Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C7 |
| prime radical | C22×C7×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C4×C7 | C2 | semidirect |
| 3. C7×#123 | C2 | direct (N × H) |
| 4. C2×C7×C4⋊C4 | C2 | semidirect |
| 5. C2×C7×C22⋊C4 | C2 | semidirect |
| 6. C2×C7×C22⋊C4 | C2 | semidirect |
| 15. C22×C4×C7 | C22 | semidirect |
| 16. C7×C4⋊C4 | C22 | semidirect |
| 17. C7×C22⋊C4 | C22 | semidirect |
| 18. C7×C22⋊C4 | C22 | semidirect |
| 19. C2×#123 | C7 | direct (N × H) |
| 36. C23×C4 | C2×C7 | semidirect |
| 37. #123 | C2×C7 | direct (N × H) |
| 38. C2×C4⋊C4 | C2×C7 | semidirect |
| 39. C2×C22⋊C4 | C2×C7 | semidirect |
| 40. C2×C22⋊C4 | C2×C7 | semidirect |
| 61. C22×C4 | C22×C7 | semidirect |
| 62. C4⋊C4 | C22×C7 | semidirect |
| 63. C22⋊C4 | C22×C7 | semidirect |
| 64. C22⋊C4 | C22×C7 | semidirect |
| 67. C2×C7 | #123 | direct (N × H) |
| 85. C7 | C2×#123 | direct (N × H) |
| 100. C2 | C7×#123 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×D4 | 2 |
| C23×C4×C7 | 2 |
| C7×#123 | 2 |
| C2×C7×C4⋊C4 | 2 |
| C2×C7×C22⋊C4 | 2 |
| C2×#123 | 7 |