Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
164 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 22 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 | #10598887 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10598520 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10598517 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #64984 | — | 2 | 1 | no | — |
| 5 | #64984 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #64983 | — | 2 | 1 | no | — |
| 7 | #64982 | C22 | 1 | 1 | yes | characteristic |
| 8 | #64855 | — | 2 | 1 | no | — |
| 9 | #64853 | — | 2 | 1 | no | — |
| 10 | C5×#322 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #7527 | — | 5 | 1 | no | maximal, Sylow |
| 12 | C2×D40 | — | 4 | 1 | no | — |
| 13 | C2×D40 | — | 4 | 2 | no | — |
| 14 | C2×#677 | — | 4 | 1 | no | — |
| 15 | #4020 | — | 2 | 1 | no | — |
| 16 | #4020 | D4 | 1 | 1 | yes | characteristic |
| 17 | #4015 | D4 | 1 | 1 | yes | characteristic |
| 18 | #4005 | — | 4 | 1 | no | — |
| 19 | #3999 | — | 4 | 1 | no | — |
| 20 | C4×C5×C8 | — | 2 | 1 | no | — |
| 21 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | #3953 | — | 4 | 2 | no | — |
| 23 | #1721 | — | 5 | 1 | no | — |
| 24 | #1410 | — | 5 | 1 | no | — |
| 25 | #1409 | — | 5 | 1 | no | — |
| 26 | C2×D20 | — | 2 | 1 | no | — |
| 27 | C2×D20 | — | 4 | 1 | no | — |
| 28 | C2×D20 | — | 4 | 1 | no | — |
| 29 | C2×D20 | — | 4 | 1 | no | — |
| 30 | C2×D20 | — | 4 | 1 | no | — |
| 31 | C2×Dic10 | — | 2 | 1 | no | — |
| 32 | C22⋊AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 33 | #704 | — | 8 | 1 | no | — |
| 34 | C2×C5×C8 | — | 2 | 1 | no | — |
| 35 | C2×C5×C8 | — | 2 | 2 | no | — |
| 36 | C42×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 37 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 38 | D40 | — | 8 | 2 | no | — |
| 39 | D40 | — | 8 | 4 | no | — |
| 40 | #677 | — | 8 | 2 | no | — |
| 41 | #493 | — | 5 | 1 | no | — |
| 42 | #493 | — | 10 | 1 | no | — |
| 43 | #492 | — | 5 | 1 | no | — |
| 44 | #486 | — | 10 | 1 | no | — |
| 45 | #356 | — | 10 | 1 | no | — |
| 46 | #353 | — | 10 | 1 | no | — |
| 47 | #322 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 48 | C22×D5 | — | 4 | 1 | no | — |
| 49 | C22×D5 | — | 8 | 1 | no | — |
| 50 | C2×AGL(1,𝔽5) | — | 16 | 1 | no | — |
| 51 | C2×C4×C5 | — | 2 | 1 | no | — |
| 52 | C2×C4×C5 | #99 | 1 | 1 | yes | characteristic |
| 53 | C2×Dic5 | — | 4 | 1 | no | — |
| 54 | D20 | — | 8 | 1 | no | — |
| 55 | D20 | — | 8 | 2 | no | — |
| 56 | D20 | — | 8 | 2 | no | — |
| 57 | D20 | — | 8 | 2 | no | — |
| 58 | D20 | — | 8 | 2 | no | — |
| 59 | Dic10 | — | 8 | 1 | no | — |
| 60 | #170 | — | 16 | 1 | no | — |
| 61 | C5×C8 | — | 2 | 2 | no | — |
| 62 | C5×C8 | — | 4 | 2 | no | — |
| 63 | C2×SD16 | — | 20 | 1 | no | — |
| 64 | C2×D8 | — | 20 | 1 | no | — |
| 65 | C2×D8 | — | 20 | 2 | no | — |
| 66 | C4⋊Q8 | — | 5 | 1 | no | — |
| 67 | #127 | — | 5 | 1 | no | — |
| 68 | #127 | — | 10 | 1 | no | — |
| 69 | #102 | — | 20 | 2 | no | — |
| 70 | #101 | — | 20 | 1 | no | — |
| 71 | #99 | — | 20 | 1 | no | — |
| 72 | C4×C8 | — | 2 | 1 | no | — |
| 73 | C4×C8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 74 | C22×C5 | #353 | 1 | 1 | yes | characteristic |
| 75 | C2×D5 | — | 8 | 1 | no | — |
| 76 | C2×D5 | — | 16 | 1 | no | — |
| 77 | C2×D5 | — | 16 | 1 | no | — |
| 78 | C2×D5 | — | 16 | 1 | no | — |
| 79 | C2×D5 | — | 16 | 1 | no | — |
| 80 | AGL(1,𝔽5) | — | 32 | 1 | no | — |
| 81 | C4×C5 | — | 2 | 2 | no | — |
| 82 | C4×C5 | — | 2 | 1 | no | — |
| 83 | Dic5 | — | 8 | 1 | no | — |
| 84 | C2×Q8 | — | 10 | 1 | no | — |
| 85 | C2×D4 | — | 10 | 1 | no | — |
| 86 | C2×D4 | — | 20 | 1 | no | — |
| 87 | C2×D4 | — | 20 | 1 | no | — |
| 88 | C2×D4 | — | 20 | 1 | no | — |
| 89 | C2×D4 | — | 20 | 1 | no | — |
| 90 | SD16 | — | 40 | 2 | no | — |
| 91 | D8 | — | 40 | 2 | no | — |
| 92 | D8 | — | 40 | 4 | no | — |
| 93 | M16 | — | 40 | 1 | no | — |
| 94 | C2×C8 | — | 2 | 1 | no | — |
| 95 | C2×C8 | — | 2 | 2 | no | — |
| 96 | C4⋊C4 | — | 20 | 1 | no | — |
| 97 | C22⋊C4 | — | 40 | 1 | no | — |
| 98 | C42 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 99 | C2×C5 | — | 2 | 1 | no | — |
| 100 | C2×C5 | #1408 | 1 | 1 | yes | characteristic |
| 101 | D5 | — | 16 | 1 | no | — |
| 102 | D5 | — | 32 | 1 | no | — |
| 103 | C23 | — | 20 | 1 | no | — |
| 104 | C23 | — | 40 | 1 | no | — |
| 105 | Q8 | — | 40 | 1 | no | — |
| 106 | D4 | — | 40 | 1 | no | — |
| 107 | D4 | — | 40 | 2 | no | — |
| 108 | D4 | — | 40 | 2 | no | — |
| 109 | D4 | — | 40 | 2 | no | — |
| 110 | D4 | — | 40 | 2 | no | — |
| 111 | C2×C4 | — | 2 | 1 | no | — |
| 112 | C2×C4 | — | 20 | 1 | no | — |
| 113 | C2×C4 | — | 80 | 1 | no | — |
| 114 | C2×C4 | #3999 | 1 | 1 | yes | characteristic |
| 115 | C8 | — | 2 | 2 | no | — |
| 116 | C8 | — | 4 | 2 | no | — |
| 117 | C8 | — | 80 | 1 | no | — |
| 118 | C5 | #7527 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 119 | C22 | — | 40 | 1 | no | — |
| 120 | C22 | — | 80 | 1 | no | — |
| 121 | C22 | — | 80 | 1 | no | — |
| 122 | C22 | — | 80 | 1 | no | — |
| 123 | C22 | — | 80 | 1 | no | — |
| 124 | C22 | #64853 | 1 | 1 | yes | characteristic |
| 125 | C4 | — | 2 | 2 | no | — |
| 126 | C4 | — | 2 | 1 | no | — |
| 127 | C4 | — | 40 | 1 | no | — |
| 128 | C4 | — | 160 | 1 | no | — |
| 129 | C2 | — | 2 | 1 | no | — |
| 130 | C2 | — | 80 | 1 | no | — |
| 131 | C2 | — | 160 | 1 | no | — |
| 132 | C2 | #10598516 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10598520 | C2 | semidirect |
| 3. #10598517 | C2 | semidirect |
| 5. #64984 | C4 | semidirect |
| 10. C5×#322 | C4 | semidirect |
| 47. #322 | AGL(1,𝔽5) | semidirect |
| 118. C5 | #7527 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10598887 | 2 |
| #10598520 | 2 |
| #10598517 | 2 |
| #7527 | 5 |