Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C4×C7×Q8 | C2 |
| hypercenter | C4×Q8 | D7 |
| hypocenter | C7 | #548 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1161 | C2 | semidirect |
| 2. C2×Q8×D7 | C2 | semidirect |
| 3. C4×C7×Q8 | C2 | semidirect |
| 4. #6460 | C2 | semidirect |
| 5. #6458 | C2 | semidirect |
| 6. #6438 | C2 | semidirect |
| 7. C4×D28 | C2 | semidirect |
| 25. C7×Q8 | D4 | semidirect |
| 41. C4×Q8 | D7 | semidirect |
| 49. C2×D7 | #41 | semidirect |
| 77. Q8 | D28 | semidirect |
| 87. C7 | #548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1161 | 2 |
| C2×Q8×D7 | 2 |
| C4×C7×Q8 | 2 |
| #6460 | 2 |
| #6458 | 2 |
| #6438 | 2 |
| C4×D28 | 2 |
| #548 | 7 |