  Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be
called $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain
of subgroups $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$
such that either $H_{i-1}$ is normal in $H_{i}$ or $|H_{i} : H_{i-1}|$ is a
some prime $p$ and $p-1$ is not divisible by the $(t+1)$th powers of primes for
every $i = 1,\ldots , n$. In this work, properties of
$\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups and classes of groups with
Sylow $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups are obtained.
