Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
244 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 32 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 | #10602914 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10598790 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10598781 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×#575 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #66156 | — | 2 | 1 | no | — |
| 6 | #65745 | — | 2 | 1 | no | — |
| 7 | #65523 | — | 2 | 1 | no | — |
| 8 | #65418 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22⋊#674 | — | 2 | 1 | no | — |
| 10 | #64780 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #7406 | — | 5 | 1 | no | maximal, Sylow |
| 12 | C2×C5×#41 | — | 2 | 1 | no | — |
| 13 | C2×#709 | D4 | 1 | 1 | yes | characteristic |
| 14 | #4130 | — | 4 | 1 | no | — |
| 15 | C5×#137 | — | 4 | 1 | no | — |
| 16 | C5×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 17 | C5×#135 | — | 4 | 1 | no | — |
| 18 | C2×C5×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | #4097 | — | 4 | 1 | no | — |
| 20 | #4096 | — | 4 | 2 | no | — |
| 21 | #4095 | — | 4 | 1 | no | — |
| 22 | #4088 | — | 4 | 1 | no | — |
| 23 | #4078 | — | 4 | 1 | no | — |
| 24 | C2×#681 | — | 2 | 1 | no | — |
| 25 | C22×#168 | D4 | 1 | 1 | yes | characteristic |
| 26 | #4011 | — | 2 | 2 | no | — |
| 27 | #3999 | — | 4 | 1 | no | — |
| 28 | C2×#674 | — | 4 | 1 | no | — |
| 29 | C22⋊#168 | — | 4 | 2 | no | — |
| 30 | #3952 | — | 4 | 1 | no | — |
| 31 | #2017 | — | 5 | 1 | no | — |
| 32 | #1391 | — | 5 | 1 | no | — |
| 33 | #1315 | — | 5 | 1 | no | — |
| 34 | C5×#41 | — | 4 | 1 | no | — |
| 35 | C5×#41 | — | 4 | 1 | no | — |
| 36 | C5×#41 | — | 4 | 1 | no | — |
| 37 | C5×#41 | — | 4 | 1 | no | — |
| 38 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 39 | C2×C5×D4 | — | 2 | 1 | no | — |
| 40 | C2×C5×D4 | — | 4 | 1 | no | — |
| 41 | C22×C4×C5 | — | 4 | 1 | no | — |
| 42 | C22×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C22⋊D5 | — | 2 | 1 | no | — |
| 44 | #709 | — | 4 | 1 | no | — |
| 45 | #709 | — | 4 | 1 | no | — |
| 46 | C2×D20 | D8 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×D5 | — | 2 | 1 | no | — |
| 48 | C2×Dic10 | SD16 | 1 | 1 | yes | characteristic |
| 49 | C22⋊AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 50 | C4⋊AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 51 | C4×AGL(1,𝔽5) | — | 8 | 1 | no | — |
| 52 | C5×Q16 | — | 4 | 1 | no | — |
| 53 | C5×SD16 | — | 4 | 2 | no | — |
| 54 | C5×D8 | — | 4 | 1 | no | — |
| 55 | C5×M16 | — | 4 | 1 | no | — |
| 56 | C2×C5×C8 | — | 2 | 1 | no | — |
| 57 | #689 | — | 4 | 2 | no | — |
| 58 | #688 | — | 4 | 2 | no | — |
| 59 | #687 | — | 4 | 2 | no | — |
| 60 | #686 | — | 4 | 2 | no | — |
| 61 | #681 | — | 4 | 1 | no | — |
| 62 | #681 | — | 4 | 1 | no | — |
| 63 | C2×#168 | — | 2 | 2 | no | — |
| 64 | C2×#168 | — | 2 | 1 | no | — |
| 65 | C2×#168 | — | 4 | 1 | no | — |
| 66 | #674 | — | 8 | 1 | no | — |
| 67 | #575 | — | 10 | 1 | no | — |
| 68 | #575 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 69 | #410 | — | 10 | 1 | no | — |
| 70 | #408 | — | 5 | 1 | no | — |
| 71 | #408 | — | 10 | 1 | no | — |
| 72 | C22⋊C16 | — | 10 | 1 | no | — |
| 73 | #338 | — | 5 | 1 | no | — |
| 74 | C23×C5 | — | 4 | 1 | no | — |
| 75 | C23×C5 | #104 | 1 | 1 | yes | characteristic |
| 76 | C22×D5 | — | 2 | 1 | no | — |
| 77 | C2×AGL(1,𝔽5) | — | 8 | 2 | no | — |
| 78 | C5×Q8 | — | 4 | 1 | no | — |
| 79 | C5×Q8 | — | 4 | 1 | no | — |
| 80 | C5×D4 | — | 4 | 1 | no | — |
| 81 | C5×D4 | — | 4 | 1 | no | — |
| 82 | C5×D4 | — | 8 | 1 | no | — |
| 83 | C5×D4 | — | 8 | 1 | no | — |
| 84 | C2×C4×C5 | — | 2 | 1 | no | — |
| 85 | C2×C4×C5 | — | 2 | 1 | no | — |
| 86 | C2×C4×C5 | — | 4 | 1 | no | — |
| 87 | C2×C4×C5 | — | 8 | 1 | no | — |
| 88 | C2×C4×C5 | — | 8 | 1 | no | — |
| 89 | C2×C4×C5 | #99 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C5 | #102 | 1 | 1 | yes | characteristic |
| 91 | C22⋊D5 | — | 8 | 1 | no | — |
| 92 | C2×Dic5 | — | 2 | 1 | no | — |
| 93 | D20 | — | 4 | 1 | no | — |
| 94 | C4×D5 | — | 4 | 1 | no | — |
| 95 | C4×D5 | — | 4 | 1 | no | — |
| 96 | Dic10 | — | 4 | 1 | no | — |
| 97 | C5×C8 | — | 4 | 1 | no | — |
| 98 | #168 | — | 4 | 2 | no | — |
| 99 | #168 | — | 4 | 1 | no | — |
| 100 | C2×#41 | — | 2 | 1 | no | — |
| 101 | C2×#41 | — | 5 | 1 | no | — |
| 102 | #137 | — | 4 | 1 | no | — |
| 103 | #137 | — | 20 | 1 | no | — |
| 104 | #137 | — | 20 | 1 | no | — |
| 105 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 106 | AGL(1,ℤ/8) | — | 20 | 1 | no | — |
| 107 | AGL(1,ℤ/8) | — | 20 | 1 | no | — |
| 108 | #135 | — | 4 | 1 | no | — |
| 109 | #135 | — | 20 | 2 | no | — |
| 110 | C2×M16 | — | 10 | 1 | no | — |
| 111 | C2×M16 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 112 | C22×C8 | — | 5 | 1 | no | — |
| 113 | #117 | — | 20 | 1 | no | — |
| 114 | C2×C16 | — | 20 | 1 | no | — |
| 115 | #99 | — | 10 | 2 | no | — |
| 116 | #99 | — | 20 | 1 | no | — |
| 117 | C22⋊C8 | — | 20 | 1 | no | — |
| 118 | C22⋊C8 | — | 20 | 2 | no | — |
| 119 | C22×C5 | — | 2 | 1 | no | — |
| 120 | C22×C5 | — | 4 | 1 | no | — |
| 121 | C22×C5 | — | 8 | 2 | no | — |
| 122 | C22×C5 | — | 8 | 1 | no | — |
| 123 | C22×C5 | #327 | 1 | 1 | yes | characteristic |
| 124 | C2×D5 | — | 4 | 1 | no | — |
| 125 | C2×D5 | — | 8 | 1 | no | — |
| 126 | AGL(1,𝔽5) | — | 16 | 2 | no | — |
| 127 | C4×C5 | — | 2 | 1 | no | — |
| 128 | C4×C5 | — | 8 | 1 | no | — |
| 129 | C4×C5 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 130 | C4×C5 | #353 | 1 | 1 | yes | characteristic |
| 131 | Dic5 | — | 4 | 1 | no | — |
| 132 | #41 | — | 4 | 1 | no | — |
| 133 | #41 | — | 4 | 1 | no | — |
| 134 | #41 | — | 4 | 1 | no | — |
| 135 | #41 | — | 4 | 1 | no | — |
| 136 | #41 | — | 20 | 1 | no | — |
| 137 | #41 | — | 20 | 1 | no | — |
| 138 | C2×Q8 | — | 2 | 1 | no | — |
| 139 | C2×Q8 | — | 5 | 1 | no | — |
| 140 | C2×D4 | — | 2 | 1 | no | — |
| 141 | C2×D4 | — | 4 | 1 | no | — |
| 142 | C2×D4 | — | 5 | 1 | no | — |
| 143 | C2×D4 | — | 10 | 1 | no | — |
| 144 | C22×C4 | — | 4 | 1 | no | — |
| 145 | C22×C4 | — | 10 | 1 | no | — |
| 146 | C22×C4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 147 | Q16 | — | 4 | 1 | no | — |
| 148 | Q16 | — | 20 | 2 | no | — |
| 149 | SD16 | — | 4 | 2 | no | — |
| 150 | SD16 | — | 20 | 2 | no | — |
| 151 | SD16 | — | 20 | 2 | no | — |
| 152 | D8 | — | 4 | 1 | no | — |
| 153 | D8 | — | 20 | 2 | no | — |
| 154 | M16 | — | 4 | 1 | no | — |
| 155 | M16 | — | 20 | 1 | no | — |
| 156 | M16 | — | 20 | 1 | no | — |
| 157 | C2×C8 | — | 2 | 1 | no | — |
| 158 | C2×C8 | — | 10 | 2 | no | — |
| 159 | C2×C8 | — | 10 | 1 | no | — |
| 160 | C2×C8 | — | 20 | 1 | no | — |
| 161 | C4⋊C4 | — | 40 | 1 | no | — |
| 162 | C22⋊C4 | — | 20 | 2 | no | — |
| 163 | C42 | — | 40 | 1 | no | — |
| 164 | C16 | — | 40 | 1 | no | — |
| 165 | C2×C5 | — | 2 | 1 | no | — |
| 166 | C2×C5 | — | 4 | 1 | no | — |
| 167 | C2×C5 | — | 16 | 1 | no | — |
| 168 | C2×C5 | #1343 | 1 | 1 | yes | characteristic |
| 169 | D5 | — | 8 | 1 | no | — |
| 170 | C23 | — | 4 | 1 | no | — |
| 171 | C23 | — | 10 | 1 | no | — |
| 172 | C23 | #4008 | 1 | 1 | yes | characteristic |
| 173 | Q8 | — | 4 | 1 | no | — |
| 174 | Q8 | — | 4 | 1 | no | — |
| 175 | Q8 | — | 20 | 1 | no | — |
| 176 | D4 | — | 4 | 1 | no | — |
| 177 | D4 | — | 4 | 1 | no | — |
| 178 | D4 | — | 8 | 1 | no | — |
| 179 | D4 | — | 8 | 1 | no | — |
| 180 | D4 | — | 20 | 1 | no | — |
| 181 | D4 | — | 40 | 1 | no | — |
| 182 | C2×C4 | — | 2 | 1 | no | — |
| 183 | C2×C4 | — | 2 | 1 | no | — |
| 184 | C2×C4 | — | 4 | 1 | no | — |
| 185 | C2×C4 | — | 8 | 1 | no | — |
| 186 | C2×C4 | — | 8 | 1 | no | — |
| 187 | C2×C4 | — | 10 | 1 | no | — |
| 188 | C2×C4 | — | 20 | 1 | no | — |
| 189 | C2×C4 | — | 20 | 1 | no | — |
| 190 | C2×C4 | — | 40 | 2 | no | — |
| 191 | C2×C4 | #4007 | 1 | 1 | yes | characteristic |
| 192 | C2×C4 | #4011 | 1 | 1 | yes | characteristic |
| 193 | C8 | — | 4 | 1 | no | — |
| 194 | C8 | — | 20 | 2 | no | — |
| 195 | C8 | — | 20 | 1 | no | — |
| 196 | C5 | #7406 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 197 | C22 | — | 2 | 1 | no | — |
| 198 | C22 | — | 4 | 1 | no | — |
| 199 | C22 | — | 8 | 2 | no | — |
| 200 | C22 | — | 8 | 1 | no | — |
| 201 | C22 | — | 20 | 1 | no | — |
| 202 | C22 | — | 40 | 1 | no | — |
| 203 | C22 | #64920 | 1 | 1 | yes | characteristic |
| 204 | C4 | — | 2 | 1 | no | — |
| 205 | C4 | — | 8 | 1 | no | — |
| 206 | C4 | — | 20 | 1 | no | — |
| 207 | C4 | — | 80 | 2 | no | — |
| 208 | C4 | #64922 | 1 | 1 | yes | characteristic |
| 209 | C4 | #64934 | 1 | 1 | yes | characteristic |
| 210 | C2 | — | 2 | 1 | no | — |
| 211 | C2 | — | 4 | 1 | no | — |
| 212 | C2 | — | 16 | 1 | no | — |
| 213 | C2 | — | 40 | 1 | no | — |
| 214 | C2 | #10598825 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10598790 | C2 | semidirect |
| 3. #10598781 | C2 | semidirect |
| 4. C5×#575 | C4 | semidirect |
| 10. #64780 | C4 | semidirect |
| 68. #575 | AGL(1,𝔽5) | semidirect |
| 196. C5 | #7406 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10602914 | 2 |
| #10598790 | 2 |
| #10598781 | 2 |
| #7406 | 5 |