PGL(3,4): |P| = 60480, k = 21, |Q| = 2880, |phibar(Q)| = 60, |Q_t| by o(t): {2: [192], 3: [144], 5: [240]}, M = 240, |P|/M = 252
  [lattice] processed 1/8 classes, closures 7, 0s
  [lattice] processed 11/82 classes, closures 2357, 9s
  [lattice] processed 21/83 classes, closures 4639, 22s
  [lattice] processed 31/91 classes, closures 6098, 28s
  [lattice] processed 41/97 classes, closures 8082, 40s
  [lattice] processed 51/97 classes, closures 9175, 47s
  [lattice] processed 61/97 classes, closures 9980, 54s
  [lattice] processed 71/97 classes, closures 11143, 64s
  [lattice] processed 81/98 classes, closures 12111, 70s
  [lattice] processed 91/99 classes, closures 12960, 76s
  [lattice] processed 100/100 classes, closures 13593, 80s
classes of subgroups: 100; subgroups: 81220; time 80s
normal subgroups (orders): [1, 20160, 60480]
order histogram of classes: {1: 1, 2: 1, 3: 3, 4: 4, 5: 1, 6: 2, 7: 1, 8: 5, 9: 3, 10: 1, 12: 6, 15: 1, 16: 7, 18: 2, 21: 4, 24: 5, 27: 1, 30: 1, 32: 3, 36: 2, 48: 8, 54: 1, 60: 3, 63: 1, 64: 1, 72: 1, 80: 2, 96: 4, 144: 2, 160: 2, 168: 1, 180: 1, 192: 4, 216: 1, 240: 2, 288: 2, 360: 1, 480: 2, 576: 1, 960: 2, 2880: 2, 20160: 1, 60480: 1}

classes with |S| >= M = 240:  (order, conjugates, [(orbit length, |S_d|, induced order on the lines through d, good at d)], good)
   60480      1  [(21, 2880, 60, False)]  not good
   20160      1  [(21, 960, 60, False)]  not good
    2880     21  [(20, 144, 12, False), (1, 2880, 60, False)]  not good
    2880     21  [(16, 180, 60, False), (5, 576, 12, False)]  not good
     960     21  [(20, 48, 12, False), (1, 960, 60, False)]  not good
     960     21  [(5, 192, 12, False), (16, 60, 60, False)]  not good
     576    105  [(4, 144, 12, False), (16, 36, 12, False), (1, 576, 12, False)]  not good
     480    126  [(16, 30, 10, False), (5, 96, 12, False)]  not good
     480    126  [(20, 24, 12, False), (1, 480, 10, False)]  not good
     360    168  [(15, 24, 6, False), (6, 60, 60, False)]  not good
     288    210  [(12, 24, 12, False), (1, 288, 6, False), (8, 36, 12, False)]  not good
     288    210  [(16, 18, 6, False), (3, 96, 12, False), (2, 144, 12, False)]  not good
     240    126  [(16, 15, 5, True), (5, 48, 12, False)]  GOOD
     240    126  [(20, 12, 12, False), (1, 240, 5, True)]  GOOD

RESULT PGL(3,4): M = 240; M_G = largest order of a good subgroup over all 100 classes = 240; classes with |S| > M: 12, of which good: 0; good classes of order M: 2; |P|/M = 252, |P|/M_G = 252; strict inequality: False
number of subgroups of order >= M that are good at the base point, by order: {240: 102}
total time 80s
