Finite Group Databasebeta

← #35224909

Subgroups — #35224909

Order 20864.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#23782221C211yesmaximal, characteristic, complemented
2#127×D163C211yesmaximal, characteristic
3#23782031C211yesmaximal, characteristic, complemented
4#23782030C211yesmaximal, characteristic, complemented
5#23782029C211yesmaximal, characteristic, complemented
6#23782010C214yesmaximal, complemented
7#23781668C214yesmaximal, complemented
8#23781617C211yesmaximal, characteristic, complemented
9#490×C163C211yesmaximal, characteristic, complemented
10C2×D4×D163—22no—
11C2×D4×D163C2212yes—
12C2×#14579682C2212yes—
13#19104734C2212yescomplemented
14#19104732C2211yescharacteristic
15#19104731—22no—
16#19104731C2212yes—
17#19104727C2211yescharacteristic, complemented
18#19104713—22no—
19#19104700—22no—
20#19104693C2212yes—
21C42×D163C2211yescharacteristic
22#19104657C2214yescomplemented
23#19104656C2214yescomplemented
24#19104655C2214yescomplemented
25#19104646C2211yescharacteristic, complemented
26C8×Dic163C2212yescomplemented
27#127×C163C2211yescharacteristic
28#125×C163C2211yescharacteristic, complemented
29#102×C163C2214yescomplemented
30#97×C163C2211yescharacteristic, complemented
31C23×D163—22no—
32C2×Dic326—22no—
33C2×D652—22no—
34C2×D652C2312yes—
35C2×C22⋊D163—24no—
36C2×C22⋊D163—22no—
37D4×D163—28no—
38D4×D163—48no—
39C2×C4×D163D412yes—
40C2×C4×D163C2311yescharacteristic
41#14579682—24no—
42C2×C163⋊C8C2312yes—
43C4⋊Dic163C2312yes—
44Dic163⋊C4—22no—
45Dic163⋊C4—22no—
46(C2×D163)⋊C4—22no—
47C4×Dic163—22no—
48C4×Dic163D412yes—
49C4×Dic163C2311yescharacteristic
50C2×D4×C163—22no—
51C2×D4×C163C2312yes—
52C2×C8×C163C2312yes—
53C4⋊C4×C163—22no—
54C4⋊C4×C163C2312yes—
55C42×C163C2311yescharacteristic
56C22×D163—22no—
57C22×D163—48no—
58C22×D163—44no—
59C22×D163C2×D411yescharacteristic
60Dic326—44no—
61D652—24no—
62D652—44no—
63C22⋊D163—44no—
64C22⋊D163—48no—
65C2×Dic163—22no—
66C2×Dic163C2×D411yescharacteristic
67C2×Dic163C2×D412yes—
68C4×D163—24no—
69C4×D163#4114yes—
70C163⋊C8—22no—
71C23×C163—22no—
72D4×C163—24no—
73D4×C163—44no—
74C2×C4×C163—22no—
75C2×C4×C163C2×D412yes—
76C2×C4×C163C2411yescharacteristic
77C8×C163—22no—
78C22×C163—42no—
79C22×C163—44no—
80C22×C163C22×D411yescharacteristic
81C2×D163—22no—
82C2×D163—42no—
83C2×D163—44no—
84C2×D163—88no—
85C2×D163#12412yes—
86C4×C163—22no—
87C4×C163—42no—
88C4×C163C2×#4112yes—
89Dic163—42no—
90Dic163#12412yes—
91Dic163AGL(1,ℤ/8)14yes—
92C2×C163—82no—
93C2×C163C2×#12411yescharacteristic
94C2×C163C2×AGL(1,ℤ/8)12yes—
95D163—22no—
96D163—82no—
97C163#313811yescharacteristic, complemented, Sylow
98#3138—1631nomaximal, Sylow
99#533—1631no—
100C2×#127—1631no—
101#490—1631no—
102#490—1631no—
103#490—1631no—
104#490D16311yescharacteristic, complemented
105#486—1634no—
106#418—1634no—
107C4×M16—1631no—
108C22×D4—3262no—
109C22×D4—1632no—
110C2×M16—1632no—
111C4⋊Q8—1632no—
112#127—3262no—
113#127—1632no—
114#127—1631no—
115#127C2×D16311yescharacteristic
116#125—1631no—
117#125C2×D16311yescharacteristic, complemented
118C22⋊Q8—3262no—
119C4×Q8—3262no—
120#117—1632no—
121C2×C42—1631no—
122#102—1634no—
123#102—1634no—
124#102—1634no—
125#102C2×D16314yescomplemented
126#97—1631no—
127#97C2×D16311yescharacteristic, complemented
128C4×C8—1632no—
129C24—3262no—
130C2×Q8—3262no—
131C2×D4—22no—
132C2×D4—3268no—
133C2×D4—3262no—
134C2×D4—3264no—
135C2×D4—3262no—
136C2×D4—6528no—
137C2×D4—1632no—
138C2×D4C22×D16312yes—
139C22×C4—1632no—
140C22×C4—1631no—
141M16—3264no—
142C2×C8—1632no—
143C2×C8C22×D16312yes—
144C4⋊C4—22no—
145C4⋊C4—3262no—
146C4⋊C4—3262no—
147C4⋊C4—1632no—
148C4⋊C4C22×D16312yes—
149C22⋊C4—3262no—
150C42—3262no—
151C42—1632no—
152C42—1631no—
153C42C22×D16311yescharacteristic
154C23—22no—
155C23—3262no—
156C23—6528no—
157C23—6524no—
158C23—1631no—
159Q8—6524no—
160D4—24no—
161D4—44no—
162D4—3264no—
163D4—6524no—
164D4—6524no—
165D4—6528no—
166C2×C4—22no—
167C2×C4—3264no—
168C2×C4—3262no—
169C2×C4—1631no—
170C2×C4—1632no—
171C2×C4—1634no—
172C2×C4D4×D16312yes—
173C2×C4C23×D16311yescharacteristic
174C8—22no—
175C8—3262no—
176C22—42no—
177C22—44no—
178C22—13048no—
179C22—3262no—
180C22—6522no—
181C22—6524no—
182C22—1632no—
183C22C2×D4×D16311yescharacteristic
184C4—22no—
185C4—42no—
186C4—6522no—
187C4—1632no—
188C4—1634no—
189C4#41×D16312yes—
190C2—82no—
191C2—13042no—
192C2—3262no—
193C2#124×D16311yescharacteristic
194C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#490×C163C2
socleC22×C163C22×D4
prime radical#127×D163C2
hypercenter#490D163
hypocenterC163#3138
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #23782221C2semidirect
3. #23782031C2semidirect
4. #23782030C2semidirect
5. #23782029C2semidirect
6. #23782010C2semidirect
7. #23781668C2semidirect
8. #23781617C2semidirect
9. #490×C163C2semidirect
13. #19104734C22semidirect
17. #19104727C22semidirect
22. #19104657C22semidirect
23. #19104656C22semidirect
24. #19104655C22semidirect
25. #19104646C22semidirect
26. C8×Dic163C22semidirect
28. #125×C163C22semidirect
29. #102×C163C22semidirect
30. #97×C163C22semidirect
97. C163#3138semidirect
104. #490D163semidirect
117. #125C2×D163semidirect
125. #102C2×D163semidirect
127. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822212
#127×D1632
#237820312
#237820302
#237820292
#237820102
#237816682
#237816172
#490×C1632
#3138163