Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 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 | C4 | C24×C7 |
| prime radical | C2×C7×#41 | C2 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. C22×C7×Q8 | C2 | semidirect |
| 3. C7×#137 | C2 | direct (N × H) |
| 4. C2×C7×Q16 | C2 | semidirect |
| 5. C2×C7×SD16 | C2 | semidirect |
| 6. C2×C7×M16 | C2 | semidirect |
| 13. C2×C7×Q8 | C22 | semidirect |
| 19. C7×Q16 | C22 | semidirect |
| 20. C7×SD16 | C22 | semidirect |
| 21. C7×M16 | C22 | semidirect |
| 23. C2×#137 | C7 | direct (N × H) |
| 44. C22×Q8 | C2×C7 | semidirect |
| 45. #137 | C2×C7 | direct (N × H) |
| 46. C2×Q16 | C2×C7 | semidirect |
| 47. C2×SD16 | C2×C7 | semidirect |
| 48. C2×M16 | C2×C7 | semidirect |
| 65. C2×Q8 | C22×C7 | semidirect |
| 71. Q16 | C22×C7 | semidirect |
| 72. SD16 | C22×C7 | semidirect |
| 73. M16 | C22×C7 | semidirect |
| 77. C2×C7 | #137 | direct (N × H) |
| 98. C7 | C2×#137 | direct (N × H) |
| 111. C2 | C7×#137 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41 | 2 |
| C22×C7×Q8 | 2 |
| C7×#137 | 2 |
| C2×C7×Q16 | 2 |
| C2×C7×SD16 | 2 |
| C2×C7×M16 | 2 |
| C2×#137 | 7 |