A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.

Let $G$ be a finite group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup _{i\in I}\unicode[STIX]{x1D70E}_{i}$ and $\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}=\emptyset$ for all $i\neq j$. We say that $G$ is $\unicode[STIX]{x1D70E}$-primary if $G$ is a $\unicode[STIX]{x1D70E}_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $\unicode[STIX]{x1D70E}$-subnormal in$G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D70E}$-primary for all $i=1,\ldots ,n$; modular in$G$ if the following conditions hold: (i) $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ for all $X\leq G,Z\leq G$ such that $X\leq Z$ and (ii) $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ for all $Y\leq G,Z\leq G$ such that $A\leq Z$; and $\unicode[STIX]{x1D70E}$-quasinormal in$G$ if $A$ is modular and $\unicode[STIX]{x1D70E}$-subnormal in $G$. We study $\unicode[STIX]{x1D70E}$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\unicode[STIX]{x1D70E}$-quasinormal in $G$, then every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ is $\unicode[STIX]{x1D70E}$-central in$G$, that is, the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $\unicode[STIX]{x1D70E}$-primary.

Let $G$ be a group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup $A$ of $G$ is $\unicode[STIX]{x1D70E}$ -subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\unicode[STIX]{x1D70E}_{j}$ -group for some $j=j(i)$ for $i=1,\ldots ,m$ , and it is modular in $G$ if $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when $X\leq Z\leq G$ and $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when $Y\leq G$ and $A\leq Z\leq G$ . The group $G$ is called $\unicode[STIX]{x1D70E}$ -soluble if every chief factor $H/K$ of $G$ is a finite $\unicode[STIX]{x1D70E}_{i}$ -group for some $i=i(H/K)$ . In this paper, we describe finite $\unicode[STIX]{x1D70E}$ -soluble groups in which every $\unicode[STIX]{x1D70E}$ -subnormal subgroup is modular.