Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
212 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 150 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 |
| prime radical | C2×C7×D4 | C22 |
| 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. C7×C4⋊Q8 | C2 | semidirect |
| 2. C7×#126 | C2 | semidirect |
| 4. C7×C22⋊Q8 | C2 | semidirect |
| 5. C4×C7×Q8 | C2 | semidirect |
| 6. C4×C7×Q8 | C2 | semidirect |
| 8. C7×#117 | C2 | semidirect |
| 25. #556 | C7 | direct (N × H) |
| 38. C4⋊Q8 | C2×C7 | semidirect |
| 39. #126 | C2×C7 | semidirect |
| 41. C22⋊Q8 | C2×C7 | semidirect |
| 42. C4×Q8 | C2×C7 | semidirect |
| 43. C4×Q8 | C2×C7 | semidirect |
| 45. #117 | C2×C7 | semidirect |
| 89. C7 | #556 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C4⋊Q8 | 2 |
| C7×#126 | 2 |
| C7×#124 | 2 |
| C7×C22⋊Q8 | 2 |
| C4×C7×Q8 | 2 |
| C4×C7×D4 | 2 |
| C7×#117 | 2 |
| #556 | 7 |