Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
744 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 136 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 | C52×C22⋊C4 | C22 |
| hypercenter | C22⋊C4 | D52 |
| hypocenter | C52 | #535 |
| 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. C22×#6037 | C2 | semidirect |
| 2. C2×#87880 | C2 | semidirect |
| 3. #11717197 | C2 | semidirect |
| 4. C5×#4028 | C2 | semidirect |
| 5. #11717094 | C2 | semidirect |
| 6. #11717033 | C2 | semidirect |
| 7. #11717032 | C2 | semidirect |
| 8. #11717004 | C2 | semidirect |
| 12. C2×C5×D20 | C22 | semidirect |
| 13. C2×#6039 | C22 | semidirect |
| 17. C2×#6037 | C22 | semidirect |
| 18. C2×#6037 | C22 | semidirect |
| 20. C2×D5×Dic5 | C22 | semidirect |
| 23. #87806 | C22 | semidirect |
| 24. C4×#1039 | C22 | semidirect |
| 25. C5×#685 | C22 | semidirect |
| 26. #87778 | C22 | semidirect |
| 34. C2×#1039 | D4 | semidirect |
| 56. #4028 | D5 | semidirect |
| 88. C2×D20 | C2×D5 | semidirect |
| 98. #685 | C2×D5 | semidirect |
| 151. C52 | #535 | semidirect |
| 228. C5 | #65926 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6037 | 2 |
| C2×#87880 | 2 |
| #11717197 | 2 |
| C5×#4028 | 2 |
| #11717094 | 2 |
| #11717033 | 2 |
| #11717032 | 2 |
| #11717004 | 2 |
| #65926 | 5 |