The Irony of Ethical vs. Physical Power
Stalin and the gulag prisoners form an opposition of wills. To Stalin, obsessed with power, the acquisition of power by force is the mark of success, but he is unable to take the ethical power of the prisoners. They possess strength of will which cannot be stolen or crushed, so in the end they, as prisoners, are more powerful than even Stalin.
The Irony of Sologdin's Machine
Sologdin has secretly been developing plans for a secret cryptography machine which would allow him to send messages to the outside, but he is discovered. The government officials then force him to complete his machine and use it to help them, making his machine of opposition yet another tool of corruption.
The Irony of Doronin's Punishment
Doronin is punished for revealing the names of other state informers to the prisoners. This punishment is ironic, however, because the prisoners supposedly have no power to do anything with this information. They're immobilized sources, already imprisoned for their traitorous actions. Doronin only accomplishes his own destruction by confiding in the prisoners.
The Irony of the Sharashka
Prisoners in the sharashka are granted special favors in exchange for their expertise. Although most are political prisoners, they are asked to help their government once again in order to benefit themselves. This is a tricky test for the traitor. In order to honor their original defiance, whichever crime of disloyalty which landed them in jail, ironically, they must continue to defy the state.
The Irony of Refusing to Help the Government
Under Volodin's leadership, many of the prisoners who are involved with the sharashka stop helping the state. They prefer the less comfortable prisons than to serve the corruption of the state. This defiance is ironic because they really don't have much to lose by being sent to different gulags. They are still just as much prisoners elsewhere as among the sharashka. The choice becomes a clear test of loyalty.