Question: does the milky way belong to a galaxy cluster

Answer: Yes, The Virgo Supercluster (Virgo SC) or the Local Supercluster (LSC or LS) is a mass concentration of galaxies containing the Virgo Cluster and Local Group, which in turn contains the Milky Way and Andromeda galaxies. At least 100 galaxy groups and clusters are located within its diameter of 33 megaparsecs (110 million light-years). The Virgo SC is one of about 10 million superclusters in the observable universe and is in the Pisces--Cetus Supercluster Complex, a galaxy filament.