Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
125 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#6037 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #87880 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #87867 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #87838 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #87837 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×C5×D5 | — | 2 | 1 | no | — |
| 7 | C2×#1039 | C22 | 1 | 1 | yes | characteristic |
| 8 | C5×C22⋊D5 | — | 2 | 1 | no | — |
| 9 | #6039 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #6037 | — | 2 | 1 | no | — |
| 11 | #6037 | C22 | 1 | 2 | yes | — |
| 12 | #6037 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | D5×Dic5 | — | 2 | 1 | no | — |
| 14 | #6034 | C22 | 1 | 2 | yes | complemented |
| 15 | C22×C52 | D4 | 1 | 1 | yes | characteristic |
| 16 | C2×C5×D5 | — | 2 | 1 | no | — |
| 17 | C2×C5×D5 | — | 2 | 2 | no | — |
| 18 | C2×C5×D5 | — | 4 | 2 | no | — |
| 19 | #1039 | D4 | 1 | 1 | yes | characteristic |
| 20 | #1039 | C23 | 1 | 1 | yes | characteristic |
| 21 | C5×Dic5 | — | 2 | 1 | no | — |
| 22 | C2×C22⋊D5 | — | 10 | 1 | no | — |
| 23 | #711 | — | 10 | 1 | no | — |
| 24 | #704 | — | 10 | 1 | no | — |
| 25 | C2×C52 | — | 2 | 1 | no | — |
| 26 | C2×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 27 | C5×D5 | — | 4 | 1 | no | — |
| 28 | C5×D5 | — | 4 | 2 | no | — |
| 29 | C23×C5 | — | 10 | 1 | no | — |
| 30 | C22×D5 | — | 2 | 1 | no | — |
| 31 | C5×D4 | — | 10 | 1 | no | — |
| 32 | C22⋊D5 | — | 2 | 1 | no | — |
| 33 | C22⋊D5 | — | 10 | 2 | no | — |
| 34 | C22⋊D5 | — | 10 | 1 | no | — |
| 35 | C22⋊D5 | — | 10 | 2 | no | — |
| 36 | C2×Dic5 | — | 10 | 1 | no | — |
| 37 | C2×Dic5 | — | 10 | 1 | no | — |
| 38 | C2×Dic5 | — | 10 | 1 | no | — |
| 39 | C2×Dic5 | — | 10 | 1 | no | — |
| 40 | C4×D5 | — | 10 | 1 | no | — |
| 41 | Dic10 | — | 10 | 1 | no | — |
| 42 | #170 | — | 10 | 2 | no | — |
| 43 | AGL(1,ℤ/8) | — | 25 | 1 | no | maximal, Sylow |
| 44 | C52 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 45 | C22×C5 | — | 2 | 1 | no | — |
| 46 | C22×C5 | — | 2 | 1 | no | — |
| 47 | C22×C5 | — | 2 | 1 | no | — |
| 48 | C22×C5 | — | 10 | 1 | no | — |
| 49 | C22×C5 | — | 10 | 2 | no | — |
| 50 | C22×C5 | — | 20 | 2 | no | — |
| 51 | C2×D5 | — | 2 | 1 | no | — |
| 52 | C2×D5 | — | 2 | 2 | no | — |
| 53 | C2×D5 | — | 4 | 2 | no | — |
| 54 | C4×C5 | — | 10 | 1 | no | — |
| 55 | Dic5 | — | 2 | 1 | no | — |
| 56 | Dic5 | — | 10 | 1 | no | — |
| 57 | Dic5 | — | 10 | 1 | no | — |
| 58 | Dic5 | — | 10 | 1 | no | — |
| 59 | Dic5 | — | 10 | 1 | no | — |
| 60 | Dic5 | — | 10 | 1 | no | — |
| 61 | Dic5 | — | 10 | 1 | no | — |
| 62 | #41 | — | 25 | 1 | no | — |
| 63 | C2×D4 | — | 25 | 1 | no | — |
| 64 | SD16 | — | 25 | 2 | no | — |
| 65 | D8 | — | 25 | 2 | no | — |
| 66 | M16 | — | 25 | 1 | no | — |
| 67 | C2×C5 | — | 2 | 2 | no | — |
| 68 | C2×C5 | — | 2 | 1 | no | — |
| 69 | C2×C5 | — | 2 | 1 | no | — |
| 70 | C2×C5 | — | 2 | 1 | no | — |
| 71 | C2×C5 | — | 4 | 1 | no | — |
| 72 | C2×C5 | — | 4 | 1 | no | — |
| 73 | C2×C5 | — | 20 | 2 | no | — |
| 74 | C2×C5 | — | 20 | 1 | no | — |
| 75 | D5 | — | 4 | 2 | no | — |
| 76 | D5 | — | 4 | 1 | no | — |
| 77 | C23 | — | 10 | 1 | no | — |
| 78 | Q8 | — | 25 | 1 | no | — |
| 79 | D4 | — | 10 | 1 | no | — |
| 80 | D4 | — | 25 | 1 | no | — |
| 81 | D4 | — | 25 | 2 | no | — |
| 82 | D4 | — | 50 | 1 | no | — |
| 83 | C2×C4 | — | 25 | 1 | no | — |
| 84 | C2×C4 | — | 50 | 1 | no | — |
| 85 | C8 | — | 25 | 2 | no | — |
| 86 | C5 | — | 2 | 1 | no | — |
| 87 | C5 | — | 2 | 1 | no | — |
| 88 | C5 | — | 2 | 1 | no | — |
| 89 | C22 | — | 10 | 1 | no | — |
| 90 | C22 | — | 10 | 2 | no | — |
| 91 | C22 | — | 20 | 2 | no | — |
| 92 | C22 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 93 | C4 | — | 10 | 1 | no | — |
| 94 | C4 | — | 25 | 1 | no | — |
| 95 | C4 | — | 25 | 1 | no | — |
| 96 | C2 | — | 2 | 1 | no | — |
| 97 | C2 | — | 20 | 2 | no | — |
| 98 | C2 | — | 20 | 1 | no | — |
| 99 | C2 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52⋊C2 |
| commutator, cosocle | #1039 | C23 |
| Fitting | C22×C52 | D4 |
| socle | C2×C52 | C2×D4 |
| hypercenter | C22 | D52⋊C2 |
| hypocenter | C52 | AGL(1,ℤ/8) |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6037 | C2 | semidirect |
| 2. #87880 | C2 | semidirect |
| 3. #87867 | C2 | semidirect |
| 4. #87838 | C2 | semidirect |
| 5. #87837 | C2 | semidirect |
| 9. #6039 | C22 | semidirect |
| 12. #6037 | C22 | semidirect |
| 14. #6034 | C22 | semidirect |
| 44. C52 | AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6037 | 2 |
| #87880 | 2 |
| #87867 | 2 |
| #87838 | 2 |
| #87837 | 2 |
| AGL(1,ℤ/8) | 25 |