Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D7 |
| commutator, cosocle | C2×C7 | C25 |
| Frattini | C2 | C24×D7 |
| Fitting | C22×C7×D4 | C2 |
| socle | C23×C7 | C23 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C23×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D7 | C2 | semidirect |
| 2. C22×C7×D4 | C2 | semidirect |
| 3. C22×C22⋊D7 | C2 | semidirect |
| 4. C2×D4×D7 | C2 | direct (N × H) |
| 5. C22×D28 | C2 | semidirect |
| 6. C22×C4×D7 | C2 | semidirect |
| 7. C24×C7 | C22 | semidirect |
| 10. C23×D7 | C22 | semidirect |
| 11. C23×D7 | C22 | semidirect |
| 12. C2×C7×D4 | C22 | semidirect |
| 13. C22×C4×C7 | C22 | semidirect |
| 14. C2×C22⋊D7 | C22 | semidirect |
| 15. C22×Dic7 | C22 | semidirect |
| 16. D4×D7 | C22 | direct (N × H) |
| 17. C2×D28 | C22 | semidirect |
| 18. C2×C4×D7 | C22 | semidirect |
| 22. C23×C7 | C23 | semidirect |
| 25. C22×D7 | D4 | direct (N × H) |
| 27. C22×D7 | C23 | semidirect |
| 28. C22×D7 | C23 | semidirect |
| 29. C7×D4 | C23 | semidirect |
| 30. C2×C4×C7 | C23 | semidirect |
| 31. C22⋊D7 | C23 | semidirect |
| 32. C2×Dic7 | C23 | semidirect |
| 33. D28 | C23 | semidirect |
| 34. C4×D7 | C23 | semidirect |
| 39. C22×D4 | D7 | direct (N × H) |
| 42. C22×C7 | C2×D4 | semidirect |
| 44. C22×C7 | C24 | semidirect |
| 47. C2×D7 | C2×D4 | direct (N × H) |
| 49. C2×D7 | C24 | semidirect |
| 50. C4×C7 | C24 | semidirect |
| 51. Dic7 | C24 | semidirect |
| 56. C24 | C2×D7 | semidirect |
| 60. C2×D4 | C2×D7 | direct (N × H) |
| 63. C22×C4 | C2×D7 | semidirect |
| 65. C2×C7 | C22×D4 | semidirect |
| 68. D7 | C22×D4 | direct (N × H) |
| 77. C23 | C22×D7 | semidirect |
| 80. D4 | C22×D7 | direct (N × H) |
| 83. C2×C4 | C22×D7 | semidirect |
| 84. C7 | C23×D4 | semidirect |
| 91. C22 | D4×D7 | direct (N × H) |
| 93. C22 | C23×D7 | semidirect |
| 95. C4 | C23×D7 | semidirect |
| 99. C2 | C2×D4×D7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D7 | 2 |
| C22×C7×D4 | 2 |
| C22×C22⋊D7 | 2 |
| C2×D4×D7 | 2 |
| C22×D28 | 2 |
| C22×C4×D7 | 2 |
| C23×D4 | 7 |