Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×D7 |
| commutator, socle | C2×C7 | C23×C4 |
| Fitting | C7×#131 | C2 |
| prime radical | D7×#41 | C2 |
| cosocle | C4×C7 | C24 |
| hypercenter | #131 | D7 |
| hypocenter | C7 | #568 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#131 | C2 | semidirect |
| 3. #6511 | C2 | semidirect |
| 4. #6470 | C2 | semidirect |
| 5. D7×M16 | C2 | semidirect |
| 6. #6464 | C2 | semidirect |
| 7. C2×#1131 | C2 | semidirect |
| 15. C7×M16 | C22 | semidirect |
| 16. C2×C7×C8 | C22 | semidirect |
| 17. #1136 | C22 | semidirect |
| 18. C2×C7⋊C8 | C22 | semidirect |
| 21. #1131 | C22 | semidirect |
| 22. #1131 | C22 | semidirect |
| 23. C8×D7 | C22 | semidirect |
| 45. #131 | D7 | semidirect |
| 68. M16 | C2×D7 | semidirect |
| 71. C2×C8 | C2×D7 | semidirect |
| 92. C7 | #568 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7×#41 | 2 |
| C7×#131 | 2 |
| #6511 | 2 |
| #6470 | 2 |
| D7×M16 | 2 |
| #6464 | 2 |
| C2×#1131 | 2 |
| #568 | 7 |