Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
792 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C52 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C23×C52 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×H(5,𝔽2) | C2 | semidirect |
| 2. C52×#120 | C2 | semidirect |
| 3. C52×#99 | C2 | semidirect |
| 4. C24×C52 | C22 | semidirect |
| 10. C52×C22⋊C4 | C22 | semidirect |
| 11. C5×B(4,𝔽2) | C5 | direct (N × H) |
| 15. C23×C52 | D4 | semidirect |
| 23. C2×C4×C52 | D4 | semidirect |
| 24. C5×H(5,𝔽2) | C2×C5 | semidirect |
| 25. C5×#120 | C2×C5 | semidirect |
| 26. C5×#99 | C2×C5 | semidirect |
| 37. C24×C5 | C22×C5 | semidirect |
| 43. C5×C22⋊C4 | C22×C5 | semidirect |
| 44. B(4,𝔽2) | C52 | direct (N × H) |
| 52. C23×C5 | C5×D4 | semidirect |
| 60. C2×C4×C5 | C5×D4 | semidirect |
| 61. H(5,𝔽2) | C2×C52 | semidirect |
| 62. #120 | C2×C52 | semidirect |
| 63. #99 | C2×C52 | semidirect |
| 64. C52 | B(4,𝔽2) | direct (N × H) |
| 75. C24 | C22×C52 | semidirect |
| 81. C22⋊C4 | C22×C52 | semidirect |
| 89. C23 | C52×D4 | semidirect |
| 97. C2×C4 | C52×D4 | semidirect |
| 98. C5 | C5×B(4,𝔽2) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×H(5,𝔽2) | 2 |
| C52×#120 | 2 |
| C52×#99 | 2 |
| C5×B(4,𝔽2) | 5 |