Finite Group Databasebeta

← #35226336

Subgroups — #35226336

Order 20864.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#23782470C212yesmaximal, complemented
2#23782467C212yesmaximal, complemented
3#23782042C211yesmaximal, characteristic, complemented
4#23781936C212yesmaximal, complemented
5#23781935C212yesmaximal, complemented
6#23781834C214yesmaximal, complemented
7#23781783C211yesmaximal, characteristic, complemented
8#443×C163C211yesmaximal, characteristic, complemented
9#19104781C2214yescomplemented
10#19104763—24no—
11#19104763C2214yescomplemented
12#19104762—24no—
13#19104760C2214yescomplemented
14#19104758—22no—
15#19104757—22no—
16#19104757C2211yescharacteristic
17C2×D1304—22no—
18C2×D1304C2212yescomplemented
19C2×D1304C2212yescomplemented
20D8×D163—22no—
21#19104741C2212yescomplemented
22#19104732C2212yescomplemented
23#19104671C2211yescharacteristic, complemented
24#19104663C2212yescomplemented
25#19104651C2214yescomplemented
26#135×C163C2211yescharacteristic
27#131×C163C2212yescomplemented
28#108×C163C2211yescharacteristic, complemented
29#104×C163C2212yescomplemented
30C4×C8×C163C2211yescharacteristic, complemented
31#14579700—24no—
32#14579699—24no—
33#14579699C2312yes—
34C2×D652—24no—
35C2×D652—42no—
36C2×D652C2314yes—
37D4×D163—24no—
38D4×D163—44no—
39Dic652D411yescharacteristic, complemented
40#14579688—24no—
41#14579688D412yescomplemented
42#14579686—24no—
43D1304—24no—
44D1304—22no—
45D1304—22no—
46D1304—22no—
47D1304—22no—
48D1304D411yescharacteristic, complemented
49#14579684—24no—
50#14579683C2312yes—
51#14579682—22no—
52C8×D163—22no—
53#41×C163C2312yes—
54Q16×C163D411yescharacteristic, complemented
55SD16×C163D412yescomplemented
56D8×C163D411yescharacteristic, complemented
57M16×C163—22no—
58M16×C163C2312yes—
59C2×C8×C163—22no—
60C2×C8×C163C2311yescharacteristic
61C2×C8×C163C2311yescharacteristic
62C42×C163C2311yescharacteristic
63C22×D163—24no—
64C22×D163—44no—
65Dic326C2×D412yes—
66D652—24no—
67D652—24no—
68D652—44no—
69D652—84no—
70D652C2×D412yes—
71C22⋊D163—22no—
72C22⋊D163—44no—
73C4×D163—22no—
74C4×D163—44no—
75C163⋊C8—22no—
76Q8×C163C2×D412yes—
77D4×C163—22no—
78D4×C163C2×D412yes—
79C2×C4×C163—21no—
80C2×C4×C163—22no—
81C2×C4×C163C2411yescharacteristic
82C8×C163—22no—
83C8×C163C2×D411yescharacteristic
84C8×C163C2×D411yescharacteristic
85C8×C163C2×D411yescharacteristic
86C8×C163C2×D411yescharacteristic
87C22×C163—22no—
88C22×C163H(5,𝔽2)11yescharacteristic
89C2×D163—22no—
90C2×D163—44no—
91C2×D163—84no—
92C2×D163—164no—
93C4×C163—22no—
94C4×C163—22no—
95C4×C163C22×D411yescharacteristic
96C4×C163C22×D411yescharacteristic
97Dic163—22no—
98C2×C163—21no—
99C2×C163—42no—
100C2×C163D4211yescharacteristic
101D163—42no—
102D163—84no—
103C163#329211yescharacteristic, complemented, Sylow
104#3292—1631nomaximal, Sylow
105#576—1632no—
106#576—1632no—
107#493—1631no—
108#472—1632no—
109#472—1632no—
110#453—1634no—
111#443—1631no—
112#443D16311yescharacteristic, complemented
113H(5,𝔽2)—1634no—
114AGL(1,ℤ/8)—3264no—
115AGL(1,ℤ/8)—3264no—
116AGL(1,ℤ/8)—1634no—
117AGL(1,ℤ/8)—1634no—
118#135—3262no—
119#135—3262no—
120#135—1631no—
121#135C2×D16311yescharacteristic
122C2×D8—3262no—
123C2×D8—3262no—
124C2×D8—1632no—
125C2×D8—1632no—
126#131—1632no—
127#131C2×D16312yescomplemented
128#127—1632no—
129#108—1631no—
130#108C2×D16311yescharacteristic, complemented
131#104—1632no—
132#104C2×D16312yescomplemented
133#100—1634no—
134C4×C8C2×D16311yescharacteristic, complemented
135#41—3264no—
136#41—3264no—
137#41—1632no—
138#41C22×D16312yes—
139C2×D4—3264no—
140C2×D4—3264no—
141C2×D4—6522no—
142C2×D4—6524no—
143C2×D4—1634no—
144Q16—1631no—
145Q16D65211yescharacteristic, complemented
146SD16—3264no—
147SD16—3264no—
148SD16—1632no—
149SD16D65212yescomplemented
150D8—3264no—
151D8—3264no—
152D8—3262no—
153D8—3262no—
154D8—3262no—
155D8—3262no—
156D8—1631no—
157D8D65211yescharacteristic, complemented
158M16—22no—
159M16—3262no—
160M16—1632no—
161M16C22×D16312yes—
162C2×C8—22no—
163C2×C8—3262no—
164C2×C8C22×D16311yescharacteristic
165C2×C8C22×D16311yescharacteristic
166C42C22×D16311yescharacteristic
167C23—3264no—
168C23—6524no—
169Q8—1632no—
170Q8C2×D65212yes—
171D4—22no—
172D4—13044no—
173D4—3262no—
174D4—3264no—
175D4—3264no—
176D4—6524no—
177D4—6524no—
178D4—1632no—
179D4C2×D65212yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3262no—
183C2×C4—6524no—
184C2×C4C23×D16311yescharacteristic
185C8—22no—
186C8—3262no—
187C8D4×D16311yescharacteristic
188C8D4×D16311yescharacteristic
189C8C2×D65211yescharacteristic
190C8C2×D65211yescharacteristic
191C22—22no—
192C22—13044no—
193C22—26084no—
194C22—3262no—
195C22—6524no—
196C22#1910478111yescharacteristic
197C4—22no—
198C4—22no—
199C4—3262no—
200C4C2×D4×D16311yescharacteristic
201C4C22×D65211yescharacteristic
202C2—21no—
203C2—42no—
204C2—13044no—
205C2—6522no—
206C2D4×D65211yescharacteristic
subgroupG/H
centerC2D4×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#443×C163C2
socleC2×C163D42
hypercenter#443D163
hypocenterC163#3292
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #23782470C2semidirect
2. #23782467C2semidirect
3. #23782042C2semidirect
4. #23781936C2semidirect
5. #23781935C2semidirect
6. #23781834C2semidirect
7. #23781783C2semidirect
8. #443×C163C2semidirect
9. #19104781C22semidirect
11. #19104763C22semidirect
13. #19104760C22semidirect
18. C2×D1304C22semidirect
19. C2×D1304C22semidirect
21. #19104741C22semidirect
22. #19104732C22semidirect
23. #19104671C22semidirect
24. #19104663C22semidirect
25. #19104651C22semidirect
27. #131×C163C22semidirect
28. #108×C163C22semidirect
29. #104×C163C22semidirect
30. C4×C8×C163C22semidirect
39. Dic652D4semidirect
41. #14579688D4semidirect
48. D1304D4semidirect
54. Q16×C163D4semidirect
55. SD16×C163D4semidirect
56. D8×C163D4semidirect
103. C163#3292semidirect
112. #443D163semidirect
127. #131C2×D163semidirect
130. #108C2×D163semidirect
132. #104C2×D163semidirect
134. C4×C8C2×D163semidirect
145. Q16D652semidirect
149. SD16D652semidirect
157. D8D652semidirect

Maximal subgroups

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

subgroupindex
#237824702
#237824672
#237820422
#237819362
#237819352
#237818342
#237817832
#443×C1632
#3292163