# 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
The bounds of Theorem 12 for abelian chief factors are attained in groups which are direct products of copies of a primitive group of type \(1\) with non-cyclic socle or in \((D_{2p})^{r-1}\times C_{p}\) and \((D_{2p})^{r-2}\times C_{p}\times C_{p}\). In order to simplify the statement of the main theorem of this section, we propose the following definition. For a group \(G\) and a chief factor \(A\cong S^{k}\) of \(G\) with \(S\) a simple group and \(k\) natural,  
\[f(G,A)= \begin{cases}\frac{\log\lceil(2/3)\tau_{A}(G)\rceil}{2\log \lfloor(A)\rceil}&\text{if }S\in\mathfrak{X},\\ \frac{\log\tau_{A}(G)}{2\log 1(A)}&\text{if }S\in\mathfrak{Y},\\ \frac{\log\bigl{(}\tau_{A}(G)-1+\frac{1}{p}\bigr{)}}{\log p}&\text{if }A\cong C _{p},\,p\in\mathbb{P}\,\text{and }\zeta_{p}(G)\in\{1,2\},\\ \frac{\log\tau_{A}(G)}{\log 1(A)}&\text{otherwise.}\end{cases} \]  
As a consequence of Theorem 4.1, we obtain a lower bound for \(\mathcal{V}(G)\), the inequality of Theorem 4.1. Let \(G\) be a \(d\)-generated group. Then,  
\[\mathcal{V}(G)\geq\max\{d,\max_{A}f(G,A)-2.5\}\]  
where \(A\) runs over the non-Frattini chief factors of \(G\). Let  
\[\mathcal{M}(G)=\max_{n\geq 5}\frac{\log\mathrm{m}_{n}(G)}{\log n}.\]  
By Proposition 1.2 in [2], \(\mathcal{M}(G)-3.5\leq\mathcal{V}(G)\). Let \(B\) be a non-Frattini chief factor of \(G\), such that \(\max_{A}f(G,A)=f(G,B)\). Let \(B\cong T^{k}\) with \(T\) a simple group. Suppose that \(B\in\mathfrak{X}\). Then, \(\mathrm{m}_{x}(G)\geq\mathrm{m}_{x,B}(G)\geq x\lceil(2/3)\mathrm{r}_{B}(G)\rceil\) for \(x\in\{1(B),v_{T}^{k}\}\) and \(v_{T}^{k}\leq 1(B)^{2}\). In this case,  
\[\mathcal{M}(G)\geq\frac{\log\mathrm{m}_{x}(G)}{\log x}\geq 1+\frac{\log \lceil(2/3)\mathrm{r}_{B}(G)\rceil}{2\log 1(B)}=1+f(G,B).\]  
Suppose that \(B\in\mathfrak{Y}\). Then, \(\mathrm{m}_{x}(G)\geq\mathrm{m}_{x,B}(G)\geq x\mathrm{r}_{B}(G)\) for some \(x\leq 1(B)^{2}\). Hence,  
\[\mathcal{M}(G)\geq\frac{\log\mathrm{m}_{x}(G)}{\log x}\geq 1+\frac{\log \mathrm{r}_{B}(G)}{2\log 1(B)}=1+f(G,B).\]  
Suppose that \(B\cong C_{p}\) for a prime \(p\) and that \(G\) has exactly one or two central chief factors isomorphic to \(B\) in a given chief series. Then, \(\mathrm{m}_{p}(G)\geq\mathrm{m}_{p,B}(G)\geq pr_{B}(G)+1-p=p\bigl{(}\mathrm{r} _{B}(G)-1+(1/p)\bigr{)}\) and so  
\[\mathcal{M}(G)\geq\frac{\log\mathrm{m}_{p}(G)}{\log p}\geq 1+\frac{\log \bigl{(}\mathrm{r}_{B}(G)-1+(1/p)\bigr{)}}{\log p}=1+f(G,B).\]  
Suppose that \(B\) does not satisfy any of the previous properties. Then, for \(n=1(B)\), \(\mathrm{m}_{n}(G)\geq\mathrm{m}_{n,B}(G)\geq n\mathrm{r}_{B}(G)\) and so  
\[\mathcal{M}(G)\geq\frac{\log\mathrm{m}_{n}(G)}{\log n}\geq 1+\frac{\log \mathrm{r}_{B}(G)}{\log n}=1+f(G,B).\]  
Consequently,  
\[\mathcal{V}(G)\geq\mathcal{M}(G)-3.5\geq 1+f(G,B)-3.5\geq\max_{A}f(G,A)-2.5.\]The inequality \(\mathcal{V}(G)\geq d\) holds trivially. **Remark 3**. : _Since the smallest index of a maximal subgroup of a non-abelian simple group is at least \(5\), if \(B=T^{k}\) with \(T\in\mathfrak{X}\),_  
\[\mathcal{V}(G) \geq\frac{\log\lceil(2/3)\mathrm{r}_{B}(G)\rceil}{2\log\mathrm{I} (B)}-2.5\geq\frac{\log\mathrm{r}_{B}(G)}{2\log\mathrm{I}(B)}+\frac{\log(2/3)}{2 \log 5}-2.5\] \[\geq\frac{\log\mathrm{r}_{B}(G)}{2\log\mathrm{I}(B)}-2.63.\]  
_Consequently,_  
\[\mathcal{V}(G)\geq\max_{A}\frac{\log\mathrm{r}_{A}(G)}{2\log\mathrm{I}(A)}-2.63\]  
_where \(A\) runs over the set of all non-abelian chief factors in a given chief series of \(G\).