Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
157 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 12 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⋊#6732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5⋊#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#6729 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C5×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | #91894 | — | 5 | 1 | no | maximal |
| 6 | #91872 | — | 5 | 1 | no | maximal |
| 7 | #87829 | — | 6 | 1 | no | maximal |
| 8 | #6790 | — | 10 | 1 | no | maximal |
| 9 | C5⋊(#41⋊C3) | — | 5 | 1 | no | — |
| 10 | C5⋊GL(2,𝔽3) | — | 5 | 1 | no | — |
| 11 | C5×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 12 | #6732 | — | 5 | 1 | no | — |
| 13 | #6729 | — | 5 | 1 | no | — |
| 14 | #6729 | D5 | 1 | 1 | yes | characteristic, complemented |
| 15 | #6036 | — | 6 | 1 | no | — |
| 16 | #6032 | — | 6 | 1 | no | — |
| 17 | C5×#170 | — | 6 | 1 | no | — |
| 18 | #4060 | — | 15 | 1 | no | — |
| 19 | C22×D15 | — | 10 | 1 | no | — |
| 20 | C2×D3×D5 | — | 10 | 1 | no | — |
| 21 | C5×#67 | — | 10 | 1 | no | — |
| 22 | C3×C22⋊D5 | — | 10 | 1 | no | — |
| 23 | C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 24 | #1203 | — | 10 | 1 | no | — |
| 25 | #1202 | — | 10 | 1 | no | — |
| 26 | #1200 | — | 10 | 1 | no | — |
| 27 | SL(2,𝔽5) | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×C5⋊D5 | — | 6 | 1 | no | — |
| 29 | C5×Dic5 | — | 6 | 1 | no | — |
| 30 | C5×Dic5 | — | 6 | 1 | no | — |
| 31 | #986 | — | 25 | 1 | no | Hall |
| 32 | #713 | — | 5 | 1 | no | — |
| 33 | D4×D5 | — | 15 | 1 | no | — |
| 34 | #700 | — | 30 | 1 | no | — |
| 35 | C5×SD16 | — | 15 | 1 | no | — |
| 36 | #688 | — | 15 | 1 | no | — |
| 37 | #686 | — | 15 | 1 | no | — |
| 38 | D40 | — | 15 | 1 | no | — |
| 39 | #676 | — | 15 | 1 | no | — |
| 40 | C22×C3×C5 | — | 10 | 1 | no | — |
| 41 | C2×D15 | — | 10 | 1 | no | — |
| 42 | C2×D15 | — | 10 | 1 | no | — |
| 43 | C2×D15 | — | 20 | 2 | no | — |
| 44 | C2×C5×D3 | — | 10 | 1 | no | — |
| 45 | C2×C3×D5 | — | 10 | 1 | no | — |
| 46 | D3×D5 | — | 20 | 2 | no | — |
| 47 | C3×Dic5 | — | 10 | 1 | no | — |
| 48 | C5×Dic3 | — | 10 | 1 | no | — |
| 49 | C2×C52 | — | 6 | 1 | no | — |
| 50 | C5⋊D5 | — | 12 | 1 | no | — |
| 51 | D3×D4 | — | 50 | 1 | no | — |
| 52 | #41⋊C3 | — | 25 | 1 | no | — |
| 53 | GL(2,𝔽3) | — | 5 | 1 | no | — |
| 54 | GL(2,𝔽3) | — | 25 | 1 | no | — |
| 55 | C22×D5 | — | 30 | 1 | no | — |
| 56 | C5×Q8 | — | 5 | 1 | no | — |
| 57 | C5×D4 | — | 15 | 1 | no | — |
| 58 | C22⋊D5 | — | 10 | 1 | no | — |
| 59 | D20 | — | 15 | 1 | no | — |
| 60 | D20 | — | 15 | 1 | no | — |
| 61 | C4×D5 | — | 15 | 1 | no | — |
| 62 | C4×D5 | — | 30 | 1 | no | — |
| 63 | #170 | — | 6 | 1 | no | — |
| 64 | #170 | — | 30 | 1 | no | — |
| 65 | C5×C8 | — | 15 | 1 | no | — |
| 66 | #168 | — | 15 | 1 | no | — |
| 67 | AGL(1,ℤ/8) | — | 75 | 1 | no | Sylow |
| 68 | C2×C3×C5 | — | 10 | 1 | no | — |
| 69 | C2×C3×C5 | — | 20 | 1 | no | — |
| 70 | D15 | — | 10 | 2 | no | — |
| 71 | D15 | — | 20 | 1 | no | — |
| 72 | C3×D5 | — | 20 | 1 | no | — |
| 73 | C5×D3 | — | 20 | 1 | no | — |
| 74 | C52 | — | 6 | 1 | no | Sylow |
| 75 | C22×D3 | — | 50 | 1 | no | — |
| 76 | C22×D3 | — | 50 | 1 | no | — |
| 77 | C3×D4 | — | 50 | 1 | no | — |
| 78 | #67 | — | 10 | 1 | no | — |
| 79 | #67 | — | 50 | 1 | no | — |
| 80 | D12 | — | 50 | 1 | no | — |
| 81 | C4×D3 | — | 50 | 1 | no | — |
| 82 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 83 | C22×C5 | — | 10 | 1 | no | — |
| 84 | C2×D5 | — | 10 | 1 | no | — |
| 85 | C2×D5 | — | 15 | 1 | no | — |
| 86 | C2×D5 | — | 30 | 1 | no | — |
| 87 | C2×D5 | — | 60 | 2 | no | — |
| 88 | C2×D5 | — | 120 | 1 | no | — |
| 89 | C4×C5 | — | 15 | 1 | no | — |
| 90 | C4×C5 | — | 30 | 1 | no | — |
| 91 | Dic5 | — | 6 | 1 | no | — |
| 92 | Dic5 | S5 | 1 | 1 | yes | characteristic |
| 93 | #41 | — | 25 | 1 | no | — |
| 94 | C2×D4 | — | 75 | 1 | no | — |
| 95 | SD16 | — | 15 | 1 | no | — |
| 96 | SD16 | — | 75 | 1 | no | — |
| 97 | D8 | — | 75 | 1 | no | — |
| 98 | D8 | — | 75 | 1 | no | — |
| 99 | M16 | — | 75 | 1 | no | — |
| 100 | C3×C5 | — | 10 | 1 | no | — |
| 101 | C22×C3 | — | 10 | 1 | no | — |
| 102 | C22×C3 | — | 50 | 1 | no | — |
| 103 | C2×D3 | — | 10 | 1 | no | — |
| 104 | C2×D3 | — | 50 | 1 | no | — |
| 105 | C2×D3 | — | 50 | 1 | no | — |
| 106 | C2×D3 | — | 100 | 2 | no | — |
| 107 | C2×D3 | — | 100 | 2 | no | — |
| 108 | C3×C4 | — | 50 | 1 | no | — |
| 109 | Dic3 | — | 10 | 1 | no | — |
| 110 | C2×C5 | — | 6 | 1 | no | — |
| 111 | C2×C5 | — | 20 | 1 | no | — |
| 112 | C2×C5 | — | 24 | 1 | no | — |
| 113 | C2×C5 | C2×S5 | 1 | 1 | yes | characteristic |
| 114 | D5 | — | 20 | 1 | no | — |
| 115 | D5 | — | 30 | 1 | no | — |
| 116 | D5 | — | 60 | 1 | no | — |
| 117 | D5 | — | 120 | 2 | no | — |
| 118 | C23 | — | 150 | 1 | no | — |
| 119 | Q8 | — | 5 | 1 | no | — |
| 120 | D4 | — | 15 | 1 | no | — |
| 121 | D4 | — | 50 | 1 | no | — |
| 122 | D4 | — | 75 | 1 | no | — |
| 123 | D4 | — | 75 | 1 | no | — |
| 124 | C2×C4 | — | 75 | 1 | no | — |
| 125 | C8 | — | 15 | 1 | no | — |
| 126 | C8 | — | 75 | 1 | no | — |
| 127 | C2×C3 | — | 10 | 1 | no | — |
| 128 | C2×C3 | — | 20 | 1 | no | — |
| 129 | C2×C3 | — | 100 | 1 | no | — |
| 130 | D3 | — | 20 | 1 | no | — |
| 131 | D3 | — | 50 | 2 | no | — |
| 132 | D3 | — | 100 | 1 | no | — |
| 133 | C5 | — | 6 | 1 | no | — |
| 134 | C5 | — | 24 | 1 | no | — |
| 135 | C5 | #91872 | 1 | 1 | yes | characteristic, complemented |
| 136 | C22 | — | 10 | 1 | no | — |
| 137 | C22 | — | 300 | 2 | no | — |
| 138 | C22 | — | 50 | 1 | no | — |
| 139 | C22 | — | 75 | 1 | no | — |
| 140 | C4 | — | 5 | 1 | no | — |
| 141 | C4 | — | 15 | 1 | no | — |
| 142 | C3 | — | 10 | 1 | no | Sylow |
| 143 | C2 | — | 20 | 1 | no | — |
| 144 | C2 | — | 100 | 1 | no | — |
| 145 | C2 | — | 150 | 1 | no | — |
| 146 | C2 | D5×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×S5 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽5) | C22 |
| Fitting, socle | C2×C5 | C2×S5 |
| LCS | G ▷ C5×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5⋊#6732 | C2 | semidirect |
| 2. C5⋊#6729 | C2 | semidirect |
| 3. C5×#6729 | C2 | semidirect |
| 4. C5×SL(2,𝔽5) | C22 | semidirect |
| 14. #6729 | D5 | semidirect |
| 27. SL(2,𝔽5) | C2×D5 | semidirect |
| 135. C5 | #91872 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5⋊#6732 | 2 |
| C5⋊#6729 | 2 |
| C5×#6729 | 2 |
| #91894 | 5 |
| #91872 | 5 |
| #87829 | 6 |
| #6790 | 10 |