Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
5872 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D83 |
| commutator, cosocle | C2×C83 | C26 |
| Frattini | C2 | C25×D83 |
| Fitting | C2×2−1+4×C83 | C2 |
| socle | C22×C83 | C25 |
| hypercenter | C2×2−1+4 | D83 |
| hypocenter | C83 | C22×2−1+4 |
| LCS | G ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14581840 | C2 | semidirect |
| 2. C2×#14581839 | C2 | semidirect |
| 3. 2−1+4×D83 | C2 | direct (N × H) |
| 4. C2×#41×D83 | C2 | semidirect |
| 5. C22×Q8×D83 | C2 | semidirect |
| 6. C2×2−1+4×C83 | C2 | semidirect |
| 7. #14581840 | C22 | semidirect |
| 8. #14581839 | C22 | semidirect |
| 9. C2×#13050591 | C22 | semidirect |
| 10. C2×#13050590 | C22 | semidirect |
| 11. C2×#13050589 | C22 | semidirect |
| 12. #41×D83 | C22 | semidirect |
| 13. C22×Dic166 | C22 | semidirect |
| 15. C2×Q8×D83 | C22 | semidirect |
| 16. C2×Q8×D83 | C22 | semidirect |
| 19. 2−1+4×C83 | C22 | semidirect |
| 20. C2×#41×C83 | C22 | semidirect |
| 21. C22×Q8×C83 | C22 | semidirect |
| 23. #13050591 | C23 | semidirect |
| 24. #13050590 | C23 | semidirect |
| 25. #13050589 | C23 | semidirect |
| 27. C2×Dic166 | C23 | semidirect |
| 29. Q8×D83 | C23 | semidirect |
| 37. #41×C83 | C23 | semidirect |
| 39. C2×Q8×C83 | C23 | semidirect |
| 63. C2×D83 | 2−1+4 | direct (N × H) |
| 69. C2×C83 | C2×2−1+4 | semidirect |
| 72. D83 | C2×2−1+4 | direct (N × H) |
| 74. C83 | C22×2−1+4 | semidirect |
| 78. C2×2−1+4 | D83 | direct (N × H) |
| 83. 2−1+4 | C2×D83 | direct (N × H) |
| 88. C2×#41 | C2×D83 | semidirect |
| 93. C22×Q8 | C2×D83 | semidirect |
| 100. #41 | C22×D83 | semidirect |
| 106. C2×Q8 | C22×D83 | semidirect |
| 145. C2 | 2−1+4×D83 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#14581840 | 2 |
| C2×#14581839 | 2 |
| 2−1+4×D83 | 2 |
| C2×#41×D83 | 2 |
| C22×Q8×D83 | 2 |
| C2×2−1+4×C83 | 2 |
| C22×2−1+4 | 83 |