I understand that they compromised with the Hungarians but how were they able to run their empire with all the various groups involved?
nationalist conflict only became a serious problem following the Napoleonic wars when Napoleon was spreading the idea of...well...nationalism. Emperor Franz Joseph II, who served from 1848 to 1916 was able to keep most of populous from revolted by delegating autonomy little by little to the different nationalities. The most important of these was the Ausgleich which gave greater autonomy to Hungary and effectively created a dual Monarch and changed the name to Austro-Hungary. From this Hungary was given its own parliament. The Hungarians was the only group that the Emperor could placate. The Czech Question created the most trouble in the empire primarily because of how outspoken the particularists were. The Franz Joseph did attempt to improve the situation with the other nationalities in in 1860 with the October Diploma and later in 1861 the February Patent. Both created a more diverse parliament that allowed non-germans(Czechs and Hungarians) to participate. The slavs were quite happy with the terms of both Patents but both Patents eventually failed due to resistance from the Magyars. A few years a compromise was reached in 1867 when Hungary was given even more autonomy: parliament, fiscal control, administrative rights over Hungary and Croatia. During the same year, a new constitution was prepared that "recognized equality of all nationalities, the right of each to preserve its own language and culture".[1] This was good in paper but the slavs, Czechs, and HUngarians were not equally represented in the Reichsrat. Austrian defeats against Prussia caused some panic in the Austrian government which then led to Franz Joseph's 1871 Cabinet to grant numerous privileges to the Poles which then guarenteed their loyalty. Additionally, the political maneuvering of MP's like Count Taaffe would give these ethnic groups more autonomy and representation in the Reichsrat in exchance for political support. Since 1890, the Austrian government was only able to maintain control because of its close economic relations with Germany which had quelled the middle-class' nationalistic tendencies
Reich, Norman. "The Age of Nationalism and Reform, 1850-1890" 2nd Edition, 1977. 206