Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
520 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C5×#401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #64962 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #64960 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #64954 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #64954 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #64953 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C5⋊#401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×D20 | C22 | 1 | 3 | yes | — |
| 9 | C22×C4×D5 | C22 | 1 | 3 | yes | — |
| 10 | C22×Dic10 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#685 | C22 | 1 | 3 | yes | — |
| 12 | C2×#685 | C22 | 1 | 3 | yes | — |
| 13 | C2×#685 | C22 | 1 | 3 | yes | — |
| 14 | C2×#685 | C22 | 1 | 3 | yes | — |
| 15 | C5×#95 | C22 | 1 | 1 | yes | characteristic |
| 16 | C5×#95 | C22 | 1 | 3 | yes | — |
| 17 | C5×#95 | C22 | 1 | 3 | yes | — |
| 18 | #3963 | C22 | 1 | 3 | yes | — |
| 19 | #3963 | C22 | 1 | 3 | yes | — |
| 20 | #3963 | C22 | 1 | 3 | yes | — |
| 21 | #2760 | — | 5 | 1 | no | maximal, Sylow |
| 22 | C23×D5 | C23 | 1 | 3 | yes | — |
| 23 | C22×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 24 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 25 | C22×C4×C5 | C23 | 1 | 3 | yes | — |
| 26 | C22×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 27 | C22×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 28 | C22×Dic5 | C23 | 1 | 3 | yes | — |
| 29 | C2×D20 | — | 2 | 6 | no | — |
| 30 | C2×D20 | — | 2 | 6 | no | — |
| 31 | C2×D20 | — | 4 | 3 | no | — |
| 32 | C2×C4×D5 | — | 2 | 6 | no | — |
| 33 | C2×C4×D5 | — | 4 | 3 | no | — |
| 34 | C2×C4×D5 | — | 4 | 3 | no | — |
| 35 | C2×Dic10 | — | 4 | 3 | no | — |
| 36 | #685 | — | 4 | 3 | no | — |
| 37 | #685 | — | 4 | 3 | no | — |
| 38 | #685 | — | 4 | 3 | no | — |
| 39 | #685 | — | 4 | 3 | no | — |
| 40 | #401 | — | 5 | 1 | no | — |
| 41 | #401 | D5 | 1 | 1 | yes | characteristic, complemented |
| 42 | #393 | — | 5 | 3 | no | — |
| 43 | #392 | — | 5 | 1 | no | — |
| 44 | #386 | — | 5 | 3 | no | — |
| 45 | #386 | — | 5 | 3 | no | — |
| 46 | #381 | — | 5 | 3 | no | — |
| 47 | C23×C5 | C24 | 1 | 1 | yes | characteristic |
| 48 | C22×D5 | — | 2 | 3 | no | — |
| 49 | C22×D5 | — | 2 | 3 | no | — |
| 50 | C22×D5 | — | 2 | 3 | no | — |
| 51 | C22×D5 | — | 2 | 3 | no | — |
| 52 | C22×D5 | — | 2 | 3 | no | — |
| 53 | C22×D5 | — | 2 | 3 | no | — |
| 54 | C22×D5 | #41 | 1 | 6 | yes | — |
| 55 | C2×C4×C5 | — | 2 | 3 | no | — |
| 56 | C2×C4×C5 | — | 2 | 3 | no | — |
| 57 | C2×C4×C5 | — | 2 | 3 | no | — |
| 58 | C2×C4×C5 | — | 2 | 3 | no | — |
| 59 | C2×C4×C5 | — | 2 | 3 | no | — |
| 60 | C2×C4×C5 | — | 2 | 3 | no | — |
| 61 | C2×C4×C5 | — | 2 | 3 | no | — |
| 62 | C2×Dic5 | — | 2 | 3 | no | — |
| 63 | C2×Dic5 | — | 2 | 3 | no | — |
| 64 | C2×Dic5 | — | 2 | 3 | no | — |
| 65 | C2×Dic5 | — | 2 | 3 | no | — |
| 66 | C2×Dic5 | — | 2 | 3 | no | — |
| 67 | D20 | — | 4 | 12 | no | — |
| 68 | C4×D5 | — | 4 | 12 | no | — |
| 69 | Dic10 | — | 4 | 4 | no | — |
| 70 | C22×Q8 | — | 5 | 1 | no | — |
| 71 | C22×D4 | — | 5 | 3 | no | — |
| 72 | C23×C4 | — | 5 | 3 | no | — |
| 73 | C2×C22⋊C4 | — | 5 | 3 | no | — |
| 74 | C2×C22⋊C4 | — | 5 | 3 | no | — |
| 75 | C2×C22⋊C4 | — | 5 | 3 | no | — |
| 76 | C2×C22⋊C4 | — | 5 | 3 | no | — |
| 77 | #95 | — | 5 | 3 | no | — |
| 78 | #95 | — | 5 | 3 | no | — |
| 79 | #95 | — | 5 | 3 | no | — |
| 80 | #95 | C2×D5 | 1 | 3 | yes | — |
| 81 | #95 | C2×D5 | 1 | 3 | yes | — |
| 82 | #95 | C2×D5 | 1 | 1 | yes | characteristic |
| 83 | C22×C5 | C2×#41 | 1 | 3 | yes | — |
| 84 | C22×C5 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 85 | C22×C5 | H(5,𝔽2) | 1 | 3 | yes | — |
| 86 | C2×D5 | — | 2 | 6 | no | — |
| 87 | C2×D5 | — | 2 | 6 | no | — |
| 88 | C2×D5 | — | 2 | 6 | no | — |
| 89 | C2×D5 | — | 4 | 3 | no | — |
| 90 | C2×D5 | — | 4 | 3 | no | — |
| 91 | C2×D5 | — | 4 | 3 | no | — |
| 92 | C2×D5 | — | 4 | 3 | no | — |
| 93 | C4×C5 | — | 4 | 3 | no | — |
| 94 | C4×C5 | — | 4 | 3 | no | — |
| 95 | C4×C5 | — | 4 | 1 | no | — |
| 96 | Dic5 | — | 4 | 3 | no | — |
| 97 | Dic5 | — | 4 | 1 | no | — |
| 98 | Dic5 | — | 4 | 1 | no | — |
| 99 | C24 | — | 5 | 3 | no | — |
| 100 | C2×Q8 | — | 20 | 3 | no | — |
| 101 | C2×D4 | — | 10 | 6 | no | — |
| 102 | C2×D4 | — | 10 | 6 | no | — |
| 103 | C2×D4 | — | 20 | 3 | no | — |
| 104 | C22×C4 | — | 5 | 3 | no | — |
| 105 | C22×C4 | — | 5 | 1 | no | — |
| 106 | C22×C4 | — | 5 | 1 | no | — |
| 107 | C22×C4 | — | 10 | 6 | no | — |
| 108 | C22×C4 | — | 20 | 3 | no | — |
| 109 | C22×C4 | — | 20 | 3 | no | — |
| 110 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 111 | C22×C4 | C22×D5 | 1 | 3 | yes | — |
| 112 | C22×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 113 | C22⋊C4 | — | 20 | 3 | no | — |
| 114 | C22⋊C4 | — | 20 | 3 | no | — |
| 115 | C22⋊C4 | — | 20 | 3 | no | — |
| 116 | C22⋊C4 | — | 20 | 3 | no | — |
| 117 | C2×C5 | #538 | 1 | 3 | yes | — |
| 118 | C2×C5 | #551 | 1 | 3 | yes | — |
| 119 | C2×C5 | #560 | 1 | 1 | yes | characteristic |
| 120 | D5 | — | 4 | 6 | no | — |
| 121 | C23 | — | 5 | 6 | no | — |
| 122 | C23 | — | 10 | 3 | no | — |
| 123 | C23 | — | 10 | 3 | no | — |
| 124 | C23 | — | 10 | 3 | no | — |
| 125 | C23 | — | 10 | 3 | no | — |
| 126 | C23 | — | 10 | 3 | no | — |
| 127 | C23 | — | 10 | 3 | no | — |
| 128 | C23 | C23×D5 | 1 | 1 | yes | characteristic |
| 129 | Q8 | — | 20 | 4 | no | — |
| 130 | D4 | — | 20 | 12 | no | — |
| 131 | C2×C4 | — | 2 | 3 | no | — |
| 132 | C2×C4 | — | 2 | 3 | no | — |
| 133 | C2×C4 | — | 2 | 3 | no | — |
| 134 | C2×C4 | — | 2 | 3 | no | — |
| 135 | C2×C4 | — | 2 | 3 | no | — |
| 136 | C2×C4 | — | 2 | 3 | no | — |
| 137 | C2×C4 | — | 2 | 3 | no | — |
| 138 | C2×C4 | — | 10 | 3 | no | — |
| 139 | C2×C4 | — | 10 | 3 | no | — |
| 140 | C2×C4 | — | 10 | 3 | no | — |
| 141 | C2×C4 | — | 10 | 3 | no | — |
| 142 | C2×C4 | — | 10 | 3 | no | — |
| 143 | C2×C4 | — | 20 | 12 | no | — |
| 144 | C5 | #2760 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C22 | — | 10 | 6 | no | — |
| 146 | C22 | — | 10 | 6 | no | — |
| 147 | C22 | — | 10 | 6 | no | — |
| 148 | C22 | — | 20 | 3 | no | — |
| 149 | C22 | — | 20 | 3 | no | — |
| 150 | C22 | — | 20 | 3 | no | — |
| 151 | C22 | — | 20 | 3 | no | — |
| 152 | C22 | D5×#41 | 1 | 3 | yes | — |
| 153 | C22 | #4149 | 1 | 1 | yes | characteristic |
| 154 | C22 | #4149 | 1 | 3 | yes | — |
| 155 | C4 | — | 4 | 3 | no | — |
| 156 | C4 | — | 4 | 3 | no | — |
| 157 | C4 | — | 4 | 1 | no | — |
| 158 | C4 | — | 20 | 3 | no | — |
| 159 | C4 | — | 20 | 1 | no | — |
| 160 | C4 | — | 20 | 1 | no | — |
| 161 | C2 | — | 20 | 6 | no | — |
| 162 | C2 | #66038 | 1 | 3 | yes | — |
| 163 | C2 | #66039 | 1 | 3 | yes | — |
| 164 | C2 | #66045 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#401 | C2 |
| prime radical | #64960 | C2 |
| hypercenter | #401 | D5 |
| hypocenter | C5 | #2760 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #401 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#401 | C2 | semidirect |
| 2. #64962 | C2 | semidirect |
| 4. #64954 | C2 | semidirect |
| 5. #64954 | C2 | semidirect |
| 6. #64953 | C2 | semidirect |
| 7. C5⋊#401 | C2 | semidirect |
| 41. #401 | D5 | semidirect |
| 144. C5 | #2760 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#401 | 2 |
| #64962 | 2 |
| #64960 | 2 |
| #64954 | 2 |
| #64953 | 2 |
| C5⋊#401 | 2 |
| #2760 | 5 |