Finite Group DatabasebetaFinite Group DatabasebetaOrder 17536.
368 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C137 |
| socle | C22×C137 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C22×C4×C137 ◁ C22×C8×C137 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#135×C137 | C2 | semidirect |
| 2. #508×C137 | C2 | direct (N × H) |
| 3. C2×Q32×C137 | C2 | semidirect |
| 4. C2×SD32×C137 | C2 | semidirect |
| 5. C2×D16×C137 | C2 | semidirect |
| 6. C22×C16×C137 | C2 | semidirect |
| 9. #135×C137 | C22 | semidirect |
| 14. Q32×C137 | C22 | semidirect |
| 15. SD32×C137 | C22 | semidirect |
| 16. D16×C137 | C22 | semidirect |
| 19. C2×C16×C137 | C22 | semidirect |
| 65. C2×C137 | #508 | direct (N × H) |
| 67. C137 | C2×#508 | direct (N × H) |
| 68. C2×#508 | C137 | direct (N × H) |
| 69. C2×#135 | C2×C137 | semidirect |
| 70. #508 | C2×C137 | direct (N × H) |
| 71. C2×Q32 | C2×C137 | semidirect |
| 72. C2×SD32 | C2×C137 | semidirect |
| 73. C2×D16 | C2×C137 | semidirect |
| 74. C22×C16 | C2×C137 | semidirect |
| 77. #135 | C22×C137 | semidirect |
| 82. Q32 | C22×C137 | semidirect |
| 83. SD32 | C22×C137 | semidirect |
| 84. D16 | C22×C137 | semidirect |
| 87. C2×C16 | C22×C137 | semidirect |
| 133. C2 | #508×C137 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C137 | 2 |
| #508×C137 | 2 |
| C2×Q32×C137 | 2 |
| C2×SD32×C137 | 2 |
| C2×D16×C137 | 2 |
| C22×C16×C137 | 2 |
| C2×#508 | 137 |