Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
694 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 138 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×C4⋊C4 | C22 |
| hypercenter | C4⋊C4 | D52 |
| hypocenter | C52 | #550 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#87870 | C2 | semidirect |
| 2. #11717207 | C2 | semidirect |
| 3. C5×#4041 | C2 | semidirect |
| 4. #11717032 | C2 | semidirect |
| 5. #11717032 | C2 | semidirect |
| 6. #11717020 | C2 | semidirect |
| 7. C4×#6037 | C2 | semidirect |
| 8. #11717004 | C2 | semidirect |
| 34. #6045 | D4 | semidirect |
| 51. #4041 | D5 | semidirect |
| 67. #1039 | #41 | semidirect |
| 144. C52 | #550 | semidirect |
| 216. C5 | #65933 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87870 | 2 |
| #11717207 | 2 |
| C5×#4041 | 2 |
| #11717032 | 2 |
| #11717020 | 2 |
| C4×#6037 | 2 |
| #11717004 | 2 |
| #65933 | 5 |