Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 228 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 |
| 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×#125 | C2 | semidirect |
| 3. C7×C22⋊Q8 | C2 | semidirect |
| 4. C7×C22⋊Q8 | C2 | semidirect |
| 5. C4×C7×Q8 | C2 | semidirect |
| 8. C2×C7×C4⋊C4 | C2 | semidirect |
| 25. #554 | C7 | direct (N × H) |
| 29. C7×D4 | Q8 | semidirect |
| 39. C4⋊Q8 | C2×C7 | semidirect |
| 40. #125 | C2×C7 | semidirect |
| 41. C22⋊Q8 | C2×C7 | semidirect |
| 42. C22⋊Q8 | C2×C7 | semidirect |
| 43. C4×Q8 | C2×C7 | semidirect |
| 46. C2×C4⋊C4 | C2×C7 | semidirect |
| 81. D4 | C7×Q8 | semidirect |
| 91. C7 | #554 | 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×#125 | 2 |
| C7×C22⋊Q8 | 2 |
| C4×C7×Q8 | 2 |
| C4×C7×D4 | 2 |
| C2×C7×C4⋊C4 | 2 |
| #554 | 7 |