Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C7 |
| prime radical | C7×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#137 | C2 | semidirect |
| 3. C2×C7×SD16 | C2 | semidirect |
| 4. C7×#131 | C2 | semidirect |
| 5. C7×#108 | C2 | semidirect |
| 6. C7×#101 | C2 | semidirect |
| 7. C7×#100 | C2 | semidirect |
| 23. #471 | C7 | direct (N × H) |
| 40. #137 | C2×C7 | semidirect |
| 42. C2×SD16 | C2×C7 | semidirect |
| 43. #131 | C2×C7 | semidirect |
| 44. #108 | C2×C7 | semidirect |
| 45. #101 | C2×C7 | semidirect |
| 46. #100 | C2×C7 | semidirect |
| 90. C7 | #471 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#137 | 2 |
| C7×AGL(1,ℤ/8) | 2 |
| C2×C7×SD16 | 2 |
| C7×#131 | 2 |
| C7×#108 | 2 |
| C7×#101 | 2 |
| C7×#100 | 2 |
| #471 | 7 |