Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
594 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C22×C4×C52 | C22 |
| hypercenter | C22×C4 | D52 |
| hypocenter | C52 | #538 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717250 | C2 | semidirect |
| 2. C2×C5×#685 | C2 | semidirect |
| 3. C5×#4070 | C2 | semidirect |
| 4. #11717095 | C2 | semidirect |
| 5. #11717070 | C2 | semidirect |
| 6. #11717028 | C2 | semidirect |
| 7. #11717025 | C2 | semidirect |
| 8. #11716977 | C2 | semidirect |
| 9. #11716972 | C2 | semidirect |
| 13. C2×C4⋊(C5⋊D5) | C22 | semidirect |
| 15. C2×C5×Dic10 | C22 | semidirect |
| 18. C2×#6036 | C22 | semidirect |
| 19. #87808 | C22 | semidirect |
| 20. C4⋊#1039 | C22 | semidirect |
| 24. C5×C4⋊Dic5 | C22 | semidirect |
| 25. C5×#683 | C22 | semidirect |
| 26. #87781 | C22 | semidirect |
| 27. #87779 | C22 | semidirect |
| 30. Dic52 | C22 | semidirect |
| 61. C2×#685 | D5 | semidirect |
| 62. #4070 | D5 | semidirect |
| 109. C2×Dic10 | C2×D5 | semidirect |
| 130. C4⋊Dic5 | C2×D5 | semidirect |
| 133. #683 | C2×D5 | semidirect |
| 200. C52 | #538 | semidirect |
| 304. C5 | #65828 | semidirect |
| 305. C5 | #65978 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717250 | 2 |
| C2×C5×#685 | 2 |
| C5×#4070 | 2 |
| #11717095 | 2 |
| #11717070 | 2 |
| #11717028 | 2 |
| #11717025 | 2 |
| #11716977 | 2 |
| #11716972 | 2 |
| #65978 | 5 |
| #65828 | 5 |