7-limit note groups of finite index

Finite index means there is an integer n such that 1/n of the 7-limit
intervals belong to the subgroup. Below I took integer generators up
to 50 and looked at index 2, 3, 4, and 5. Most interesting are the
index two subgroups, it seems to me, and those I have done an ordering
for.

Index 2

{3, 4, 5, 7}, {4, 5, 6, 7}, {2, 5, 7, 9}, {3, 4, 7, 10},
{4, 6, 7, 10}, {3, 4, 5, 14}, {4, 5, 6, 14}, {3, 4, 10, 14},
{4, 6, 10, 14}, {2, 7, 9, 15}, {2, 5, 9, 21}, {2, 9, 15, 21},
{2, 3, 7, 25}, {2, 3, 25, 35}, {2, 3, 5, 49}

Index 3

{8, 10, 12, 28}, {3, 8, 10, 14}, {2, 7, 27, 45}, {3, 7, 8, 10},
{8, 12, 14, 20}, {7, 8, 10, 12}, {8, 10, 12, 14}, {3, 7, 8, 20},
{3, 5, 7, 8}, {5, 6, 8, 14}, {2, 15, 21, 27}, {3, 8, 14, 20},
{7, 8, 12, 20}, {2, 5, 21, 27}, {5, 8, 12, 28}, {6, 7, 8, 20},
{2, 21, 27, 45}, {5, 7, 8, 12}, {3, 5, 8, 14}, {2, 5, 7, 27},
{6, 8, 10, 14}, {8, 12, 20, 28}, {3, 8, 20, 28}, {6, 8, 10, 28},
{5, 8, 12, 14}, {6, 8, 20, 28}, {2, 7, 15, 27}, {3, 5, 8, 28},
{5, 6, 8, 28}, {6, 7, 8, 10}, {5, 6, 7, 8}, {2, 15, 27, 63},
{3, 8, 10, 28}, {6, 8, 14, 20}

Index 4

{4, 9, 10, 14}, {5, 12, 16, 28}, {12, 16, 20, 56}, {4, 5, 7, 18},
{2, 45, 63, 81}, {4, 9, 21, 30}, {10, 14, 16, 24}, {5, 6, 16, 56},
{5, 7, 12, 16}, {2, 9, 25, 35}, {3, 14, 16, 20}, {4, 5, 18, 42},
{10, 12, 16, 56}, {4, 7, 9, 10}, {3, 4, 5, 49}, {6, 14, 16, 40},
{5, 6, 14, 16}, {4, 5, 9, 42}, {3, 5, 7, 16}, {4, 6, 35, 50},
{6, 16, 20, 56}, {12, 16, 40, 56}, {12, 14, 16, 40}, {7, 12, 16, 40},
{4, 9, 30, 42}, {4, 7, 10, 18}, {4, 14, 18, 30}, {4, 18, 21, 30},
{2, 21, 45, 81}, {4, 9, 10, 42}, {4, 10, 14, 18}, {4, 9, 10, 21},
{5, 6, 16, 28}, {3, 10, 14, 16}, {6, 10, 16, 56}, {4, 6, 14, 50},
{4, 15, 18, 42}, {3, 5, 16, 28}, {4, 7, 18, 30}, {10, 16, 24, 56},
{3, 7, 16, 20}, {4, 7, 9, 15}, {6, 7, 16, 20}, {14, 16, 20, 24},
{3, 4, 35, 50}, {4, 6, 7, 25}, {4, 9, 15, 21}, {12, 16, 28, 40},
{3, 16, 20, 28}, {2, 9, 21, 25}, {6, 16, 20, 28}, {16, 20, 24, 28},
{14, 16, 24, 40}, {3, 5, 14, 16}, {4, 5, 9, 14}, {5, 12, 14, 16},
{4, 10, 18, 21}, {4, 10, 18, 42}, {4, 9, 14, 15}, {3, 14, 16, 40},
{2, 3, 25, 49}, {10, 12, 14, 16}, {5, 7, 16, 24}, {6, 7, 16, 40},
{3, 4, 14, 50}, {4, 9, 15, 42}, {4, 6, 25, 35}, {4, 7, 9, 30},
{4, 7, 15, 18}, {16, 24, 28, 40}, {5, 16, 24, 28}, {7, 16, 24, 40},
{6, 10, 16, 28}, {6, 10, 14, 16}, {3, 16, 28, 40}, {4, 18, 30, 42},
{3, 10, 16, 28}, {5, 14, 16, 24}, {5, 6, 7, 16}, {3, 4, 25, 35},
{3, 4, 7, 25}, {12, 16, 20, 28}, {3, 7, 16, 40}, {10, 12, 16, 28},
{4, 5, 6, 49}, {4, 6, 14, 25}, {2, 7, 45, 81}, {7, 10, 12, 16},
{4, 9, 14, 30}, {2, 5, 9, 49}, {3, 10, 16, 56}, {6, 16, 40, 56},
{4, 5, 18, 21}, {4, 15, 18, 21}, {4, 6, 7, 50}, {12, 14, 16, 20},
{4, 5, 9, 21}, {7, 10, 16, 24}, {7, 12, 16, 20}, {4, 5, 14, 18},
{4, 6, 10, 49}, {4, 5, 7, 9}, {2, 9, 15, 49}, {6, 7, 10, 16},
{5, 12, 16, 56}, {7, 16, 20, 24}, {10, 16, 24, 28}, {3, 4, 14, 25},
{2, 7, 9, 25}, {3, 7, 10, 16}, {6, 14, 16, 20}, {3, 4, 7, 50},
{3, 4, 10, 49}, {6, 16, 28, 40}, {4, 14, 15, 18}

Index 5

{6, 10, 28, 32}, {7, 12, 32, 40}, {5, 14, 32, 48}, {3, 28, 32, 40}, {3,
10, 28, 32}, {14, 20, 32, 48}, {6, 28, 32, 40}, {6, 10, 14, 32}, {5,
6, 7, 32}
, {12, 14, 32, 80}, {28, 32, 40, 48}, {12, 20, 28, 32}, {14, 32, 40,
48}, {12,
32, 56, 80}, {6, 10, 32, 112}, {5, 28, 32, 48}, {24, 28, 32, 80}, {6,
14, 32,
40}, {7, 10, 24, 32}, {7, 10, 12, 32}, {5, 24, 28, 32}, {24, 32, 40,
56}, {3,
20, 28, 32}, {10, 14, 32, 48}, {12, 14, 32, 40}, {10, 12, 32, 112},
{7, 20, 32
, 48}, {5, 24, 32, 56}, {10, 28, 32, 48}, {10, 12, 28, 32}, {20, 28,
32, 48},
{3, 14, 32, 80}, {7, 12, 20, 32}, {6, 20, 32, 112}, {3, 5, 7, 32},
{10, 32, 48
, 112}, {10, 14, 24, 32}, {7, 20, 24, 32}, {6, 20, 28, 32}, {7, 10,
32, 48}, {
14, 20, 24, 32}, {12, 20, 32, 112}, {5, 7, 32, 48}, {6, 7, 10, 32},
{3, 7, 10,
32}, {5, 7, 24, 32}, {10, 24, 32, 112}, {3, 5, 28, 32}, {20, 24, 32,
56}, {5,
6, 32, 112}, {5, 14, 24, 32}, {7, 32, 40, 48}, {3, 14, 32, 40}, {7,
12, 32, 80
}, {6, 14, 20, 32}, {7, 24, 32, 40}, {6, 28, 32, 80}, {5, 12, 28, 32},
{12, 32
, 40, 112}, {24, 32, 56, 80}, {3, 7, 32, 40}, {12, 20, 32, 56}, {12,
32, 40,
56}, {5, 6, 14, 32}, {5, 7, 12, 32}, {10, 12, 14, 32}, {12, 14, 20,
32}, {6,
32, 80, 112}, {20, 24, 28, 32}, {20, 32, 48, 56}, {12, 32, 80, 112},
{5, 12,
14, 32}, {5, 12, 32, 112}, {7, 24, 32, 80}, {14, 24, 32, 40}, {24, 28,
32, 40}
, {3, 20, 32, 56}, {5, 6, 28, 32}, {5, 12, 32, 56}, {3, 5, 14, 32},
{3, 14, 20
, 32}, {6, 7, 32, 80}, {6, 20, 32, 56}, {3, 10, 14, 32}, {6, 14, 32,
80}, {14,
24, 32, 80}, {10, 24, 32, 56}, {10, 12, 32, 56}, {10, 24, 28, 32},
{12, 28, 32
, 80}, {3, 10, 32, 112}, {6, 32, 40, 112}, {3, 7, 20, 32}, {14, 32,
48, 80}, {
6, 7, 32, 40}, {6, 7, 20, 32}, {12, 28, 32, 40}}