Cleary, 20, confessed to launching a string of attacks on major institutions in Britain and the US with fellow hacker Jake Davis, 19.
The two targeted sites including the National Health Service, News International, Sony, Nintendo, Arizona State police, film studio 20th Century Fox and other sites in a series of so-called distributed denial of service (DDoS) attacks, where websites are flooded with traffic to make them crash.
Cleary, of Wickford, Essex, and Davis, of Lerwick, Shetland, plotted to carry out the attacks with other unknown members of the internet groups Anonymous, Internet Feds and LulzSec.
Other websites targeted by the pair were Westboro Baptist Church, Bethesda, Eve Online, HBGary, HBGary Federal, PBS Inc and Infragard.
Cleary also confessed to four separate charges, including hacking into US Air Force Agency computers, based at the Pentagon.
Both men appeared in the dock at Southwark crown court on Monday to enter guilty pleas to a series of charges brought against them.