Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
292 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×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×#123 | C2 | semidirect |
| 4. C7×C22⋊Q8 | C2 | semidirect |
| 5. C7×#121 | C2 | semidirect |
| 6. C4×C7×D4 | C2 | semidirect |
| 7. C4×C7×D4 | C2 | semidirect |
| 8. C2×C7×C4⋊C4 | C2 | semidirect |
| 26. #547 | C7 | direct (N × H) |
| 32. C7×D4 | D4 | semidirect |
| 42. C2×#41 | C2×C7 | semidirect |
| 43. C4⋊Q8 | C2×C7 | semidirect |
| 44. #123 | C2×C7 | semidirect |
| 45. C22⋊Q8 | C2×C7 | semidirect |
| 46. #121 | C2×C7 | semidirect |
| 47. C4×D4 | C2×C7 | semidirect |
| 48. C4×D4 | C2×C7 | semidirect |
| 49. C2×C4⋊C4 | C2×C7 | semidirect |
| 62. C4×C7 | #41 | semidirect |
| 90. D4 | C7×D4 | semidirect |
| 100. C7 | #547 | direct (N × H) |
| 113. 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×#123 | 2 |
| C7×C22⋊Q8 | 2 |
| C7×#121 | 2 |
| C4×C7×D4 | 2 |
| C2×C7×C4⋊C4 | 2 |
| #547 | 7 |