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The generalized Fitting height $h^*(G)$ of a finite group $G$ is the leastnumber $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) =1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalizedFitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime,$1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in whichfor $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for$i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabeliansimple groups. Then $h=\lambda_p(G)$ is called the non-$p$-soluble length of agroup $G$. We proved that if a finite group $G$ is a mutually permutableproduct of of subgroups $A$ and $B$ then $\max\{h^*(A), h^*(B)\}\leq h^*(G)\leq\max\{h^*(A), h^*(B)\}+1$ and $\max\{\lambda_p(A), \lambda_p(B)\}=\lambda_p(G)$. Also we introduced and studied the non-Frattini length.

On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups