According to New Advent thirty:
Catholic Encyclopedia
Just like After the Peloponnesian War the Thirty Tyrants replaced Democracy with Oligarchic rule in Athens. But that is just an historical coincidence.
Wikipedia link has this:
Antipope
An antipope is a person who makes a widely accepted claim to be the lawful Pope, in opposition to the Pope recognized by the Roman Catholic Church. Antipopes are typically those supported by a fairly significant faction of cardinals. Persons who claim to be the Pope but have few followers, such as the modern Sedevacantist antipopes, are not generally counted as antipopes (though they technically are), and therefore are ignored for regnal numbering.
In several cases it is hard to tell which was, in fact, the lawful Pope and which was the antipope.
Antipope Felix V, the last historically significant Antipope.In its list of the Popes, the Holy See's annual directory, Annuario Pontificio, attaches to the name of Pope Leo VIII (963-965) the following note: "At this point, as again in the mid-eleventh century, we come across elections in which problems of harmonizing historical criteria and those of theology and canon law make it impossible to decide clearly which side possessed the legitimacy whose factual existence guarantees the unbroken lawful succession of the Successors of Saint Peter. The uncertainty that in some cases results has made it advisable to abandon the assignation of successive numbers in the list of the Popes." In all cases it is clear that whichever was the Pope, the other was an antipope, since the claim of each was widely accepted."
"The period when antipopes were most numerous was during the struggles between the Popes and the Holy Roman Emperors of the 11th and 12th centuries. The emperors frequently imposed their own nominees, in order to further their cause. (The popes, likewise, sometimes sponsored rival imperial claimants in Germany in order to overcome a particular emperor.)
The Great Western Schism, which, on the grounds of the allegedly invalid election of Pope Urban VI, began in 1378 with the election of Clement VII, who took up residence in Avignon, France, led to two, and eventually three, rival lines of claimants to papacy: the Roman line, the Avignon line, and the Pisan line. "
++++++++++++++++++++++++++++++++++++++++++++++++++ ++++
This whole concept of apostolic succession and unbroken line seems suspect.
Also it should be mentioned that the papacy was not only influenced and held captive by political intrigues it was also the pawn of powerful Italian families, the Medici, a Florentine family, being a prime example.
And then there is this interesting female character that had some special influence on the Papacy:
Marzoia
"At the age of fifteen, she became the mistress of Theophylact's cousin Pope Sergius III, whom she knew when he was bishop of Portus. The two had a son, John.
In 909, when she was pregnant by Sergius, Marozia married Alberic I, duke of Spoleto. Besides legitimating John, Alberic had another son by Marozia, Alberic II, born in 911 or 912. By the time Alberic I was killed at Orte in 924, the Roman landowners had won complete victory over the traditional bureaucracy represented by the papal curia and Rome was virtually under secular control, the historic nadir of the papacy.
In order to counter the influence of Pope John X (whom the hostile chronicler Liutprand of Cremona alleges was another of her lovers) she subsequently married his opposer Guy of Tuscany, who loved his beautiful wife as much as he loved power. Together they attacked Rome, arrested Pope John X in the Lateran, and jailed him in the Castel Sant'Angelo. Guy had him smothered with a pillow in 928, and Marozia seized power in Rome in a coup d'état. The following popes, Leo VI and Stephen VIII, were both her puppets. In 931 she even managed to impose her son as pontiff, under the name of John XI. John was only twenty-one at the time. "
God works in mysterious ways.
-Zeno, The Antipope