Finite Group Databasebeta

← #46830916

Subgroups — #46830916

Order 34624.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31772625C211yesmaximal, characteristic, complemented
2#31772622C211yesmaximal, characteristic, complemented
3#31772613C213yesmaximal, complemented
4#31772603C213yesmaximal, complemented
5#31772601C213yesmaximal, complemented
6#31772598C213yesmaximal, complemented
7#126×C541C211yesmaximal, characteristic, complemented
8C23×D541C2211yescharacteristic
9C2×D2164C2213yes—
10C2×D2164C2213yes—
11C2×D2164C2213yescomplemented
12C2×C22⋊D541C2213yes—
13C2×C4×D541C2213yescomplemented
14#22575815C2213yescomplemented
15#22575813C2213yes—
16#22575813C2213yes—
17#22575813C2213yescomplemented
18C4⋊C4×C541C2213yescomplemented
19C22⋊C4×C541C2213yes—
20C42×C541C2211yescharacteristic, complemented
21C22×D541—43no—
22C22×D541C2311yescharacteristic
23C22×D541C2311yescharacteristic
24C22×D541C2313yes—
25D2164—43no—
26D2164—43no—
27D2164—43no—
28C22⋊D541—43no—
29C2×Dic541C2313yes—
30C4×D541—43no—
31C23×C541C2311yescharacteristic
32C2×C4×C541C2313yes—
33C2×C4×C541C2313yes—
34C22×C541—23no—
35C22×C541C2411yescharacteristic
36C2×D541—23no—
37C2×D541—23no—
38C2×D541—23no—
39C2×D541—23no—
40C2×D541—23no—
41C2×D541—44no—
42C4×C541—23no—
43C4×C541—23no—
44Dic541—23no—
45C2×C541—41no—
46C2×C541H(5,𝔽2)13yes—
47D541—43no—
48D541—41no—
49D541—41no—
50C541#56011yescharacteristic, complemented, Sylow
51#560—5411nomaximal, Sylow
52#127—5411no—
53#126—5411no—
54#126D54111yescharacteristic, complemented
55#123—5413no—
56#121—5413no—
57#121—5413no—
58#120—5413no—
59C24—5411no—
60C2×D4—5413no—
61C2×D4—5413no—
62C2×D4—5413no—
63C2×D4—5413no—
64C22×C4—5413no—
65C4⋊C4—5413no—
66C4⋊C4C2×D54113yescomplemented
67C22⋊C4—5413no—
68C22⋊C4—5413no—
69C22⋊C4—5413no—
70C22⋊C4C2×D54113yes—
71C42C2×D54111yescharacteristic, complemented
72C23—5411no—
73C23—5411no—
74C23—5413no—
75C23—21643no—
76C23C22×D54111yescharacteristic
77D4—21643no—
78D4—21643no—
79D4—21643no—
80D4—21643no—
81C2×C4—5413no—
82C2×C4—21643no—
83C2×C4C22×D54113yes—
84C2×C4C22×D54113yes—
85C22—23no—
86C22—10823no—
87C22—10823no—
88C22—10823no—
89C22—10823no—
90C22—10823no—
91C22—21644no—
92C22C23×D54111yescharacteristic
93C4—23no—
94C4—23no—
95C4—10823no—
96C2—41no—
97C2—21641no—
98C2—21641no—
99C2—21643no—
100C2#3177267413yes—
subgroupG/H
center, FrattiniC22C23×D541
commutator, socle, cosocleC22×C541C24
Fitting#126×C541C2
hypercenter#126D541
hypocenterC541#560
LCSG ▷ C22×C541 ▷ C541
UCSC1 ◁ C22 ◁ #126

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #31772625C2semidirect
2. #31772622C2semidirect
3. #31772613C2semidirect
4. #31772603C2semidirect
5. #31772601C2semidirect
6. #31772598C2semidirect
7. #126×C541C2semidirect
11. C2×D2164C22semidirect
13. C2×C4×D541C22semidirect
14. #22575815C22semidirect
17. #22575813C22semidirect
18. C4⋊C4×C541C22semidirect
20. C42×C541C22semidirect
50. C541#560semidirect
54. #126D541semidirect
66. C4⋊C4C2×D541semidirect
71. C42C2×D541semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317726252
#317726222
#317726132
#317726032
#317726012
#317725982
#126×C5412
#560541