By Jeremy Gaunt and Lefteris Papadimas

ATHENS (Reuters) - Greece's infrastructure cannot handle the thousands of migrants landing on its shores from places like war-torn Syria and Afghanistan and needs European Union help, Prime Minister Alexis Tsipras said on Friday.

"Now is the time to see if the EU is the EU of solidarity or an EU that has everyone trying to protect their borders," he said after a meeting with ministers dealing with the influx.

The United Nations refugee agency earlier called on Greece to take control of the "total chaos" on Mediterranean islands, where thousands of migrants have landed.

European Union member states must also do more to share the burden of Greece, where 50,000 people arrived in July alone, said Vincent Cochetel, UNHCR director for Europe, after visiting the Greek islands of Lesbos, Kos and Chios.

"The immigrant flow to Greece is beyond of what our state infrastructure can handle," Tsipras said. "We have significant problems to face it and that’s why we have asked help from EU."

Greece, along with Italy, has been on front line of a huge wave of migrants seeking safety and a better future in Europe.

But Greece's economy is falling into recession again after having only just recovered from six years of depression brought on by its debt crisis.

It is currently negotiating with EU and the International Monetary Fund for a third bailout of as much as 86 billion euros for stave of financial collapse.

With that background, it cannot handle the increasing pressure from thousands of penniless refugees from war and poverty.

Within just one hour on Thursday, Reuters journalists on the island of Lesbos saw two boatloads of refugees landing on the Greek territory, one carrying roughly 40 Syrians, the other the same number of Afghans.

The European Union has sought to share the burden of the refugees across it countries, but the response has been mixed.

Britain has said it will not participate, but is currently struggling with its own crisis as thousands of migrants seek to enter via the Channel Tunnel.

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Hungary is also preparing to build a fence along its border, where migrants from the east seek to enter.

Hopes were fading, meanwhile, of finding more survivors from a shipwreck in which 200 migrants are feared drowned near Sicily.

(Editing by Tom Heneghan)