CAIRO (Reuters) - Security forces must do more to protect Egypt's Christian minority in the turmoil following the military overthrow of Islamist president Mohamed Mursi, rights groups said on Tuesday, citing the mob killing of four near the southern city of Luxor.

Coptic Christians account for about a tenth of Egypt's 84 million people. They have suffered discrimination for decades, but communal tensions and attacks rose sharply under Mursi, who was elected president a year ago following the fall of strongman Hosni Mubarak in 2011.

The army deposed Mursi on July 3, unleashing violent street clashes and exposing deep fissures in the Arab world's most populous nation.

Two days later, a mob beat to death four Christians and destroyed at least 24 Christian-owned properties after a Muslim was found dead in the village of Naga Hassan, near Luxor, New York-based Human Rights Watch (HRW) said.

Amnesty International said security forces in the area "stood by and failed to intervene" while more than 100 Christian homes were attacked, scores of them looted or torched.

Both rights groups quoted witnesses as saying they had begged police and local officials to intervene, but to no avail.

Amnesty said Luxor prosecutors were investigating the attack and at least 18 men had been detained. A military spokesman was not contactable for comment on the criticism of the security forces. Tuesday was a public holiday in Egypt.

HRW said it had registered at least six attacks on Christians across Egypt since the ouster of Mursi, the country's first democratically-elected president.

Many Christians feared the ascendance to power of Mursi and his Islamist Muslim Brotherhood. When he was ousted, Coptic Pope Tawadros II gave his public backing, standing with other leaders beside armed forces chief Abdel Fattah al-Sisi when he announced Mursi's removal.

HRW said that in only one of the attacks against Christians had police intervened effectively.

"The Egyptian government should make ending sectarian violence a priority, or risk letting this deadly problem spiral out of control," said Nadim Houry, acting Middle East director at HRW.

Amnesty said security forces in Hagba Nassan had evacuated some women and children trapped inside a house surrounded by an angry mob, but left six men behind, "apparently following demands from the crowd that the men remain." Four of the men were later stabbed or beaten to death, it said.

"The attack went on for 18 hours," Amnesty quoted local priest, Father Barsilious, as saying. "And there was not a door on which I did not knock: police, army, local leaders, the Central Security Forces, the Governate. Nothing was done."

While in power, Mursi's government said it was committed to protecting minorities, but the Muslim Brotherhood strongly criticized the Coptic pope for backing the president's overthrow and anti-Christian sentiment has been on display at pro-Mursi rallies. The army-backed authorities who replaced him have said little about attacks on Christians.

Site of Egypt's greatest Pharaonic temples, Luxor made headlines in 1997, when Islamist militants killed 62 people, 58 of them foreign tourists in a temple in the Valley of the Queens.

(Writing by Matt Robinson; Editing by Paul Taylor)