Finite Group Databasebeta

← #23782226

Subgroups — #23782226

Order 10432.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×D652C211yesmaximal, characteristic, complemented
2C2×D4×D163C211yesmaximal, characteristic, complemented
3#19104720C212yesmaximal, complemented
4#19104718C211yesmaximal, characteristic, complemented
5#19104710C212yesmaximal, complemented
6#19104708C212yesmaximal, complemented
7#19104705C211yesmaximal, characteristic, complemented
8#19104705C212yesmaximal, complemented
9#19104703C211yesmaximal, characteristic, complemented
10C22⋊C4×D163C211yesmaximal, characteristic, complemented
11#123×C163C211yesmaximal, characteristic, complemented
12C23×D163C2211yescharacteristic
13C23×D163C2211yescharacteristic, complemented
14C23×D163C2211yescharacteristic, complemented
15C2×D652—24no—
16C2×D652C2212yes—
17C2×D652C2211yescharacteristic, complemented
18C2×D652C2212yescomplemented
19C2×D652C2212yescomplemented
20C2×C22⋊D163C2211yescharacteristic
21C2×C22⋊D163C2211yescharacteristic, complemented
22C2×C22⋊D163C2212yescomplemented
23D4×D163—24no—
24C2×C4×D163C2211yescharacteristic, complemented
25C2×C4×D163C2212yescomplemented
26Dic163⋊C4C2212yescomplemented
27C22⋊Dic163C2211yescharacteristic, complemented
28(C2×D163)⋊C4C2212yescomplemented
29(C2×D163)⋊C4C2212yescomplemented
30(C2×D163)⋊C4C2212yescomplemented
31(C2×D163)⋊C4C2212yescomplemented
32C2×D4×C163C2211yescharacteristic
33C22×C4×C163C2211yescharacteristic, complemented
34C4⋊C4×C163C2212yescomplemented
35C22⋊C4×C163C2211yescharacteristic, complemented
36C22⋊C4×C163C2212yescomplemented
37C22×D163—22no—
38C22×D163—22no—
39C22×D163—42no—
40C22×D163—42no—
41C22×D163—42no—
42C22×D163D414yescomplemented
43C22×D163C2311yescharacteristic
44C22×D163C2312yes—
45C22×D163C2311yescharacteristic
46C22×D163C2311yescharacteristic
47D652—22no—
48D652—24no—
49D652—24no—
50D652—42no—
51C22⋊D163—22no—
52C22⋊D163—22no—
53C22⋊D163—42no—
54C2×Dic163C2312yes—
55C2×Dic163C2311yescharacteristic
56C4×D163—22no—
57C4×D163—42no—
58C23×C163C2311yescharacteristic
59C23×C163C2311yescharacteristic
60D4×C163—22no—
61C2×C4×C163—22no—
62C2×C4×C163C2312yes—
63C2×C4×C163C2312yes—
64C2×C4×C163C2311yescharacteristic
65C22×C163—22no—
66C22×C163—21no—
67C22×C163—22no—
68C22×C163C2×D412yes—
69C22×C163C2411yescharacteristic
70C2×D163—22no—
71C2×D163—22no—
72C2×D163—21no—
73C2×D163—22no—
74C2×D163—22no—
75C2×D163—22no—
76C2×D163—22no—
77C2×D163—44no—
78C2×D163—44no—
79C2×D163—44no—
80C2×D163C2×D412yes—
81C2×D163C2×D412yes—
82C4×C163—22no—
83C4×C163—21no—
84C4×C163—22no—
85Dic163—21no—
86Dic163—22no—
87C2×C163—22no—
88C2×C163—41no—
89C2×C163C22×D411yescharacteristic
90C2×C163H(5,𝔽2)12yes—
91D163—22no—
92D163—22no—
93D163—41no—
94D163—42no—
95C163#53411yescharacteristic, complemented, Sylow
96#534—1631nomaximal, Sylow
97C22×D4—1631no—
98C22×D4—1631no—
99#123—1631no—
100#123—1632no—
101#123D16311yescharacteristic, complemented
102#121—1632no—
103#121—1632no—
104#120—1631no—
105#120—1632no—
106#120—1631no—
107C2×C22⋊C4—1631no—
108C24—1631no—
109C24—1631no—
110C24—1631no—
111C2×D4—3264no—
112C2×D4—3264no—
113C2×D4—1631no—
114C2×D4—1631no—
115C2×D4—1632no—
116C2×D4—1632no—
117C2×D4—1631no—
118C2×D4—1632no—
119C2×D4—1632no—
120C2×D4C2×D16311yescharacteristic
121C22×C4—1631no—
122C22×C4—1632no—
123C22×C4C2×D16311yescharacteristic, complemented
124C4⋊C4—1632no—
125C4⋊C4C2×D16312yescomplemented
126C22⋊C4—1631no—
127C22⋊C4—1632no—
128C22⋊C4—1632no—
129C22⋊C4—1632no—
130C22⋊C4—1632no—
131C22⋊C4C2×D16311yescharacteristic, complemented
132C22⋊C4C2×D16312yescomplemented
133C23—3262no—
134C23—3262no—
135C23—6522no—
136C23—6522no—
137C23—6522no—
138C23—1631no—
139C23—1632no—
140C23—1631no—
141C23—1631no—
142C23—1634no—
143C23C22×D16311yescharacteristic
144C23C22×D16311yescharacteristic
145D4—22no—
146D4—3264no—
147D4—3264no—
148D4—3262no—
149D4—3262no—
150D4—3262no—
151D4—6522no—
152D4—6522no—
153C2×C4—22no—
154C2×C4—3262no—
155C2×C4—6522no—
156C2×C4—1632no—
157C2×C4—1631no—
158C2×C4C22×D16312yes—
159C2×C4C22×D16311yescharacteristic
160C2×C4C22×D16312yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—3262no—
165C22—3262no—
166C22—3262no—
167C22—3261no—
168C22—3262no—
169C22—3262no—
170C22—3262no—
171C22—6524no—
172C22—6524no—
173C22—6524no—
174C22—1632no—
175C22—1632no—
176C22D4×D16312yes—
177C22C23×D16311yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—3261no—
182C4—3262no—
183C2—22no—
184C2—41no—
185C2—3262no—
186C2—3262no—
187C2—6522no—
188C2—6521no—
189C2C2×D4×D16311yescharacteristic
190C2#1910478112yes—
subgroupG/H
center, FrattiniC22C23×D163
commutator, socle, cosocleC22×C163C24
Fitting#123×C163C2
hypercenter#123D163
hypocenterC163#534
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D652C2semidirect
2. C2×D4×D163C2semidirect
3. #19104720C2semidirect
4. #19104718C2semidirect
5. #19104710C2semidirect
6. #19104708C2semidirect
7. #19104705C2semidirect
8. #19104705C2semidirect
9. #19104703C2semidirect
10. C22⋊C4×D163C2semidirect
11. #123×C163C2semidirect
13. C23×D163C22semidirect
14. C23×D163C22semidirect
17. C2×D652C22semidirect
18. C2×D652C22semidirect
19. C2×D652C22semidirect
21. C2×C22⋊D163C22semidirect
22. C2×C22⋊D163C22semidirect
24. C2×C4×D163C22semidirect
25. C2×C4×D163C22semidirect
26. Dic163⋊C4C22semidirect
27. C22⋊Dic163C22semidirect
28. (C2×D163)⋊C4C22semidirect
29. (C2×D163)⋊C4C22semidirect
30. (C2×D163)⋊C4C22semidirect
31. (C2×D163)⋊C4C22semidirect
33. C22×C4×C163C22semidirect
34. C4⋊C4×C163C22semidirect
35. C22⋊C4×C163C22semidirect
36. C22⋊C4×C163C22semidirect
42. C22×D163D4semidirect
95. C163#534semidirect
101. #123D163semidirect
123. C22×C4C2×D163semidirect
125. C4⋊C4C2×D163semidirect
131. C22⋊C4C2×D163semidirect
132. C22⋊C4C2×D163semidirect