Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
552 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C42×C52 | C22 |
| prime radical | C22×D52 | C22 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | C2×#126 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717196 | C2 | semidirect |
| 2. C5×#4022 | C2 | semidirect |
| 3. C42×C5×D5 | C2 | semidirect |
| 5. D5×#683 | C2 | semidirect |
| 6. #11716987 | C2 | semidirect |
| 7. #11716978 | C2 | semidirect |
| 11. C42×C52 | C22 | semidirect |
| 13. #87806 | C22 | semidirect |
| 15. C5×#683 | C22 | semidirect |
| 16. C4×C5×Dic5 | C22 | semidirect |
| 17. #87781 | C22 | semidirect |
| 18. #87780 | C22 | semidirect |
| 46. #4022 | D5 | direct (N × H) |
| 47. C42×D5 | D5 | semidirect |
| 67. C42×C5 | C2×D5 | semidirect |
| 68. C42×C5 | C2×D5 | semidirect |
| 84. #683 | C2×D5 | semidirect |
| 85. C4×Dic5 | C2×D5 | semidirect |
| 90. C5×D5 | #126 | semidirect |
| 121. C52 | C2×#126 | semidirect |
| 156. C42 | D52 | semidirect |
| 166. D5 | #4022 | direct (N × H) |
| 178. C5 | C2×#4022 | semidirect |
| 179. C5 | D5×#126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717196 | 2 |
| C5×#4022 | 2 |
| C42×C5×D5 | 2 |
| D5×#685 | 2 |
| D5×#683 | 2 |
| #11716987 | 2 |
| #11716978 | 2 |
| D5×#126 | 5 |
| C2×#4022 | 5 |