Siman 136
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א בְּשַׁבָּת וְיוֹם טוֹב וְיוֹם הַכִּפּוּרִים, קוֹרִים אַחַר לֵּוִי תַּלְמִידֵי חֲכָמִים הַמְמֻנִּין עַל הַצִּבּוּר, וְאַחֲרֵיהֶם תַּלְמִידֵי חֲכָמִים הָרְאוּיִם לְמַנּוֹתָם פַּרְנָסִים עַל הַצִּבּוּר (שֶׁשּׁוֹאֲלִים אוֹתוֹ דְּבַר הֲלָכָה בְּכָל מָקוֹם וְאוֹמֵר), וְאַחַר כָּךְ בְּנֵי תַּלְמִידֵי חֲכָמִים שֶׁאֲבוֹתֵיהֶם מְמֻנִּים עַל הַצִּבּוּר, וְאַחַר כָּךְ רָאשֵׁי כְּנֵסִיּוֹת וְכָל הָעָם.
1 Who Gets Called Up To The Torah On Shabbos. Containing 1 Se'if: On Shabbos, Yom Tov, and Yom Kippur, after the Levi we call up Torah scholars who are appointed [as leaders] over the community; and after them, Torah scholars who are fit to be appointed over the community [who can be asked a Halachic (Jewish Law) question in any area of Torah and can respond]; and after them, children of Torah scholars whose fathers are appointed over the community; and after them, the heads of synagogues, and the rest of the people.