Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D28 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×#131 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | #131 | D7 |
| hypocenter | C7 | #577 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6554 | C2 | semidirect |
| 2. #6553 | C2 | semidirect |
| 3. C7×#131 | C2 | semidirect |
| 4. #6472 | C2 | semidirect |
| 5. #6471 | C2 | semidirect |
| 6. #6467 | C2 | semidirect |
| 7. C2×#1132 | C2 | semidirect |
| 14. #1157 | C22 | semidirect |
| 16. C2×Dic14 | C22 | semidirect |
| 17. C7×M16 | C22 | semidirect |
| 18. C2×C7×C8 | C22 | semidirect |
| 19. Dic28 | C22 | semidirect |
| 20. D56 | C22 | semidirect |
| 21. #1132 | C22 | semidirect |
| 22. #1132 | C22 | semidirect |
| 23. #1132 | C22 | semidirect |
| 48. #131 | D7 | semidirect |
| 71. M16 | C2×D7 | semidirect |
| 72. C2×C8 | C2×D7 | semidirect |
| 94. C7 | #577 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6554 | 2 |
| #6553 | 2 |
| C7×#131 | 2 |
| #6472 | 2 |
| #6471 | 2 |
| #6467 | 2 |
| C2×#1132 | 2 |
| #577 | 7 |