The site, which resembled Facebook but appeared unfinished, went live on Sunday then went offline again a day later and its linked Twitter account was shut down.
A message in English posted on its front page said it had temporarily suspended operations to "protect the information and details of its members and their safety".
Islamic State, which had declared its own caliphate in territory captured in Iraq and Syria, has relied heavily on social media to spread messages and publicise military victories and its beheadings of prisoners.
But mainstream social media companies have raced to remove links to footage of killings -- and many governments have called for stronger restrictions on militants' use of the Internet.
It was unclear who created -- a name based on an English transliteration of the Arabic word for caliphate 'khelafa' -- or how many members it attracted.
The message posted on the front page said it was independent and not sponsored by Islamic State.
But it went on to say the militant group was expanding across the whole world "by Allah's permission". The original website showed a map of the world dotted with Islamic State's trademark Arabic insignia.
Data online showed it had been built using Socialkit, a program that lets users produce do-it-yourself social networks. The site was registered with webservices company on March 3 and cited its home address as IS-controlled Mosul in Iraq but its home country as Egypt with an apparently false phone number there.
Islamic State supporters held a debate in a separate web forum on whether platforms like 5elafabook could be trusted or whether they could be used by IS's enemies to gain intelligence, according to the militancy watchdog SITE Intelligence Group.