Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #10617378 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604663 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10604019 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10599275 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10598583 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #66059 | C22 | 1 | 1 | yes | characteristic |
| 7 | #65787 | — | 2 | 2 | no | — |
| 8 | C2×#4010 | — | 2 | 2 | no | — |
| 9 | #65705 | — | 2 | 2 | no | — |
| 10 | #65702 | — | 2 | 2 | no | — |
| 11 | C42×AGL(1,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 12 | #65681 | C22 | 1 | 1 | yes | characteristic |
| 13 | #65680 | C22 | 1 | 2 | yes | — |
| 14 | #65122 | C22 | 1 | 2 | yes | — |
| 15 | #64884 | — | 2 | 2 | no | — |
| 16 | #64881 | — | 2 | 2 | no | — |
| 17 | C5×#333 | C4 | 1 | 1 | yes | characteristic, complemented |
| 18 | #64766 | C4 | 1 | 1 | yes | characteristic, complemented |
| 19 | #64705 | C4 | 1 | 2 | yes | complemented |
| 20 | #8935 | — | 5 | 1 | no | maximal, Sylow |
| 21 | C2×#713 | D4 | 1 | 2 | yes | — |
| 22 | C2×C4×AGL(1,𝔽5) | — | 2 | 1 | no | — |
| 23 | C2×C4×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 24 | C2×#700 | — | 2 | 1 | no | — |
| 25 | C2×#700 | — | 2 | 2 | no | — |
| 26 | C2×D5⋊C8 | — | 2 | 2 | no | — |
| 27 | C5×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #4093 | Q8 | 1 | 2 | yes | — |
| 29 | #4041 | — | 2 | 2 | no | — |
| 30 | #4039 | — | 2 | 2 | no | — |
| 31 | #4020 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C42×D5 | C23 | 1 | 1 | yes | characteristic |
| 33 | #4010 | — | 4 | 4 | no | — |
| 34 | #4003 | — | 2 | 4 | no | — |
| 35 | #4003 | — | 2 | 2 | no | — |
| 36 | C4⋊#170 | — | 2 | 2 | no | — |
| 37 | C4⋊#170 | D4 | 1 | 2 | yes | — |
| 38 | C4⋊#170 | D4 | 1 | 2 | yes | — |
| 39 | C4×#170 | — | 2 | 2 | no | — |
| 40 | C5×C4⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 41 | #3952 | — | 2 | 2 | no | — |
| 42 | C8×Dic5 | — | 2 | 2 | no | — |
| 43 | C4⋊#168 | C2×C4 | 1 | 2 | yes | — |
| 44 | #2979 | — | 5 | 1 | no | — |
| 45 | #1962 | — | 5 | 1 | no | — |
| 46 | #1574 | — | 5 | 2 | no | — |
| 47 | #1543 | — | 5 | 1 | no | — |
| 48 | #1276 | — | 5 | 2 | no | — |
| 49 | C22×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 50 | C2×C5×Q8 | C4⋊C4 | 1 | 2 | yes | — |
| 51 | #713 | — | 4 | 2 | no | — |
| 52 | #713 | — | 4 | 2 | no | — |
| 53 | C2×D20 | — | 2 | 2 | no | — |
| 54 | C2×D20 | — | 2 | 2 | no | — |
| 55 | C2×D20 | C4⋊C4 | 1 | 2 | yes | — |
| 56 | C2×C4×D5 | — | 2 | 2 | no | — |
| 57 | C2×C4×D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C2×C4×D5 | #41 | 1 | 2 | yes | — |
| 59 | C2×#170 | — | 2 | 2 | no | — |
| 60 | C2×#170 | — | 2 | 2 | no | — |
| 61 | C2×#170 | — | 4 | 1 | no | — |
| 62 | C4×AGL(1,𝔽5) | — | 2 | 4 | no | — |
| 63 | C4×AGL(1,𝔽5) | — | 2 | 2 | no | — |
| 64 | C4×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 65 | C4×AGL(1,𝔽5) | — | 4 | 1 | no | — |
| 66 | #700 | — | 4 | 2 | no | — |
| 67 | #700 | — | 8 | 2 | no | — |
| 68 | D5⋊C8 | — | 8 | 2 | no | — |
| 69 | C2×C5×C8 | — | 2 | 2 | no | — |
| 70 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 71 | C42×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | #685 | — | 2 | 4 | no | — |
| 73 | C4×Dic5 | C2×D4 | 1 | 2 | yes | — |
| 74 | C4×Dic5 | C2×Q8 | 1 | 1 | yes | characteristic |
| 75 | C2×#168 | — | 2 | 2 | no | — |
| 76 | #532 | — | 5 | 1 | no | — |
| 77 | #436 | — | 10 | 2 | no | — |
| 78 | C8×D4 | — | 10 | 2 | no | — |
| 79 | #431 | — | 5 | 2 | no | — |
| 80 | #431 | — | 5 | 2 | no | — |
| 81 | #423 | — | 5 | 1 | no | — |
| 82 | C2×#104 | — | 10 | 2 | no | — |
| 83 | #408 | — | 10 | 2 | no | — |
| 84 | C43 | — | 5 | 1 | no | — |
| 85 | #335 | — | 10 | 2 | no | — |
| 86 | #334 | — | 10 | 2 | no | — |
| 87 | #333 | — | 5 | 1 | no | — |
| 88 | #333 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 89 | #331 | — | 5 | 2 | no | — |
| 90 | C22×D5 | — | 2 | 2 | no | — |
| 91 | C22×D5 | #123 | 1 | 1 | yes | characteristic |
| 92 | C2×AGL(1,𝔽5) | — | 4 | 2 | no | — |
| 93 | C2×AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 94 | C5×Q8 | — | 4 | 2 | no | — |
| 95 | C2×C4×C5 | — | 2 | 2 | no | — |
| 96 | C2×C4×C5 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C5 | C4×D4 | 1 | 2 | yes | — |
| 98 | C2×Dic5 | #120 | 1 | 1 | yes | characteristic |
| 99 | C2×Dic5 | C22⋊Q8 | 1 | 2 | yes | — |
| 100 | D20 | — | 4 | 2 | no | — |
| 101 | D20 | — | 4 | 2 | no | — |
| 102 | D20 | — | 4 | 2 | no | — |
| 103 | D20 | — | 8 | 2 | no | — |
| 104 | C4×D5 | — | 2 | 2 | no | — |
| 105 | C4×D5 | — | 2 | 2 | no | — |
| 106 | C4×D5 | — | 2 | 1 | no | — |
| 107 | C4×D5 | — | 2 | 1 | no | — |
| 108 | C4×D5 | — | 4 | 2 | no | — |
| 109 | C4×D5 | — | 4 | 2 | no | — |
| 110 | #170 | — | 4 | 2 | no | — |
| 111 | #170 | — | 4 | 2 | no | — |
| 112 | #170 | — | 8 | 1 | no | — |
| 113 | C5×C8 | — | 4 | 2 | no | — |
| 114 | #168 | — | 4 | 2 | no | — |
| 115 | C2×#41 | — | 5 | 2 | no | — |
| 116 | C2×M16 | — | 10 | 2 | no | — |
| 117 | C2×M16 | — | 10 | 1 | no | — |
| 118 | C22×C8 | — | 10 | 2 | no | — |
| 119 | C4⋊Q8 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 120 | #127 | — | 5 | 1 | no | — |
| 121 | #124 | — | 5 | 2 | no | — |
| 122 | #121 | — | 10 | 2 | no | — |
| 123 | C4×D4 | — | 10 | 2 | no | — |
| 124 | C2×C42 | — | 5 | 1 | no | — |
| 125 | C2×C42 | — | 10 | 2 | no | — |
| 126 | C2×C42 | — | 10 | 1 | no | — |
| 127 | C4⋊C8 | — | 5 | 2 | no | — |
| 128 | C4⋊C8 | — | 5 | 2 | no | — |
| 129 | C4⋊C8 | — | 5 | 2 | no | — |
| 130 | C4⋊C8 | — | 10 | 2 | no | — |
| 131 | C4⋊C8 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 132 | #104 | — | 20 | 4 | no | — |
| 133 | C22⋊C8 | — | 10 | 2 | no | — |
| 134 | C22⋊C8 | — | 10 | 2 | no | — |
| 135 | C22⋊C8 | — | 10 | 4 | no | — |
| 136 | C4×C8 | — | 10 | 2 | no | — |
| 137 | C4×C8 | — | 10 | 2 | no | — |
| 138 | C22×C5 | #385 | 1 | 1 | yes | characteristic |
| 139 | C2×D5 | — | 2 | 1 | no | — |
| 140 | C2×D5 | — | 2 | 2 | no | — |
| 141 | C2×D5 | — | 4 | 2 | no | — |
| 142 | C2×D5 | — | 4 | 4 | no | — |
| 143 | AGL(1,𝔽5) | — | 4 | 4 | no | — |
| 144 | C4×C5 | — | 2 | 1 | no | — |
| 145 | C4×C5 | — | 4 | 2 | no | — |
| 146 | C4×C5 | #443 | 1 | 2 | yes | — |
| 147 | C4×C5 | #444 | 1 | 2 | yes | — |
| 148 | Dic5 | — | 2 | 2 | no | — |
| 149 | Dic5 | #453 | 1 | 1 | yes | characteristic |
| 150 | Dic5 | #455 | 1 | 1 | yes | characteristic |
| 151 | #41 | — | 20 | 2 | no | — |
| 152 | #41 | — | 20 | 2 | no | — |
| 153 | C2×Q8 | C4⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 154 | C2×D4 | — | 5 | 2 | no | — |
| 155 | C2×D4 | — | 10 | 2 | no | — |
| 156 | C2×D4 | — | 10 | 2 | no | — |
| 157 | C22×C4 | — | 5 | 1 | no | — |
| 158 | C22×C4 | — | 5 | 2 | no | — |
| 159 | C22×C4 | — | 10 | 2 | no | — |
| 160 | C22×C4 | — | 20 | 1 | no | — |
| 161 | M16 | — | 20 | 2 | no | — |
| 162 | M16 | — | 40 | 2 | no | — |
| 163 | C2×C8 | — | 2 | 2 | no | — |
| 164 | C2×C8 | — | 10 | 2 | no | — |
| 165 | C2×C8 | — | 10 | 2 | no | — |
| 166 | C2×C8 | — | 10 | 2 | no | — |
| 167 | C2×C8 | — | 20 | 1 | no | — |
| 168 | C2×C8 | — | 40 | 2 | no | — |
| 169 | C4⋊C4 | — | 2 | 2 | no | — |
| 170 | C22⋊C4 | — | 10 | 4 | no | — |
| 171 | C42 | — | 5 | 1 | no | — |
| 172 | C42 | — | 5 | 2 | no | — |
| 173 | C42 | — | 10 | 2 | no | — |
| 174 | C42 | — | 10 | 4 | no | — |
| 175 | C42 | — | 20 | 1 | no | — |
| 176 | C42 | — | 20 | 2 | no | — |
| 177 | C42 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 178 | C2×C5 | #1856 | 1 | 1 | yes | characteristic |
| 179 | C2×C5 | #1866 | 1 | 2 | yes | — |
| 180 | D5 | — | 4 | 1 | no | — |
| 181 | D5 | — | 8 | 2 | no | — |
| 182 | C23 | — | 5 | 1 | no | — |
| 183 | C23 | — | 10 | 2 | no | — |
| 184 | Q8 | — | 4 | 2 | no | — |
| 185 | D4 | — | 20 | 2 | no | — |
| 186 | D4 | — | 20 | 2 | no | — |
| 187 | D4 | — | 20 | 2 | no | — |
| 188 | D4 | — | 40 | 2 | no | — |
| 189 | C2×C4 | — | 2 | 2 | no | — |
| 190 | C2×C4 | — | 5 | 1 | no | — |
| 191 | C2×C4 | — | 5 | 2 | no | — |
| 192 | C2×C4 | — | 10 | 1 | no | — |
| 193 | C2×C4 | — | 10 | 1 | no | — |
| 194 | C2×C4 | — | 10 | 2 | no | — |
| 195 | C2×C4 | — | 10 | 2 | no | — |
| 196 | C2×C4 | — | 20 | 2 | no | — |
| 197 | C2×C4 | — | 20 | 2 | no | — |
| 198 | C2×C4 | — | 20 | 2 | no | — |
| 199 | C2×C4 | — | 20 | 4 | no | — |
| 200 | C2×C4 | C2×C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 202 | C8 | — | 4 | 2 | no | — |
| 203 | C8 | — | 20 | 2 | no | — |
| 204 | C8 | — | 20 | 2 | no | — |
| 205 | C8 | — | 20 | 2 | no | — |
| 206 | C8 | — | 40 | 1 | no | — |
| 207 | C5 | #8935 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 208 | C22 | — | 10 | 1 | no | — |
| 209 | C22 | — | 10 | 2 | no | — |
| 210 | C22 | — | 20 | 2 | no | — |
| 211 | C22 | — | 20 | 4 | no | — |
| 212 | C22 | #65699 | 1 | 1 | yes | characteristic |
| 213 | C4 | — | 2 | 1 | no | — |
| 214 | C4 | — | 4 | 2 | no | — |
| 215 | C4 | — | 5 | 1 | no | — |
| 216 | C4 | — | 5 | 1 | no | — |
| 217 | C4 | — | 10 | 2 | no | — |
| 218 | C4 | — | 20 | 4 | no | — |
| 219 | C4 | #65736 | 1 | 2 | yes | — |
| 220 | C4 | #65741 | 1 | 2 | yes | — |
| 221 | C2 | — | 20 | 1 | no | — |
| 222 | C2 | — | 40 | 2 | no | — |
| 223 | C2 | #10603925 | 1 | 1 | yes | characteristic |
| 224 | C2 | #10603997 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10604019 | C2 | semidirect |
| 5. #10598583 | C2 | semidirect |
| 17. C5×#333 | C4 | semidirect |
| 18. #64766 | C4 | semidirect |
| 19. #64705 | C4 | semidirect |
| 88. #333 | AGL(1,𝔽5) | semidirect |
| 207. C5 | #8935 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617378 | 2 |
| #10604663 | 2 |
| #10604019 | 2 |
| #10599275 | 2 |
| #10598583 | 2 |
| #8935 | 5 |