Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
198 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C22⋊D7 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#121 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | #121 | D7 |
| hypocenter | C7 | #447 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D28 | C2 | semidirect |
| 2. C7×#121 | C2 | semidirect |
| 3. C2×#1141 | C2 | semidirect |
| 4. C22⋊(C7⋊C8) | C2 | semidirect |
| 5. #6383 | C2 | semidirect |
| 8. C2×C7×D4 | C22 | semidirect |
| 13. C7×C4⋊C4 | C22 | semidirect |
| 16. C2×C7⋊C8 | C22 | semidirect |
| 32. D28 | D4 | semidirect |
| 36. #121 | D7 | semidirect |
| 44. C22×C7 | D8 | semidirect |
| 58. C2×D4 | C2×D7 | semidirect |
| 62. C4⋊C4 | C2×D7 | semidirect |
| 86. C7 | #447 | semidirect |
| 96. C22 | #1141 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D28 | 2 |
| C7×#121 | 2 |
| C2×#1141 | 2 |
| C22⋊(C7⋊C8) | 2 |
| #6383 | 2 |
| #447 | 7 |