Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
170 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 82 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 | C22⋊C4 |
| prime radical | C2×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×M16 | C2 | semidirect |
| 3. C2×C42×C7 | C2 | semidirect |
| 4. C7×#104 | C2 | direct (N × H) |
| 7. C7×#41 | C4 | semidirect |
| 9. C2×C7×Q8 | C4 | semidirect |
| 11. C2×C7×D4 | C4 | semidirect |
| 16. C7×M16 | C22 | semidirect |
| 19. C42×C7 | C22 | semidirect |
| 20. C2×#104 | C7 | direct (N × H) |
| 24. C7×Q8 | C2×C4 | semidirect |
| 28. C7×D4 | C2×C4 | semidirect |
| 42. C2×M16 | C2×C7 | semidirect |
| 43. C2×C42 | C2×C7 | semidirect |
| 44. #104 | C2×C7 | direct (N × H) |
| 59. #41 | C4×C7 | semidirect |
| 61. C2×Q8 | C4×C7 | semidirect |
| 63. C2×D4 | C4×C7 | semidirect |
| 68. M16 | C22×C7 | semidirect |
| 71. C42 | C22×C7 | semidirect |
| 75. C2×C7 | #104 | direct (N × H) |
| 80. Q8 | C2×C4×C7 | semidirect |
| 84. D4 | C2×C4×C7 | semidirect |
| 97. C7 | C2×#104 | direct (N × H) |
| 113. C2 | C7×#104 | 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 |
| C2×C7×M16 | 2 |
| C2×C42×C7 | 2 |
| C7×#104 | 2 |
| C2×#104 | 7 |