Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
200 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#64425 |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Frattini | C2 | C22×#64425 |
| Fitting | C2×C4×C72 | C2×C3 |
| hypercenter | C2×C4 | #64425 |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×#3862 | C2 | semidirect |
| 2. #11924222 | C2 | semidirect |
| 3. C4⋊#64425 | C2 | semidirect |
| 4. C4×#64425 | C2 | semidirect |
| 5. #11924218 | C2 | semidirect |
| 6. #11716129 | C3 | semidirect |
| 12. C2×C4×C72 | C2×C3 | semidirect |
| 13. C22⋊(C7⋊D7) | C2×C3 | semidirect |
| 14. C4⋊(C7⋊D7) | C2×C3 | semidirect |
| 15. C4×C7⋊D7 | C2×C3 | semidirect |
| 16. #87648 | C2×C3 | semidirect |
| 37. #3862 | #41 | semidirect |
| 55. C2×C4×C7 | AGL(1,𝔽7) | semidirect |
| 56. C2×C4×C7 | AGL(1,𝔽7) | semidirect |
| 69. C72 | C3×#41 | semidirect |
| 118. C2×C4 | #64425 | semidirect |
| 120. C7 | #1157⋊C3 | semidirect |
| 121. C7 | #1157⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#3862 | 2 |
| #11924222 | 2 |
| C4⋊#64425 | 2 |
| C4×#64425 | 2 |
| #11924218 | 2 |
| #11716129 | 3 |
| #1157⋊C3 | 7 |