Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
170 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 82 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×C271 | D4 |
| commutator | C4 | C22×C4×C271 |
| Frattini, cosocle | C2×C4 | C23×C271 |
| socle | C22×C271 | C22⋊C4 |
| prime radical | C2×#41×C271 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C271 ◁ C22×C4×C271 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M16×C271 | C2 | semidirect |
| 3. C2×C42×C271 | C2 | semidirect |
| 4. #104×C271 | C2 | direct (N × H) |
| 7. #41×C271 | C4 | semidirect |
| 9. C2×Q8×C271 | C4 | semidirect |
| 11. C2×D4×C271 | C4 | semidirect |
| 16. M16×C271 | C22 | semidirect |
| 19. C42×C271 | C22 | semidirect |
| 23. Q8×C271 | C2×C4 | semidirect |
| 27. D4×C271 | C2×C4 | semidirect |
| 55. C2×C271 | #104 | direct (N × H) |
| 57. C271 | C2×#104 | direct (N × H) |
| 58. C2×#104 | C271 | direct (N × H) |
| 60. C2×M16 | C2×C271 | semidirect |
| 61. C2×C42 | C2×C271 | semidirect |
| 62. #104 | C2×C271 | direct (N × H) |
| 65. #41 | C4×C271 | semidirect |
| 67. C2×Q8 | C4×C271 | semidirect |
| 69. C2×D4 | C4×C271 | semidirect |
| 74. M16 | C22×C271 | semidirect |
| 77. C42 | C22×C271 | semidirect |
| 81. Q8 | C2×C4×C271 | semidirect |
| 85. D4 | C2×C4×C271 | semidirect |
| 113. C2 | #104×C271 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C271 | 2 |
| C2×M16×C271 | 2 |
| C2×C42×C271 | 2 |
| #104×C271 | 2 |
| C2×#104 | 271 |