Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
122 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D4 |
| commutator | C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C7 |
| socle | C22×C7 | C2×D4 |
| prime radical | C2×C7×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C7×Q8 | C2 | semidirect |
| 4. C7×#106 | C2 | semidirect |
| 5. C7×#103 | C2 | semidirect |
| 7. C4×C7×C8 | C2 | semidirect |
| 11. C7×SD16 | C4 | direct (N × H) |
| 20. C4×SD16 | C7 | direct (N × H) |
| 23. C7×Q8 | C2×C4 | semidirect |
| 34. C7×C8 | C2×C4 | semidirect |
| 36. C4×Q8 | C2×C7 | semidirect |
| 38. #106 | C2×C7 | semidirect |
| 39. #103 | C2×C7 | semidirect |
| 41. C4×C8 | C2×C7 | semidirect |
| 50. C4×C7 | SD16 | direct (N × H) |
| 56. SD16 | C4×C7 | direct (N × H) |
| 71. Q8 | C2×C4×C7 | semidirect |
| 82. C8 | C2×C4×C7 | semidirect |
| 83. C7 | C4×SD16 | direct (N × H) |
| 92. C4 | C7×SD16 | 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 |
| C4×C7×Q8 | 2 |
| C4×C7×D4 | 2 |
| C7×#106 | 2 |
| C7×#103 | 2 |
| C7×#102 | 2 |
| C4×C7×C8 | 2 |
| C4×SD16 | 7 |