Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
994 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 175 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 | #10613619 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #10613492 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C5×#1705 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602129 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10602055 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10598855 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10598843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#4060 | — | 2 | 6 | no | — |
| 9 | #66059 | C22 | 1 | 3 | yes | — |
| 10 | D5×#127 | C22 | 1 | 1 | yes | characteristic |
| 11 | C5×#498 | C22 | 1 | 3 | yes | complemented |
| 12 | C5×#492 | C22 | 1 | 3 | yes | — |
| 13 | #65465 | — | 2 | 6 | no | — |
| 14 | #65377 | C22 | 1 | 6 | yes | complemented |
| 15 | #65374 | C22 | 1 | 3 | yes | — |
| 16 | #65362 | C22 | 1 | 3 | yes | — |
| 17 | #65158 | — | 2 | 6 | no | — |
| 18 | #65154 | — | 2 | 6 | no | — |
| 19 | #64984 | C22 | 1 | 3 | yes | — |
| 20 | #64984 | C22 | 1 | 3 | yes | complemented |
| 21 | C4×#676 | C22 | 1 | 3 | yes | — |
| 22 | C5×C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | #64675 | C22 | 1 | 3 | yes | complemented |
| 24 | #19745 | — | 5 | 1 | no | maximal, Sylow |
| 25 | C2×#713 | — | 2 | 3 | no | — |
| 26 | C2×D4×D5 | — | 2 | 3 | no | — |
| 27 | C2×C5×SD16 | — | 2 | 6 | no | — |
| 28 | C5×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 29 | C5×#127 | C23 | 1 | 1 | yes | characteristic |
| 30 | #4093 | — | 2 | 3 | no | — |
| 31 | C2×#688 | — | 2 | 6 | no | — |
| 32 | #4086 | — | 2 | 3 | no | — |
| 33 | C2×#686 | — | 2 | 6 | no | — |
| 34 | #4060 | — | 4 | 24 | no | — |
| 35 | C2×D40 | — | 2 | 6 | no | — |
| 36 | C2×D40 | — | 2 | 6 | no | — |
| 37 | C2×#676 | D4 | 1 | 6 | yes | — |
| 38 | #4041 | — | 2 | 6 | no | — |
| 39 | #4039 | — | 2 | 6 | no | — |
| 40 | #4020 | C23 | 1 | 1 | yes | characteristic |
| 41 | #4020 | C23 | 1 | 3 | yes | — |
| 42 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 43 | C5×#106 | — | 2 | 6 | no | — |
| 44 | C4×C5×C8 | C23 | 1 | 3 | yes | — |
| 45 | #3947 | D4 | 1 | 6 | yes | — |
| 46 | #3941 | — | 2 | 12 | no | — |
| 47 | C4×#168 | C23 | 1 | 3 | yes | — |
| 48 | #3184 | — | 5 | 3 | no | — |
| 49 | #3151 | — | 5 | 3 | no | — |
| 50 | #1721 | — | 5 | 3 | no | — |
| 51 | #1715 | — | 5 | 3 | no | — |
| 52 | #1712 | — | 5 | 1 | no | — |
| 53 | #1705 | D5 | 1 | 1 | yes | characteristic, complemented |
| 54 | #1455 | — | 5 | 1 | no | — |
| 55 | C23×D5 | — | 4 | 1 | no | — |
| 56 | C2×C5×Q8 | — | 2 | 3 | no | — |
| 57 | C2×C5×D4 | — | 2 | 3 | no | — |
| 58 | C2×C22⋊D5 | — | 4 | 3 | no | — |
| 59 | #713 | — | 4 | 12 | no | — |
| 60 | D4×D5 | — | 4 | 12 | no | — |
| 61 | C2×D20 | — | 2 | 3 | no | — |
| 62 | C2×D20 | — | 2 | 3 | no | — |
| 63 | C2×D20 | — | 2 | 6 | no | — |
| 64 | C2×D20 | — | 4 | 3 | no | — |
| 65 | C2×C4×D5 | — | 4 | 3 | no | — |
| 66 | C2×C4×D5 | C2×D4 | 1 | 3 | yes | — |
| 67 | C5×SD16 | — | 4 | 12 | no | — |
| 68 | C2×C5×C8 | C2×D4 | 1 | 6 | yes | — |
| 69 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 70 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 71 | #688 | — | 4 | 12 | no | — |
| 72 | #686 | — | 4 | 12 | no | — |
| 73 | #685 | — | 4 | 6 | no | — |
| 74 | C4×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 75 | C2×#168 | C2×D4 | 1 | 6 | yes | — |
| 76 | D40 | — | 4 | 12 | no | — |
| 77 | D40 | — | 4 | 12 | no | — |
| 78 | #676 | — | 2 | 12 | no | — |
| 79 | C2×AGL(1,ℤ/8) | — | 10 | 6 | no | — |
| 80 | #532 | — | 5 | 3 | no | — |
| 81 | C2×#127 | — | 5 | 1 | no | — |
| 82 | #498 | C2×D5 | 1 | 3 | yes | complemented |
| 83 | #496 | — | 10 | 6 | no | — |
| 84 | #493 | — | 5 | 3 | no | — |
| 85 | #493 | — | 5 | 3 | no | — |
| 86 | #493 | — | 5 | 3 | no | — |
| 87 | #492 | — | 5 | 3 | no | — |
| 88 | #492 | C2×D5 | 1 | 3 | yes | — |
| 89 | #486 | — | 5 | 6 | no | — |
| 90 | #469 | — | 10 | 6 | no | — |
| 91 | #442 | — | 10 | 6 | no | — |
| 92 | C4×M16 | — | 5 | 3 | no | — |
| 93 | #322 | — | 5 | 3 | no | — |
| 94 | C82 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 95 | C23×C5 | — | 4 | 1 | no | — |
| 96 | C22×D5 | — | 4 | 3 | no | — |
| 97 | C22×D5 | — | 4 | 1 | no | — |
| 98 | C22×D5 | — | 8 | 6 | no | — |
| 99 | C22×D5 | #127 | 1 | 1 | yes | characteristic |
| 100 | C5×Q8 | — | 4 | 6 | no | — |
| 101 | C5×D4 | — | 4 | 6 | no | — |
| 102 | C2×C4×C5 | — | 4 | 3 | no | — |
| 103 | C2×C4×C5 | C22×D4 | 1 | 3 | yes | — |
| 104 | C22⋊D5 | — | 8 | 6 | no | — |
| 105 | C2×Dic5 | #127 | 1 | 3 | yes | — |
| 106 | D20 | — | 4 | 6 | no | — |
| 107 | D20 | — | 4 | 6 | no | — |
| 108 | D20 | — | 4 | 12 | no | — |
| 109 | D20 | — | 8 | 6 | no | — |
| 110 | C4×D5 | — | 2 | 6 | no | — |
| 111 | C4×D5 | — | 8 | 6 | no | — |
| 112 | C5×C8 | AGL(1,ℤ/8) | 1 | 12 | yes | complemented |
| 113 | #168 | — | 2 | 6 | no | — |
| 114 | C2×#41 | — | 10 | 3 | no | — |
| 115 | C22×D4 | — | 10 | 3 | no | — |
| 116 | AGL(1,ℤ/8) | — | 20 | 24 | no | — |
| 117 | C2×SD16 | — | 2 | 6 | no | — |
| 118 | C2×SD16 | — | 10 | 6 | no | — |
| 119 | C2×D8 | — | 10 | 6 | no | — |
| 120 | C2×D8 | — | 10 | 6 | no | — |
| 121 | C2×D8 | — | 10 | 6 | no | — |
| 122 | C2×M16 | — | 5 | 6 | no | — |
| 123 | C4⋊Q8 | C22×D5 | 1 | 3 | yes | — |
| 124 | #127 | — | 5 | 1 | no | — |
| 125 | #127 | — | 5 | 3 | no | — |
| 126 | #127 | — | 10 | 3 | no | — |
| 127 | #127 | C22×D5 | 1 | 1 | yes | characteristic |
| 128 | #124 | — | 10 | 3 | no | — |
| 129 | #121 | — | 10 | 6 | no | — |
| 130 | C4×D4 | — | 10 | 6 | no | — |
| 131 | C2×C42 | — | 5 | 1 | no | — |
| 132 | #106 | — | 2 | 6 | no | — |
| 133 | #102 | — | 10 | 12 | no | — |
| 134 | #97 | — | 5 | 6 | no | — |
| 135 | C4×C8 | — | 5 | 3 | no | — |
| 136 | C4×C8 | C22×D5 | 1 | 3 | yes | — |
| 137 | C22×C5 | — | 8 | 3 | no | — |
| 138 | C22×C5 | C2×#127 | 1 | 1 | yes | characteristic |
| 139 | C2×D5 | — | 2 | 3 | no | — |
| 140 | C2×D5 | — | 8 | 3 | no | — |
| 141 | C2×D5 | — | 8 | 3 | no | — |
| 142 | C2×D5 | — | 8 | 6 | no | — |
| 143 | C2×D5 | — | 16 | 4 | no | — |
| 144 | C4×C5 | — | 8 | 3 | no | — |
| 145 | C4×C5 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 146 | Dic5 | — | 2 | 3 | no | — |
| 147 | C24 | — | 20 | 1 | no | — |
| 148 | #41 | — | 20 | 12 | no | — |
| 149 | C2×Q8 | — | 2 | 3 | no | — |
| 150 | C2×D4 | — | 2 | 3 | no | — |
| 151 | C2×D4 | — | 10 | 6 | no | — |
| 152 | C2×D4 | — | 10 | 3 | no | — |
| 153 | C2×D4 | — | 10 | 3 | no | — |
| 154 | C2×D4 | — | 20 | 3 | no | — |
| 155 | C2×D4 | — | 20 | 3 | no | — |
| 156 | C2×D4 | — | 20 | 12 | no | — |
| 157 | C22×C4 | — | 5 | 3 | no | — |
| 158 | C22×C4 | — | 20 | 3 | no | — |
| 159 | SD16 | — | 4 | 12 | no | — |
| 160 | SD16 | — | 20 | 12 | no | — |
| 161 | D8 | — | 20 | 12 | no | — |
| 162 | D8 | — | 20 | 12 | no | — |
| 163 | D8 | — | 20 | 12 | no | — |
| 164 | M16 | — | 10 | 12 | no | — |
| 165 | C2×C8 | — | 5 | 6 | no | — |
| 166 | C2×C8 | D4×D5 | 1 | 6 | yes | — |
| 167 | C4⋊C4 | — | 2 | 6 | no | — |
| 168 | C22⋊C4 | — | 20 | 6 | no | — |
| 169 | C42 | — | 5 | 3 | no | — |
| 170 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 171 | C2×C5 | — | 16 | 1 | no | — |
| 172 | C2×C5 | #3151 | 1 | 3 | yes | — |
| 173 | D5 | — | 4 | 1 | no | — |
| 174 | D5 | — | 16 | 3 | no | — |
| 175 | D5 | — | 16 | 1 | no | — |
| 176 | C23 | — | 4 | 1 | no | — |
| 177 | C23 | — | 5 | 1 | no | — |
| 178 | C23 | — | 20 | 3 | no | — |
| 179 | C23 | — | 20 | 1 | no | — |
| 180 | C23 | — | 40 | 6 | no | — |
| 181 | Q8 | — | 4 | 6 | no | — |
| 182 | D4 | — | 4 | 6 | no | — |
| 183 | D4 | — | 20 | 6 | no | — |
| 184 | D4 | — | 20 | 6 | no | — |
| 185 | D4 | — | 20 | 12 | no | — |
| 186 | D4 | — | 40 | 6 | no | — |
| 187 | D4 | — | 40 | 6 | no | — |
| 188 | C2×C4 | — | 4 | 3 | no | — |
| 189 | C2×C4 | — | 5 | 3 | no | — |
| 190 | C2×C4 | — | 10 | 6 | no | — |
| 191 | C2×C4 | — | 40 | 6 | no | — |
| 192 | C2×C4 | C2×D4×D5 | 1 | 3 | yes | — |
| 193 | C8 | — | 10 | 6 | no | — |
| 194 | C8 | #4060 | 1 | 12 | yes | complemented |
| 195 | C5 | #19745 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 196 | C22 | — | 8 | 3 | no | — |
| 197 | C22 | — | 10 | 3 | no | — |
| 198 | C22 | — | 40 | 3 | no | — |
| 199 | C22 | — | 40 | 3 | no | — |
| 200 | C22 | — | 40 | 6 | no | — |
| 201 | C22 | — | 80 | 4 | no | — |
| 202 | C22 | D5×#127 | 1 | 1 | yes | characteristic |
| 203 | C4 | — | 8 | 3 | no | — |
| 204 | C4 | — | 10 | 3 | no | — |
| 205 | C4 | C2×#4060 | 1 | 6 | yes | — |
| 206 | C2 | — | 16 | 1 | no | — |
| 207 | C2 | — | 20 | 1 | no | — |
| 208 | C2 | — | 80 | 1 | no | — |
| 209 | C2 | — | 80 | 3 | no | — |
| 210 | C2 | #10613492 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D5×#127 |
| commutator, cosocle | C42×C5 | C24 |
| Frattini | C42 | C23×D5 |
| Fitting | C5×#1705 | C2 |
| socle | C22×C5 | C2×#127 |
| hypercenter | #1705 | D5 |
| hypocenter | C5 | #19745 |
| LCS | G ▷ C42×C5 ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1705 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10613619 | C2 | semidirect |
| 2. #10613492 | C2 | semidirect |
| 3. C5×#1705 | C2 | semidirect |
| 4. #10602129 | C2 | semidirect |
| 5. #10602055 | C2 | semidirect |
| 6. #10598855 | C2 | semidirect |
| 7. #10598843 | C2 | semidirect |
| 11. C5×#498 | C22 | semidirect |
| 14. #65377 | C22 | semidirect |
| 20. #64984 | C22 | semidirect |
| 22. C5×C82 | C22 | semidirect |
| 23. #64675 | C22 | semidirect |
| 53. #1705 | D5 | semidirect |
| 82. #498 | C2×D5 | semidirect |
| 94. C82 | C2×D5 | semidirect |
| 112. C5×C8 | AGL(1,ℤ/8) | semidirect |
| 194. C8 | #4060 | semidirect |
| 195. C5 | #19745 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10613619 | 2 |
| #10613492 | 2 |
| C5×#1705 | 2 |
| #10602129 | 2 |
| #10602055 | 2 |
| #10598855 | 2 |
| #10598843 | 2 |
| #19745 | 5 |