Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
310 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| socle | C22×C7 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41 | C2 | semidirect |
| 2. C7×C4⋊Q8 | C2 | semidirect |
| 3. C7×#127 | C2 | semidirect |
| 4. C7×#124 | C2 | semidirect |
| 5. C7×#121 | C2 | semidirect |
| 6. C4×C7×D4 | C2 | semidirect |
| 7. C2×C42×C7 | C2 | semidirect |
| 21. #532 | C7 | direct (N × H) |
| 30. C2×C4×C7 | D4 | semidirect |
| 36. C2×#41 | C2×C7 | semidirect |
| 37. C4⋊Q8 | C2×C7 | semidirect |
| 38. #127 | C2×C7 | semidirect |
| 39. #124 | C2×C7 | semidirect |
| 40. #121 | C2×C7 | semidirect |
| 41. C4×D4 | C2×C7 | semidirect |
| 42. C2×C42 | C2×C7 | semidirect |
| 52. C4×C7 | #41 | semidirect |
| 78. C2×C4 | C7×D4 | semidirect |
| 84. C7 | #532 | direct (N × H) |
| 94. C4 | C7×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41 | 2 |
| C7×C4⋊Q8 | 2 |
| C7×#127 | 2 |
| C7×#124 | 2 |
| C7×#121 | 2 |
| C4×C7×D4 | 2 |
| C2×C42×C7 | 2 |
| #532 | 7 |