The 4 were arrested during overnight raids in the Jatrabari and Khilkhet areas of the capital, Dhaka.
"Yes, the IS coordinator and the three others have been arrested," a police spokesman told PTI.
The IS coordinator has been identified as Shakhawatul Kabir who is also the regional commander of the militant group in Bangladesh. The other 3 men arrested are Md Anwar Hossain, Md Rabiul Islam and Md Nazrul Alam.
They have confessed that they are members of Islamic State, said Masudur Rahman, deputy commissioner of Detective Branch.
Kabir's brother-in-law Shamim and Baten's brother Sayem live in Pakistan and they are members of IS, bdnews quoted Detective Branch Inspector Md Abu Zafar as saying.
The two had given Kabir the responsibility to work as IS's chief coordinator in Bangladesh, Zafar said.
Kabir has been frequently travelling to India, Pakistan and other countries, the report said.
The objective of these militants was to kill important persons in Bangladesh and catch media attention, Rahman said, adding that a case has been registered against the 4.
Police also seized laptops and jihadi leaflets from their possessions, the report said.
A Britisher was arrested in September last year for allegedly attempting to recruit people for IS. He was held on suspicion as he had spent time in Syria and previously belonged to militant group al-Nusra Front.
The arrests came amid fears expressed by several security analysts that the current spate of political unrest in Bangladesh could allow Islamists militants to regroup and spread their networks taking the advantage of diverted focus of the concerned agencies and media.
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