Who are considered “most Haredim”? Which groups are they apart of?
One can generally divide Ashkenazic Haredim into 2 basic groups: Hassidim and misnagdim. Among the Hassidim there many subdivisions of Hassidic groups and courts (e.g. Lubavitch, Breslav, Ger, Belz, Vizhnitz etc.) which often break up into more sub-groups, usually when more than one son is seen by the rebbe's Hassidim as being worthy of being the next rebbe. The mitnagdim are best defined as: non-hassidic, but the difference between the two is gradually disappearing (aside from the garb), where we see the Hassidim today learning Torah like the mitnagdim, and the mitnagdim singing, dancing, adulating their teachers and wearing identical clothing, like the hassidim. Just about all of the above are included under the "Agudat Yisrael" party and their board of rabbis ("Mo'etzes Gedolei HaTorah"), except for Satmar, Neturei Karta and a handful of other vehement anti-Zionists, who even after the Holocaust, naively do not see the need and advantages of having a Jewish State and Jewish army. They make up a very small percentage of the haredim. The Sephardic haredim are much less extreme, and are generally united under the "Shas" party, where a significant percentage serve in the army, go to college, work, pay taxes, and are generally more involved with the general population (and don't cut off personal ties with their non-religious or just traditional relatives).