Finite Group Databasebeta

← #35222803

Subgroups — #35222803

Order 20864.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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#23781607C213yesmaximal
2#23781599C213yesmaximal
3#23781597C213yesmaximal
4#23781586C213yesmaximal
5#23781535C211yesmaximal, characteristic
6#23781533C211yesmaximal, characteristic
7#383×C163C211yesmaximal, characteristic
8C22×Dic326C2211yescharacteristic
9C2×C4⋊Dic163C2213yes—
10C2×C4⋊Dic163C2213yes—
11C2×C4⋊Dic163C2213yes—
12C2×Dic163⋊C4C2213yes—
13C2×Dic163⋊C4C2213yes—
14C163⋊#95C2213yes—
15C163⋊#95C2213yes—
16C163⋊#95C2213yes—
17C163⋊#95C2213yes—
18C2×C42×C163C2211yescharacteristic
19#95×C163C2213yes—
20#95×C163C2213yes—
21C2×Dic326—43no—
22C22×Dic163C2311yescharacteristic
23C22×Dic163C2311yescharacteristic
24C22×Dic163C2313yes—
25C22×Dic163C2313yes—
26C4⋊Dic163—43no—
27C4⋊Dic163—43no—
28C4⋊Dic163—43no—
29Dic163⋊C4—43no—
30Dic163⋊C4—43no—
31C22×C4×C163C2311yescharacteristic
32C22×C4×C163C2313yes—
33C22×C4×C163C2313yes—
34C42×C163Q814yescomplemented
35Dic326—44no—
36C2×Dic163—23no—
37C2×Dic163—23no—
38C2×Dic163—23no—
39C2×Dic163—23no—
40C2×Dic163—23no—
41C2×Dic163—23no—
42C2×Dic163—26no—
43C23×C163C2411yescharacteristic
44C2×C4×C163—23no—
45C2×C4×C163—23no—
46C2×C4×C163—23no—
47C2×C4×C163—23no—
48C2×C4×C163—26no—
49C2×C4×C163C2×Q816yes—
50C22×C163C22×Q811yescharacteristic
51C22×C163H(5,𝔽2)13yes—
52C22×C1632−1+413yes—
53C4×C163—26no—
54C4×C163—41no—
55C4×C163—43no—
56Dic163—43no—
57Dic163—41no—
58Dic163—41no—
59Dic163—43no—
60C2×C163#54413yes—
61C2×C163#56011yescharacteristic
62C2×C163#56313yes—
63C163#284311yescharacteristic, complemented, Sylow
64#2843—1631nomaximal, Sylow
65#400—1633no—
66#398—1633no—
67#398—1633no—
68#395—1633no—
69#384—1631no—
70#383—1631no—
71#383D16311yescharacteristic
72C22×Q8—1631no—
73C2×C4⋊C4—1633no—
74C2×C4⋊C4—1633no—
75C2×C4⋊C4—1633no—
76C2×C4⋊C4—1633no—
77C2×C4⋊C4—1633no—
78C2×C42C2×D16311yescharacteristic
79#95—1633no—
80#95—1633no—
81#95—1633no—
82#95—1633no—
83#95C2×D16313yes—
84#95C2×D16313yes—
85C2×Q8—6523no—
86C22×C4—1633no—
87C22×C4—1631no—
88C22×C4—1631no—
89C22×C4—1633no—
90C22×C4C22×D16313yes—
91C22×C4C22×D16311yescharacteristic
92C22×C4C22×D16313yes—
93C4⋊C4—6523no—
94C4⋊C4—6523no—
95C4⋊C4—6523no—
96C4⋊C4—6523no—
97C4⋊C4—6523no—
98C42Dic32614yescomplemented
99C23C23×D16311yescharacteristic
100Q8—6524no—
101C2×C4—23no—
102C2×C4—23no—
103C2×C4—23no—
104C2×C4—23no—
105C2×C4—26no—
106C2×C4—3263no—
107C2×C4—3263no—
108C2×C4—3263no—
109C2×C4—3263no—
110C2×C4—3263no—
111C2×C4—3263no—
112C2×C4—3266no—
113C2×C4C2×Dic32616yes—
114C22C22×Dic32611yescharacteristic
115C22#1910478113yes—
116C22#1910478213yes—
117C4—26no—
118C4—41no—
119C4—43no—
120C4—6521no—
121C4—6521no—
122C4—6523no—
123C4—6523no—
124C2#2378229313yes—
125C2#2378240611yescharacteristic
126C2#2378241713yes—
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, prime radical, cosocleC23×C163C24
Fitting#383×C163C2
hypercenter#383D163
hypocenterC163#2843
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
34. C42×C163Q8semidirect
63. C163#2843semidirect
98. C42Dic326semidirect

Maximal subgroups

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

subgroupindex
#237816072
#237815992
#237815972
#237815862
#237815352
#237815332
#383×C1632
#2843163