Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D7 |
| commutator, cosocle | C2×C7 | C25 |
| Frattini | C2 | C24×D7 |
| Fitting | C22×C7×D4 | C2 |
| socle | C22×C7 | C24 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×D4 | C2 | semidirect |
| 2. C22×C22⋊D7 | C2 | semidirect |
| 3. #6548 | C2 | direct (N × H) |
| 4. C2×#1159 | C2 | semidirect |
| 5. C2×D4×D7 | C2 | semidirect |
| 6. C2×#1157 | C2 | semidirect |
| 7. C24×C7 | C22 | semidirect |
| 8. C23×D7 | C22 | semidirect |
| 9. C2×C7×D4 | C22 | semidirect |
| 10. C2×C7×D4 | C22 | semidirect |
| 11. C22×C4×C7 | C22 | semidirect |
| 12. C2×C22⋊D7 | C22 | semidirect |
| 13. C2×C22⋊D7 | C22 | semidirect |
| 14. C2×C22⋊D7 | C22 | semidirect |
| 15. C22×Dic7 | C22 | semidirect |
| 16. #1159 | C22 | semidirect |
| 17. D4×D7 | C22 | semidirect |
| 18. #1157 | C22 | semidirect |
| 19. C2×D28 | C22 | semidirect |
| 20. C2×C4×D7 | C22 | semidirect |
| 21. C2×Dic14 | C22 | semidirect |
| 26. C23×C7 | C23 | semidirect |
| 29. C22×D7 | C23 | semidirect |
| 30. C7×D4 | C23 | semidirect |
| 32. C2×C4×C7 | C23 | semidirect |
| 33. C22⋊D7 | C23 | semidirect |
| 34. C22⋊D7 | C23 | semidirect |
| 36. C2×Dic7 | C23 | semidirect |
| 37. D28 | C23 | semidirect |
| 38. C4×D7 | C23 | semidirect |
| 39. Dic14 | C23 | semidirect |
| 45. C22×D4 | D7 | semidirect |
| 58. C24 | C2×D7 | semidirect |
| 67. C2×D4 | C2×D7 | semidirect |
| 68. C2×D4 | C2×D7 | semidirect |
| 71. C22×C4 | C2×D7 | semidirect |
| 74. C2×C7 | H(5,𝔽2) | semidirect |
| 83. C23 | C22×D7 | semidirect |
| 88. D4 | C22×D7 | semidirect |
| 93. C2×C4 | C22×D7 | semidirect |
| 94. C7 | C2×H(5,𝔽2) | semidirect |
| 109. C2 | #6548 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×D4 | 2 |
| C22×C22⋊D7 | 2 |
| #6548 | 2 |
| C2×#1159 | 2 |
| C2×D4×D7 | 2 |
| C2×#1157 | 2 |
| C2×H(5,𝔽2) | 7 |