This is the 8311th most frequent Turkish word.
müdürlüklerinin
"Of their directorates" or "of their managements."
Here, 'müdürlüklerinin' serves as a possessive form of 'müdürlük', meaning the directorates belonging to the institutions.
Bu şehirdeki kurumların müdürlüklerinin sorduğu sorular dikkatle incelenmelidir.
The questions asked by the directorates of the institutions in this city should be carefully examined.
In this sentence, 'müdürlüklerinin' indicates the possessive relationship regarding the directorates that belong to the companies.
Şirketlerin müdürlüklerinin sorumlulukları bazen çakışabilir.
The responsibilities of the directorates of the companies may sometimes overlap.
The word 'müdürlüklerinin' here shows that the plans belong to the education directorates, employing a genitive construction.