy classes of subgroups of small indices in almost simple groups and some arithmetical properties about the smallest index of a core-free maximal subgroup of an almost simple group. These results are needed to obtain lower bounds for the number of maximal subgroups of a given index in a group \(G\) and for \(\mathcal{V}(G)\). The results depend on some classes of simple groups that we define now. **Notation 1**. : _Let \(\mathfrak{X}\) be the class of simple groups composed of the following groups:_  
1. _The linear groups_ \(\mathrm{PSL}_{3}(q)\)_, where_ \(q=p^{f}>3\) _is a power of a prime_ \(p\) _with_ \(f\) _odd;_
2. _The linear groups_ \(\mathrm{PSL}_{n}(q)\)_, with_ \(q\) _a prime power and_ \(n=5\) _or_ \(n\geq 7\)_;_
3. _The symplectic groups_ \(\mathrm{PSp}_{\mathbf{4}}(2^{f})\)_,_ \(f\geq 2\)_._  
**Notation 2**. : _Let \(\mathfrak{Y}\) be the class of simple groups composed of the following groups:_  
1. _The [PERSON] group_ \(\mathrm{M}_{12}\)_;_
2. _The [PERSON] group_ \(\mathrm{O}^{\prime}\mathrm{N}\)_;_
3. _The [PERSON] group_ \({}^{2}\!F_{\mathbf{4}}(2)^{\prime}\)_;_
4. _The linear groups_ \(\mathrm{PSL}_{2}(7)\cong\mathrm{PSL}_{3}(2)\)_,_ \(\mathrm{PSL}_{2}(9)\cong\mathrm{Alt}(6)\)_,_ \(\mathrm{PSL}_{2}(11)\)_,_ \(\mathrm{PSL}_{3}(3)\)_;_
5. _The linear groups_ \(\mathrm{PSL}_{3}(q_{2}^{q})\)_, with_ \(q_{0}\) _a prime power;_
6. _The linear groups_ \(\mathrm{PSL}_{4}(q)\)_,_ \(q\) _a prime power;_
7. _The linear groups_ \(\mathrm{PSL}_{6}(q)\)_,_ \(q\) _a prime power;_
8. _The unitary group_ \(\mathrm{PSU}_{3}(5)\)_;_
9. _The orthogonal groups_ \(\mathrm{O}^{+}_{8}(q)\)_,_ \(q\) _a prime power;_
10. _The orthogonal groups_ \(\mathrm{O}^{+}_{n}(3)\)_,_ \(n\geq 10\)_;_
11. _The exceptional groups of Lie type_ \(G_{2}(3^{f})\)_,_ \(f\geq 1\)_;_
12. _The exceptional groups of Lie type_ \(F_{\mathbf{4}}(2^{f})\)_,_ \(f\geq 1\)_;_
13. _The exceptional groups of Lie type_ \(E_{6}(q)\)_,_ \(q\) _a prime power._  
If \(R\) is a primitive group, we denote by \(\mathrm{l}^{*}(R)\) the smallest index of a core-free maximal subgroup of \(R\). We say that a maximal subgroup of a simple group \(S\) is _ordinary_ if its conjugacy class in \(S\) coincides with its conjugacy class in \(\mathrm{Aut}(S)\). **Theorem 11** (Theorem A in [10]). : _Let \(S\) be a simple group and let \(R\) be an almost simple group, such that \(\mathrm{Soc}(R)\cong S\). We can assume that \(S\leq R\leq A=\mathrm{Aut}(S)\)._  
1. _If_ \(S\) _belongs to_ \(\mathfrak{X}\)_, then_ \(S\) _has at least two conjugacy classes of maximal subgroups of the smallest index_ \(\mathrm{l}(S)\) _and there exists a number_ \(v_{S}\leq\mathrm{l}(S)^{2}\)_, depending only on_ \(S\)_, such that_ \(R\) _has at least two conjugacy classes of core-free maximal subgroups with index_ \(\mathrm{l}(S)\) _or one conjugacy class of core-free maximal subgroups with index_ \(v_{S}\)_;_
2. _If_ \(S\) _belongs to_ \(\mathfrak{Y}\)_, then_ \(S\) _has at least two conjugacy classes of maximal subgroups of the smallest index_ \(\mathrm{l}(S)\) _and there exists a number_ \(v_{S}\leq\mathrm{l}(S)^{2}\)_, depending only on_ \(S\)_, such that has a conjugacy class of core-free maximal subgroups with index_ \(\mathrm{l}_{S}\)_;_
3. _If_ \(S\) _belongs to_ \(\mathfrak{X}\cup\mathfrak{Y}\)_, then_ \(S\) _has a conjugacy class of ordinary maximal subgroups. In particular, the smallest index of a core-free maximal subgroup of_ \(R\) _is also_ \(\mathrm{l}(S)\)_;_
4. _In all cases,_ \(\mathrm{l}(S)^{2}<|S|\) _and_ \(|\mathrm{Out}\,S|\leq 3\log\mathrm{l}(S)\)_;_
5.