Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
266 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 111 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 | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×#41 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | C2×#135 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C22×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. C2×#1157 | C2 | semidirect |
| 3. #6513 | C2 | direct (N × H) |
| 4. C2×#1144 | C2 | semidirect |
| 5. C2×#1143 | C2 | semidirect |
| 6. C2×#1142 | C2 | semidirect |
| 7. C2×#1141 | C2 | semidirect |
| 8. C22×C7⋊C8 | C2 | semidirect |
| 10. C7×#41 | C22 | semidirect |
| 18. #1157 | C22 | semidirect |
| 22. #1144 | C22 | semidirect |
| 23. #1143 | C22 | semidirect |
| 24. #1142 | C22 | semidirect |
| 25. #1141 | C22 | semidirect |
| 28. C2×C7⋊C8 | C22 | semidirect |
| 54. C2×#41 | D7 | semidirect |
| 76. #41 | C2×D7 | semidirect |
| 95. C2×C7 | #135 | semidirect |
| 121. C7 | C2×#135 | semidirect |
| 137. C2 | #6513 | 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×#1157 | 2 |
| #6513 | 2 |
| C2×#1144 | 2 |
| C2×#1143 | 2 |
| C2×#1142 | 2 |
| C2×#1141 | 2 |
| C22×C7⋊C8 | 2 |
| C2×#135 | 7 |