← C23×#62635 #19210261
Subgroups — C23×#62635 #19210261
Order 5888.
2038 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×#3584 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C23×#3417 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C23×#3416 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | C23×#3415 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C23×#3414 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×C23×#508 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×C23×#504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C23×#585 | — | 2 | 2 | no | — |
| 9 | C23×#578 | C22 | 1 | 8 | yes | complemented |
| 10 | C23×#577 | — | 2 | 8 | no | — |
| 11 | C23×#576 | C22 | 1 | 8 | yes | complemented |
| 12 | C23×#575 | — | 2 | 6 | no | — |
| 13 | C2×C23×#135 | C22 | 1 | 12 | yes | complemented |
| 14 | C2×C23×#131 | C22 | 1 | 1 | yes | characteristic |
| 15 | C23×#510 | C22 | 1 | 24 | yes | complemented |
| 16 | C23×#509 | C22 | 1 | 24 | yes | complemented |
| 17 | C23×#508 | C22 | 1 | 4 | yes | complemented |
| 18 | C23×#508 | C22 | 1 | 12 | yes | complemented |
| 19 | C23×#508 | C22 | 1 | 12 | yes | complemented |
| 20 | C23×#508 | C22 | 1 | 12 | yes | complemented |
| 21 | C2×C23×Q32 | C22 | 1 | 6 | yes | complemented |
| 22 | C2×C23×SD32 | C22 | 1 | 12 | yes | complemented |
| 23 | C2×C23×D16 | C22 | 1 | 6 | yes | complemented |
| 24 | C23×#504 | C22 | 1 | 4 | yes | complemented |
| 25 | C23×#504 | C22 | 1 | 4 | yes | complemented |
| 26 | C2×C23×M32 | C22 | 1 | 3 | yes | complemented |
| 27 | C22×C16×C23 | C22 | 1 | 3 | yes | complemented |
| 28 | C23×2−1+4 | — | 2 | 8 | no | — |
| 29 | C23×2−1+4 | — | 4 | 2 | no | — |
| 30 | C23×H(5,𝔽2) | — | 2 | 8 | no | — |
| 31 | C23×H(5,𝔽2) | — | 4 | 6 | no | — |
| 32 | C2×#41×C23 | — | 2 | 12 | no | — |
| 33 | C2×#41×C23 | — | 4 | 8 | no | — |
| 34 | C2×#41×C23 | D4 | 1 | 1 | yes | characteristic |
| 35 | C23×#137 | — | 2 | 24 | no | — |
| 36 | C23×AGL(1,ℤ/8) | — | 2 | 24 | no | — |
| 37 | C23×#135 | — | 2 | 12 | no | — |
| 38 | C23×#135 | — | 2 | 12 | no | — |
| 39 | C23×#135 | C23 | 1 | 8 | yes | — |
| 40 | C23×#135 | C23 | 1 | 24 | yes | — |
| 41 | C2×Q16×C23 | C23 | 1 | 12 | yes | — |
| 42 | C2×SD16×C23 | — | 2 | 12 | no | — |
| 43 | C2×D8×C23 | C23 | 1 | 12 | yes | — |
| 44 | C23×#131 | D4 | 1 | 4 | yes | — |
| 45 | C23×#131 | C23 | 1 | 4 | yes | — |
| 46 | C2×M16×C23 | D4 | 1 | 3 | yes | — |
| 47 | C22×C8×C23 | C23 | 1 | 3 | yes | — |
| 48 | C23×Q32 | C23 | 1 | 4 | yes | — |
| 49 | C23×Q32 | C23 | 1 | 12 | yes | — |
| 50 | C23×SD32 | C23 | 1 | 8 | yes | — |
| 51 | C23×SD32 | C23 | 1 | 24 | yes | — |
| 52 | C23×D16 | C23 | 1 | 4 | yes | — |
| 53 | C23×D16 | C23 | 1 | 12 | yes | — |
| 54 | C23×M32 | C23 | 1 | 12 | yes | — |
| 55 | C2×C16×C23 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C16×C23 | C23 | 1 | 3 | yes | — |
| 57 | C2×C16×C23 | C23 | 1 | 6 | yes | — |
| 58 | C2×C16×C23 | C23 | 1 | 6 | yes | — |
| 59 | #41×C23 | — | 2 | 8 | no | — |
| 60 | #41×C23 | — | 2 | 24 | no | — |
| 61 | #41×C23 | — | 4 | 8 | no | — |
| 62 | #41×C23 | — | 4 | 24 | no | — |
| 63 | #41×C23 | — | 4 | 12 | no | — |
| 64 | #41×C23 | — | 4 | 12 | no | — |
| 65 | #41×C23 | D8 | 1 | 4 | yes | — |
| 66 | #41×C23 | C2×D4 | 1 | 4 | yes | — |
| 67 | C2×Q8×C23 | — | 2 | 12 | no | — |
| 68 | C2×Q8×C23 | — | 4 | 8 | no | — |
| 69 | C2×Q8×C23 | D8 | 1 | 1 | yes | characteristic |
| 70 | C2×D4×C23 | — | 2 | 12 | no | — |
| 71 | C2×D4×C23 | — | 4 | 12 | no | — |
| 72 | C2×D4×C23 | — | 4 | 24 | no | — |
| 73 | C2×D4×C23 | D8 | 1 | 3 | yes | — |
| 74 | C22×C4×C23 | — | 4 | 12 | no | — |
| 75 | C22×C4×C23 | C2×D4 | 1 | 3 | yes | — |
| 76 | Q16×C23 | — | 2 | 12 | no | — |
| 77 | Q16×C23 | C24 | 1 | 8 | yes | — |
| 78 | SD16×C23 | — | 2 | 8 | no | — |
| 79 | SD16×C23 | — | 2 | 24 | no | — |
| 80 | D8×C23 | — | 2 | 12 | no | — |
| 81 | D8×C23 | C24 | 1 | 8 | yes | — |
| 82 | M16×C23 | C2×D4 | 1 | 12 | yes | — |
| 83 | C2×C8×C23 | C2×D4 | 1 | 3 | yes | — |
| 84 | C2×C8×C23 | C2×D4 | 1 | 6 | yes | — |
| 85 | C2×C8×C23 | C24 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C23 | C24 | 1 | 6 | yes | — |
| 87 | C16×C23 | C24 | 1 | 1 | yes | characteristic |
| 88 | C16×C23 | C24 | 1 | 1 | yes | characteristic |
| 89 | C16×C23 | C24 | 1 | 6 | yes | — |
| 90 | #62635 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 91 | C23×C23 | — | 4 | 12 | no | — |
| 92 | C23×C23 | C2×D8 | 1 | 3 | yes | — |
| 93 | Q8×C23 | — | 2 | 8 | no | — |
| 94 | Q8×C23 | — | 4 | 12 | no | — |
| 95 | Q8×C23 | C2×D8 | 1 | 4 | yes | — |
| 96 | D4×C23 | — | 2 | 8 | no | — |
| 97 | D4×C23 | — | 4 | 24 | no | — |
| 98 | D4×C23 | — | 4 | 8 | no | — |
| 99 | D4×C23 | — | 4 | 12 | no | — |
| 100 | D4×C23 | C2×D8 | 1 | 12 | yes | — |
| 101 | C2×C4×C23 | — | 4 | 12 | no | — |
| 102 | C2×C4×C23 | — | 4 | 24 | no | — |
| 103 | C2×C4×C23 | — | 4 | 8 | no | — |
| 104 | C2×C4×C23 | C2×D8 | 1 | 3 | yes | — |
| 105 | C2×C4×C23 | C2×D8 | 1 | 6 | yes | — |
| 106 | C2×C4×C23 | C22×D4 | 1 | 6 | yes | — |
| 107 | C2×C4×C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 108 | C8×C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 109 | C8×C23 | C22×D4 | 1 | 6 | yes | — |
| 110 | C8×C23 | C25 | 1 | 1 | yes | characteristic |
| 111 | #3584 | C2×C23 | 1 | 2 | yes | complemented |
| 112 | #3417 | C2×C23 | 1 | 4 | yes | complemented |
| 113 | #3416 | C2×C23 | 1 | 8 | yes | complemented |
| 114 | #3415 | C2×C23 | 1 | 4 | yes | complemented |
| 115 | #3414 | C2×C23 | 1 | 6 | yes | complemented |
| 116 | C2×#508 | C2×C23 | 1 | 6 | yes | complemented |
| 117 | C2×#504 | C2×C23 | 1 | 1 | yes | characteristic, complemented |
| 118 | C22×C23 | — | 4 | 3 | no | — |
| 119 | C22×C23 | — | 4 | 8 | no | — |
| 120 | C22×C23 | — | 16 | 12 | no | — |
| 121 | C22×C23 | C22×D8 | 1 | 6 | yes | — |
| 122 | C22×C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 123 | C4×C23 | — | 4 | 8 | no | — |
| 124 | C4×C23 | C22×D8 | 1 | 6 | yes | — |
| 125 | C4×C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 126 | C4×C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 127 | #585 | — | 2 | 2 | no | — |
| 128 | #578 | C22×C23 | 1 | 8 | yes | complemented |
| 129 | #577 | — | 2 | 8 | no | — |
| 130 | #576 | C22×C23 | 1 | 8 | yes | complemented |
| 131 | #575 | — | 2 | 6 | no | — |
| 132 | C2×#135 | C22×C23 | 1 | 12 | yes | complemented |
| 133 | C2×#131 | C22×C23 | 1 | 1 | yes | characteristic |
| 134 | #510 | C22×C23 | 1 | 24 | yes | complemented |
| 135 | #509 | C22×C23 | 1 | 24 | yes | complemented |
| 136 | #508 | C22×C23 | 1 | 4 | yes | complemented |
| 137 | #508 | C22×C23 | 1 | 12 | yes | complemented |
| 138 | #508 | C22×C23 | 1 | 12 | yes | complemented |
| 139 | #508 | C22×C23 | 1 | 12 | yes | complemented |
| 140 | C2×Q32 | C22×C23 | 1 | 6 | yes | complemented |
| 141 | C2×SD32 | C22×C23 | 1 | 12 | yes | complemented |
| 142 | C2×D16 | C22×C23 | 1 | 6 | yes | complemented |
| 143 | #504 | C22×C23 | 1 | 4 | yes | complemented |
| 144 | #504 | C22×C23 | 1 | 4 | yes | complemented |
| 145 | C2×M32 | C22×C23 | 1 | 3 | yes | complemented |
| 146 | C22×C16 | C22×C23 | 1 | 3 | yes | complemented |
| 147 | C2×C23 | — | 2 | 1 | no | — |
| 148 | C2×C23 | — | 2 | 6 | no | — |
| 149 | C2×C23 | — | 8 | 8 | no | — |
| 150 | C2×C23 | C23×D8 | 1 | 1 | yes | characteristic |
| 151 | 2−1+4 | — | 2 | 8 | no | — |
| 152 | 2−1+4 | — | 4 | 2 | no | — |
| 153 | H(5,𝔽2) | — | 2 | 8 | no | — |
| 154 | H(5,𝔽2) | — | 4 | 6 | no | — |
| 155 | C2×#41 | — | 2 | 12 | no | — |
| 156 | C2×#41 | — | 4 | 8 | no | — |
| 157 | C2×#41 | D4×C23 | 1 | 1 | yes | characteristic |
| 158 | #137 | — | 2 | 24 | no | — |
| 159 | AGL(1,ℤ/8) | — | 2 | 24 | no | — |
| 160 | #135 | — | 2 | 12 | no | — |
| 161 | #135 | — | 2 | 12 | no | — |
| 162 | #135 | C23×C23 | 1 | 8 | yes | — |
| 163 | #135 | C23×C23 | 1 | 24 | yes | — |
| 164 | C2×Q16 | C23×C23 | 1 | 12 | yes | — |
| 165 | C2×SD16 | — | 2 | 12 | no | — |
| 166 | C2×D8 | C23×C23 | 1 | 12 | yes | — |
| 167 | #131 | D4×C23 | 1 | 4 | yes | — |
| 168 | #131 | C23×C23 | 1 | 4 | yes | — |
| 169 | C2×M16 | D4×C23 | 1 | 3 | yes | — |
| 170 | C22×C8 | C23×C23 | 1 | 3 | yes | — |
| 171 | Q32 | C23×C23 | 1 | 4 | yes | — |
| 172 | Q32 | C23×C23 | 1 | 12 | yes | — |
| 173 | SD32 | C23×C23 | 1 | 8 | yes | — |
| 174 | SD32 | C23×C23 | 1 | 24 | yes | — |
| 175 | D16 | C23×C23 | 1 | 4 | yes | — |
| 176 | D16 | C23×C23 | 1 | 12 | yes | — |
| 177 | M32 | C23×C23 | 1 | 12 | yes | — |
| 178 | C2×C16 | C23×C23 | 1 | 1 | yes | characteristic |
| 179 | C2×C16 | C23×C23 | 1 | 3 | yes | — |
| 180 | C2×C16 | C23×C23 | 1 | 6 | yes | — |
| 181 | C2×C16 | C23×C23 | 1 | 6 | yes | — |
| 182 | C23 | #62635 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 183 | #41 | — | 2 | 8 | no | — |
| 184 | #41 | — | 2 | 24 | no | — |
| 185 | #41 | — | 4 | 8 | no | — |
| 186 | #41 | — | 4 | 24 | no | — |
| 187 | #41 | — | 4 | 12 | no | — |
| 188 | #41 | — | 4 | 12 | no | — |
| 189 | #41 | D8×C23 | 1 | 4 | yes | — |
| 190 | #41 | C2×D4×C23 | 1 | 4 | yes | — |
| 191 | C2×Q8 | — | 2 | 12 | no | — |
| 192 | C2×Q8 | — | 4 | 8 | no | — |
| 193 | C2×Q8 | D8×C23 | 1 | 1 | yes | characteristic |
| 194 | C2×D4 | — | 2 | 12 | no | — |
| 195 | C2×D4 | — | 4 | 12 | no | — |
| 196 | C2×D4 | — | 4 | 24 | no | — |
| 197 | C2×D4 | D8×C23 | 1 | 3 | yes | — |
| 198 | C22×C4 | — | 4 | 12 | no | — |
| 199 | C22×C4 | C2×D4×C23 | 1 | 3 | yes | — |
| 200 | Q16 | — | 2 | 12 | no | — |
| 201 | Q16 | C24×C23 | 1 | 8 | yes | — |
| 202 | SD16 | — | 2 | 8 | no | — |
| 203 | SD16 | — | 2 | 24 | no | — |
| 204 | D8 | — | 2 | 12 | no | — |
| 205 | D8 | C24×C23 | 1 | 8 | yes | — |
| 206 | M16 | C2×D4×C23 | 1 | 12 | yes | — |
| 207 | C2×C8 | C2×D4×C23 | 1 | 3 | yes | — |
| 208 | C2×C8 | C2×D4×C23 | 1 | 6 | yes | — |
| 209 | C2×C8 | C24×C23 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | C24×C23 | 1 | 6 | yes | — |
| 211 | C16 | C24×C23 | 1 | 1 | yes | characteristic |
| 212 | C16 | C24×C23 | 1 | 1 | yes | characteristic |
| 213 | C16 | C24×C23 | 1 | 6 | yes | — |
| 214 | C23 | — | 4 | 12 | no | — |
| 215 | C23 | C2×D8×C23 | 1 | 3 | yes | — |
| 216 | Q8 | — | 2 | 8 | no | — |
| 217 | Q8 | — | 4 | 12 | no | — |
| 218 | Q8 | C2×D8×C23 | 1 | 4 | yes | — |
| 219 | D4 | — | 2 | 8 | no | — |
| 220 | D4 | — | 4 | 24 | no | — |
| 221 | D4 | — | 4 | 8 | no | — |
| 222 | D4 | — | 4 | 12 | no | — |
| 223 | D4 | C2×D8×C23 | 1 | 12 | yes | — |
| 224 | C2×C4 | — | 4 | 12 | no | — |
| 225 | C2×C4 | — | 4 | 8 | no | — |
| 226 | C2×C4 | — | 4 | 24 | no | — |
| 227 | C2×C4 | C2×D8×C23 | 1 | 6 | yes | — |
| 228 | C2×C4 | C2×D8×C23 | 1 | 3 | yes | — |
| 229 | C2×C4 | C22×D4×C23 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | C22×D4×C23 | 1 | 6 | yes | — |
| 231 | C8 | C22×D4×C23 | 1 | 1 | yes | characteristic |
| 232 | C8 | C22×D4×C23 | 1 | 6 | yes | — |
| 233 | C8 | C25×C23 | 1 | 1 | yes | characteristic |
| 234 | C22 | — | 4 | 3 | no | — |
| 235 | C22 | — | 4 | 8 | no | — |
| 236 | C22 | — | 16 | 12 | no | — |
| 237 | C22 | C22×D8×C23 | 1 | 6 | yes | — |
| 238 | C22 | C22×D8×C23 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 4 | 8 | no | — |
| 240 | C4 | C22×D8×C23 | 1 | 1 | yes | characteristic |
| 241 | C4 | C22×D8×C23 | 1 | 6 | yes | — |
| 242 | C4 | C23×D4×C23 | 1 | 1 | yes | characteristic |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | — | 2 | 6 | no | — |
| 245 | C2 | — | 8 | 8 | no | — |
| 246 | C2 | C23×D8×C23 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta