Finite Group DatabasebetaFinite Group DatabasebetaOrder 27392.
186 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C107 | D16 |
| commutator | C16 | C22×C4×C107 |
| Frattini, cosocle | C2×C16 | C23×C107 |
| socle | C2×C107 | C4×D16 |
| prime radical | C107×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C107 ◁ C2×C8×C107 ◁ C4×C8×C107 ◁ C4×C16×C107 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C107×#2178 | C2 | semidirect |
| 3. C107×#1425 | C2 | semidirect |
| 4. C107×#1417 | C2 | semidirect |
| 5. C4×C32×C107 | C2 | semidirect |
| 8. Q64×C107 | C4 | semidirect |
| 9. SD64×C107 | C4 | semidirect |
| 10. D32×C107 | C4 | semidirect |
| 59. #13655 | C107 | direct (N × H) |
| 64. #2178 | C2×C107 | semidirect |
| 65. #1425 | C2×C107 | semidirect |
| 66. #1417 | C2×C107 | semidirect |
| 67. C4×C32 | C2×C107 | semidirect |
| 68. C107 | #13655 | direct (N × H) |
| 71. Q64 | C4×C107 | semidirect |
| 72. SD64 | C4×C107 | semidirect |
| 73. D32 | C4×C107 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C107×#2262 | 2 |
| C107×#2178 | 2 |
| C107×#1425 | 2 |
| C107×#1417 | 2 |
| C4×C32×C107 | 2 |
| #13655 | 107 |