Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C7 |
| prime radical | C7×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×SD16 | C2 | semidirect |
| 2. C2×C7×M16 | C2 | semidirect |
| 3. C7×C22⋊Q8 | C2 | semidirect |
| 5. C7×#107 | C2 | semidirect |
| 6. C7×#103 | C2 | semidirect |
| 7. C7×#102 | C2 | semidirect |
| 21. #468 | C7 | direct (N × H) |
| 37. C7×C8 | D4 | semidirect |
| 38. C2×SD16 | C2×C7 | semidirect |
| 39. C2×M16 | C2×C7 | semidirect |
| 40. C22⋊Q8 | C2×C7 | semidirect |
| 42. #107 | C2×C7 | semidirect |
| 43. #103 | C2×C7 | semidirect |
| 44. #102 | C2×C7 | semidirect |
| 91. C8 | C7×D4 | semidirect |
| 92. C7 | #468 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×SD16 | 2 |
| C2×C7×M16 | 2 |
| C7×C22⋊Q8 | 2 |
| C7×#121 | 2 |
| C7×#107 | 2 |
| C7×#103 | 2 |
| C7×#102 | 2 |
| #468 | 7 |