Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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 | C22×D7 |
| commutator, cosocle | C2×C7 | C25 |
| Frattini | C2 | C24×D7 |
| Fitting | C2×C7×#41 | C2 |
| socle | C22×C7 | C24 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | C22×#41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41 | C2 | semidirect |
| 2. D7×#41 | C2 | direct (N × H) |
| 3. C2×#1161 | C2 | semidirect |
| 4. C2×Q8×D7 | C2 | semidirect |
| 5. C2×#1159 | C2 | semidirect |
| 6. C2×D4×D7 | C2 | semidirect |
| 7. C2×#1157 | C2 | semidirect |
| 8. C22×C4×D7 | C2 | semidirect |
| 10. C7×#41 | C22 | semidirect |
| 11. C2×C7×Q8 | C22 | semidirect |
| 12. C2×C7×D4 | C22 | semidirect |
| 13. C22×C4×C7 | C22 | semidirect |
| 14. C2×C22⋊D7 | C22 | semidirect |
| 15. C22×Dic7 | C22 | semidirect |
| 16. #1161 | C22 | semidirect |
| 17. Q8×D7 | C22 | semidirect |
| 18. #1159 | C22 | semidirect |
| 19. D4×D7 | C22 | semidirect |
| 20. #1157 | C22 | semidirect |
| 21. C2×D28 | C22 | semidirect |
| 24. C2×C4×D7 | C22 | semidirect |
| 25. C2×C4×D7 | C22 | semidirect |
| 26. C2×C4×D7 | C22 | semidirect |
| 27. C2×Dic14 | C22 | semidirect |
| 34. C7×Q8 | C23 | semidirect |
| 35. C7×D4 | C23 | semidirect |
| 38. C2×C4×C7 | C23 | semidirect |
| 39. C22⋊D7 | C23 | semidirect |
| 42. C2×Dic7 | C23 | semidirect |
| 43. D28 | C23 | semidirect |
| 46. C4×D7 | C23 | semidirect |
| 47. C4×D7 | C23 | semidirect |
| 48. Dic14 | C23 | semidirect |
| 53. C2×#41 | D7 | direct (N × H) |
| 62. C2×D7 | #41 | direct (N × H) |
| 73. #41 | C2×D7 | direct (N × H) |
| 76. C2×Q8 | C2×D7 | semidirect |
| 80. C2×D4 | C2×D7 | semidirect |
| 87. C22×C4 | C2×D7 | semidirect |
| 89. C2×C7 | C2×#41 | semidirect |
| 92. D7 | C2×#41 | direct (N × H) |
| 99. Q8 | C22×D7 | semidirect |
| 102. D4 | C22×D7 | semidirect |
| 112. C2×C4 | C22×D7 | semidirect |
| 113. C7 | C22×#41 | semidirect |
| 129. C2 | D7×#41 | 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 |
| D7×#41 | 2 |
| C2×#1161 | 2 |
| C2×Q8×D7 | 2 |
| C2×#1159 | 2 |
| C2×D4×D7 | 2 |
| C2×#1157 | 2 |
| C22×C4×D7 | 2 |
| C22×#41 | 7 |