Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
696 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 136 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 | C22×#6045 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87880 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×#4075 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11717080 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11717075 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11717031 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #11717015 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #11716958 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C22×C4×C52 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×#1039 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×#1039 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#6045 | — | 2 | 4 | no | — |
| 13 | C2×#6045 | C22 | 1 | 2 | yes | — |
| 14 | C2×#6045 | C22 | 1 | 2 | yes | — |
| 15 | C2×C5×C22⋊D5 | C22 | 1 | 2 | yes | — |
| 16 | C2×#6039 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#6037 | C22 | 1 | 1 | yes | characteristic |
| 18 | #87880 | — | 2 | 4 | no | — |
| 19 | C2×D5×Dic5 | C22 | 1 | 2 | yes | — |
| 20 | C5×C22⋊Dic5 | C22 | 1 | 2 | yes | — |
| 21 | C5×#685 | C22 | 1 | 4 | yes | — |
| 22 | C5×#683 | C22 | 1 | 4 | yes | — |
| 23 | #87781 | C22 | 1 | 2 | yes | — |
| 24 | #87780 | C22 | 1 | 4 | yes | — |
| 25 | #87778 | C22 | 1 | 4 | yes | — |
| 26 | Dic52 | C22 | 1 | 2 | yes | — |
| 27 | #65982 | — | 5 | 2 | no | maximal |
| 28 | C23×C52 | C23 | 1 | 1 | yes | characteristic |
| 29 | C22×C5×D5 | C23 | 1 | 2 | yes | — |
| 30 | C2×C4×C52 | — | 2 | 2 | no | — |
| 31 | C2×C4×C52 | C23 | 1 | 2 | yes | — |
| 32 | C2×#1039 | — | 2 | 2 | no | — |
| 33 | C2×#1039 | D4 | 1 | 4 | yes | — |
| 34 | C2×#1039 | C23 | 1 | 2 | yes | — |
| 35 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 37 | #6045 | — | 2 | 4 | no | — |
| 38 | #6045 | — | 2 | 4 | no | — |
| 39 | C5×C22⋊D5 | — | 2 | 4 | no | — |
| 40 | C2×C5×Dic5 | C23 | 1 | 2 | yes | — |
| 41 | C2×C5×Dic5 | C23 | 1 | 4 | yes | — |
| 42 | #6039 | — | 2 | 2 | no | — |
| 43 | #6037 | — | 2 | 2 | no | — |
| 44 | D5×Dic5 | — | 2 | 4 | no | — |
| 45 | C2×#711 | — | 5 | 2 | no | — |
| 46 | C22×Dic10 | — | 5 | 2 | no | — |
| 47 | C22×Dic10 | — | 10 | 2 | no | — |
| 48 | C5×#123 | — | 5 | 2 | no | — |
| 49 | #4081 | — | 5 | 2 | no | — |
| 50 | #4075 | D5 | 1 | 2 | yes | complemented |
| 51 | #4042 | — | 5 | 4 | no | — |
| 52 | #4034 | — | 5 | 4 | no | — |
| 53 | #4032 | — | 5 | 2 | no | — |
| 54 | #4031 | — | 5 | 4 | no | — |
| 55 | #4024 | — | 5 | 4 | no | — |
| 56 | #4023 | — | 5 | 2 | no | — |
| 57 | C22×C52 | — | 2 | 2 | no | — |
| 58 | C22×C52 | C2×D4 | 1 | 2 | yes | — |
| 59 | C22×C52 | C24 | 1 | 1 | yes | characteristic |
| 60 | C2×C5×D5 | — | 2 | 2 | no | — |
| 61 | C2×C5×D5 | — | 2 | 4 | no | — |
| 62 | C4×C52 | — | 2 | 2 | no | — |
| 63 | #1039 | — | 2 | 2 | no | — |
| 64 | #1039 | C2×D4 | 1 | 2 | yes | — |
| 65 | #1039 | C2×D4 | 1 | 2 | yes | — |
| 66 | C5×Dic5 | — | 2 | 2 | no | — |
| 67 | C5×Dic5 | — | 2 | 4 | no | — |
| 68 | C2×C5×D4 | — | 5 | 2 | no | — |
| 69 | C22×C4×C5 | — | 2 | 2 | no | — |
| 70 | C22×C4×C5 | C2×D5 | 1 | 2 | yes | — |
| 71 | C2×C22⋊D5 | — | 5 | 2 | no | — |
| 72 | C2×C22⋊D5 | C2×D5 | 1 | 2 | yes | — |
| 73 | C22×Dic5 | — | 5 | 2 | no | — |
| 74 | C22×Dic5 | — | 5 | 2 | no | — |
| 75 | C22×Dic5 | — | 5 | 2 | no | — |
| 76 | C22×Dic5 | — | 10 | 2 | no | — |
| 77 | C22×Dic5 | — | 10 | 2 | no | — |
| 78 | #711 | — | 10 | 8 | no | — |
| 79 | C2×C4×D5 | — | 5 | 2 | no | — |
| 80 | C2×Dic10 | — | 5 | 4 | no | — |
| 81 | C2×Dic10 | — | 5 | 4 | no | — |
| 82 | C2×Dic10 | — | 5 | 2 | no | — |
| 83 | C2×Dic10 | — | 10 | 4 | no | — |
| 84 | C2×Dic10 | — | 10 | 4 | no | — |
| 85 | C2×Dic10 | — | 10 | 8 | no | — |
| 86 | C2×Dic10 | — | 10 | 16 | no | — |
| 87 | C5×C4⋊C4 | — | 5 | 4 | no | — |
| 88 | C5×C22⋊C4 | — | 5 | 2 | no | — |
| 89 | C5×C22⋊C4 | — | 5 | 4 | no | — |
| 90 | C22⋊Dic5 | — | 5 | 4 | no | — |
| 91 | C22⋊Dic5 | C2×D5 | 1 | 2 | yes | — |
| 92 | #685 | — | 5 | 4 | no | — |
| 93 | #685 | C2×D5 | 1 | 4 | yes | — |
| 94 | C4⋊Dic5 | — | 5 | 4 | no | — |
| 95 | #683 | — | 5 | 4 | no | — |
| 96 | #683 | — | 5 | 4 | no | — |
| 97 | #683 | C2×D5 | 1 | 4 | yes | — |
| 98 | C4×Dic5 | — | 5 | 4 | no | — |
| 99 | #536 | — | 25 | 1 | no | Sylow |
| 100 | C2×C52 | — | 2 | 2 | no | — |
| 101 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C52 | 2−1+4 | 1 | 2 | yes | — |
| 103 | C5×D5 | — | 4 | 2 | no | — |
| 104 | C23×C5 | — | 2 | 2 | no | — |
| 105 | C23×C5 | — | 5 | 2 | no | — |
| 106 | C23×C5 | C22×D5 | 1 | 2 | yes | — |
| 107 | C22×D5 | C22×D5 | 1 | 2 | yes | — |
| 108 | C5×D4 | — | 10 | 4 | no | — |
| 109 | C2×C4×C5 | — | 2 | 8 | no | — |
| 110 | C2×C4×C5 | — | 2 | 4 | no | — |
| 111 | C2×C4×C5 | — | 2 | 4 | no | — |
| 112 | C2×C4×C5 | — | 5 | 4 | no | — |
| 113 | C2×C4×C5 | — | 5 | 2 | no | — |
| 114 | C2×C4×C5 | C22×D5 | 1 | 4 | yes | — |
| 115 | C22⋊D5 | — | 2 | 4 | no | — |
| 116 | C22⋊D5 | — | 10 | 4 | no | — |
| 117 | C2×Dic5 | — | 5 | 8 | no | — |
| 118 | C2×Dic5 | — | 5 | 2 | no | — |
| 119 | C2×Dic5 | — | 5 | 2 | no | — |
| 120 | C2×Dic5 | — | 5 | 4 | no | — |
| 121 | C2×Dic5 | — | 10 | 8 | no | — |
| 122 | C2×Dic5 | — | 10 | 2 | no | — |
| 123 | C2×Dic5 | — | 10 | 2 | no | — |
| 124 | C2×Dic5 | — | 10 | 4 | no | — |
| 125 | C2×Dic5 | — | 10 | 4 | no | — |
| 126 | C2×Dic5 | — | 10 | 8 | no | — |
| 127 | C2×Dic5 | — | 10 | 4 | no | — |
| 128 | C2×Dic5 | C22×D5 | 1 | 4 | yes | — |
| 129 | C2×Dic5 | C22×D5 | 1 | 2 | yes | — |
| 130 | C4×D5 | — | 10 | 4 | no | — |
| 131 | Dic10 | — | 10 | 4 | no | — |
| 132 | Dic10 | — | 10 | 8 | no | — |
| 133 | Dic10 | — | 10 | 8 | no | — |
| 134 | Dic10 | — | 10 | 16 | no | — |
| 135 | Dic10 | — | 10 | 16 | no | — |
| 136 | C2×#41 | — | 25 | 1 | no | — |
| 137 | C22×Q8 | — | 25 | 1 | no | — |
| 138 | C4⋊Q8 | — | 25 | 2 | no | — |
| 139 | #124 | — | 25 | 2 | no | — |
| 140 | #123 | — | 5 | 2 | no | — |
| 141 | #123 | — | 25 | 2 | no | — |
| 142 | C22⋊Q8 | — | 25 | 4 | no | — |
| 143 | #117 | — | 25 | 1 | no | — |
| 144 | C52 | #536 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C22×C5 | — | 2 | 8 | no | — |
| 146 | C22×C5 | — | 2 | 4 | no | — |
| 147 | C22×C5 | — | 2 | 4 | no | — |
| 148 | C22×C5 | — | 2 | 2 | no | — |
| 149 | C22×C5 | — | 10 | 2 | no | — |
| 150 | C22×C5 | — | 10 | 4 | no | — |
| 151 | C22×C5 | D4×D5 | 1 | 4 | yes | — |
| 152 | C22×C5 | C23×D5 | 1 | 2 | yes | — |
| 153 | C2×D5 | — | 2 | 2 | no | — |
| 154 | C2×D5 | — | 2 | 4 | no | — |
| 155 | C4×C5 | — | 2 | 8 | no | — |
| 156 | C4×C5 | — | 2 | 4 | no | — |
| 157 | C4×C5 | — | 10 | 2 | no | — |
| 158 | C4×C5 | — | 10 | 4 | no | — |
| 159 | Dic5 | — | 2 | 2 | no | — |
| 160 | Dic5 | — | 2 | 4 | no | — |
| 161 | Dic5 | — | 5 | 4 | no | — |
| 162 | Dic5 | — | 5 | 4 | no | — |
| 163 | Dic5 | — | 10 | 4 | no | — |
| 164 | Dic5 | — | 10 | 4 | no | — |
| 165 | Dic5 | — | 10 | 8 | no | — |
| 166 | Dic5 | — | 10 | 4 | no | — |
| 167 | #41 | — | 50 | 4 | no | — |
| 168 | C2×Q8 | — | 25 | 1 | no | — |
| 169 | C2×Q8 | — | 25 | 2 | no | — |
| 170 | C2×Q8 | — | 25 | 2 | no | — |
| 171 | C2×Q8 | — | 50 | 4 | no | — |
| 172 | C2×D4 | — | 5 | 2 | no | — |
| 173 | C2×D4 | — | 25 | 1 | no | — |
| 174 | C22×C4 | — | 25 | 2 | no | — |
| 175 | C22×C4 | — | 25 | 1 | no | — |
| 176 | C22×C4 | — | 25 | 1 | no | — |
| 177 | C22×C4 | D52 | 1 | 1 | yes | characteristic |
| 178 | C4⋊C4 | — | 5 | 4 | no | — |
| 179 | C4⋊C4 | — | 25 | 2 | no | — |
| 180 | C4⋊C4 | — | 25 | 4 | no | — |
| 181 | C22⋊C4 | — | 5 | 2 | no | — |
| 182 | C22⋊C4 | — | 5 | 4 | no | — |
| 183 | C22⋊C4 | — | 25 | 4 | no | — |
| 184 | C42 | — | 25 | 2 | no | — |
| 185 | C2×C5 | — | 2 | 8 | no | — |
| 186 | C2×C5 | — | 2 | 4 | no | — |
| 187 | C2×C5 | — | 2 | 4 | no | — |
| 188 | C2×C5 | — | 2 | 2 | no | — |
| 189 | C2×C5 | — | 20 | 2 | no | — |
| 190 | C2×C5 | C2×D4×D5 | 1 | 2 | yes | — |
| 191 | C2×C5 | #4150 | 1 | 4 | yes | — |
| 192 | D5 | — | 4 | 2 | no | — |
| 193 | C23 | — | 5 | 2 | no | — |
| 194 | C23 | C2×D52 | 1 | 1 | yes | characteristic |
| 195 | Q8 | — | 50 | 4 | no | — |
| 196 | Q8 | — | 50 | 4 | no | — |
| 197 | Q8 | — | 50 | 2 | no | — |
| 198 | D4 | — | 10 | 4 | no | — |
| 199 | D4 | — | 50 | 2 | no | — |
| 200 | C2×C4 | — | 2 | 2 | no | — |
| 201 | C2×C4 | — | 5 | 4 | no | — |
| 202 | C2×C4 | — | 5 | 2 | no | — |
| 203 | C2×C4 | — | 25 | 4 | no | — |
| 204 | C2×C4 | — | 25 | 1 | no | — |
| 205 | C2×C4 | — | 25 | 1 | no | — |
| 206 | C2×C4 | — | 25 | 2 | no | — |
| 207 | C2×C4 | — | 50 | 4 | no | — |
| 208 | C2×C4 | — | 50 | 2 | no | — |
| 209 | C2×C4 | C2×D52 | 1 | 2 | yes | — |
| 210 | C5 | — | 2 | 2 | no | — |
| 211 | C5 | #65982 | 1 | 2 | yes | complemented |
| 212 | C22 | — | 2 | 2 | no | — |
| 213 | C22 | — | 10 | 4 | no | — |
| 214 | C22 | — | 10 | 2 | no | — |
| 215 | C22 | #87886 | 1 | 2 | yes | — |
| 216 | C22 | C22×D52 | 1 | 1 | yes | characteristic |
| 217 | C4 | — | 2 | 2 | no | — |
| 218 | C4 | — | 10 | 4 | no | — |
| 219 | C4 | — | 10 | 2 | no | — |
| 220 | C4 | — | 25 | 2 | no | — |
| 221 | C4 | — | 25 | 2 | no | — |
| 222 | C4 | — | 50 | 2 | no | — |
| 223 | C2 | — | 2 | 2 | no | — |
| 224 | C2 | — | 20 | 2 | no | — |
| 225 | C2 | #11717463 | 1 | 2 | yes | — |
| 226 | C2 | C2×#87886 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C22×C4×C52 | C22 |
| prime radical | C2×#87880 | C2 |
| hypercenter | C22×C4 | D52 |
| hypocenter | C52 | #536 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6045 | C2 | semidirect |
| 3. C5×#4075 | C2 | semidirect |
| 4. #11717080 | C2 | semidirect |
| 5. #11717075 | C2 | semidirect |
| 6. #11717031 | C2 | semidirect |
| 7. #11717015 | C2 | semidirect |
| 8. #11716958 | C2 | semidirect |
| 50. #4075 | D5 | semidirect |
| 144. C52 | #536 | semidirect |
| 211. C5 | #65982 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6045 | 2 |
| C2×#87880 | 2 |
| C5×#4075 | 2 |
| #11717080 | 2 |
| #11717075 | 2 |
| #11717031 | 2 |
| #11717015 | 2 |
| #11716958 | 2 |
| #65982 | 5 |