sandwich classification theorem

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theorem if $ll$ splits into a disjoint union of sandwiches $lat(f,n_g(f))$ over all subgroups $f$ such that the normal closure of $d$ in $f$ coincides with $f$. here $n_g(f)$ denotes the normalizer of $f$ in $g$. a similar notion of sandwich classification is introduced for the lattice $ll'$. if $d$ is perfect, i.,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for $ll$ and $ll'$ are equivalent. we also show how to find basic subroup $f$ of sandwiches for $ll'$ and review sandwich classification theorems in algebraic groups over rings.