Finite Group Databasebeta

← #35227674

Subgroups — #35227674

Order 20864.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#23782467C212yesmaximal, complemented
2#23782102C214yesmaximal, complemented
3#23782100C212yesmaximal, complemented
4#23782098C212yesmaximal, complemented
5C2×D2608C211yesmaximal, characteristic, complemented
6D16×D163C212yesmaximal, complemented
7#23782081C211yesmaximal, characteristic, complemented
8#508×C163C211yesmaximal, characteristic, complemented
9#19104781—22no—
10#19104762—24no—
11#19104760—22no—
12#19104758C2214yescomplemented
13C2×D1304C2212yescomplemented
14D8×D163C2214yescomplemented
15#19104741C2211yescharacteristic
16#19104678C2214yescomplemented
17D2608C2212yescomplemented
18D2608C2212yescomplemented
19#19104676C2214yescomplemented
20#19104675C2211yescharacteristic, complemented
21#19104674C2212yescomplemented
22C16×D163C2212yescomplemented
23#135×C163C2212yescomplemented
24Q32×C163C2211yescharacteristic, complemented
25SD32×C163C2212yescomplemented
26D16×C163C2211yescharacteristic, complemented
27C2×C16×C163C2211yescharacteristic, complemented
28#14579700—24no—
29#14579699—42no—
30#14579699D411yescharacteristic
31C2×D652—22no—
32C2×D652—42no—
33D4×D163—24no—
34D4×D163—44no—
35#14579686—24no—
36D1304—22no—
37D1304C2314yes—
38#14579684—24no—
39#14579683D411yescharacteristic
40#14579682D412yes—
41C8×D163C2312yes—
42C163⋊C16C2312yes—
43#41×C163—22no—
44Q16×C163C2312yes—
45SD16×C163—22no—
46D8×C163C2312yes—
47C2×C8×C163C2311yescharacteristic
48C16×C163C2311yescharacteristic
49C16×C163C2311yescharacteristic
50C22×D163—42no—
51C22×D163—44no—
52Dic326D811yescharacteristic
53D652—24no—
54D652—44no—
55D652—42no—
56D652D811yescharacteristic
57C22⋊D163—44no—
58C22⋊D163D812yes—
59C4×D163—44no—
60C4×D163C2×D412yes—
61C163⋊C8C2×D412yes—
62Q8×C163—22no—
63D4×C163—22no—
64D4×C163—42no—
65C2×C4×C163—42no—
66C2×C4×C163C2×D411yescharacteristic
67C8×C163C2×D411yescharacteristic
68C8×C163C2411yescharacteristic
69C22×C163—42no—
70C22×C163C2×D811yescharacteristic
71C2×D163—44no—
72C2×D163—164no—
73C2×D163—162no—
74C2×D163C2×D812yes—
75C4×C163—42no—
76C4×C163C2×D811yescharacteristic
77C4×C163C22×D411yescharacteristic
78Dic163C2×D812yes—
79C2×C163—21no—
80C2×C163—82no—
81C2×C163C22×D811yescharacteristic
82D163—22no—
83D163—84no—
84C163#341511yescharacteristic, complemented, Sylow
85#3415—1631nomaximal, Sylow
86#576—1632no—
87#509—1632no—
88#509—1634no—
89#508—1632no—
90#508D16311yescharacteristic, complemented
91C2×D16—1632no—
92C2×D16—1631no—
93#504—1631no—
94H(5,𝔽2)—3262no—
95AGL(1,ℤ/8)—3264no—
96AGL(1,ℤ/8)—3262no—
97#135—1634no—
98#135C2×D16312yescomplemented
99C2×D8—1634no—
100C2×D8—1632no—
101#131—1631no—
102Q32C2×D16311yescharacteristic, complemented
103SD32—1634no—
104SD32C2×D16312yescomplemented
105D16—1634no—
106D16—1632no—
107D16—1632no—
108D16C2×D16311yescharacteristic, complemented
109M32—1631no—
110M32—1632no—
111C2×C16—1632no—
112C2×C16C2×D16311yescharacteristic, complemented
113#41—22no—
114#41—3264no—
115#41—6522no—
116#41—1631no—
117C2×D4—3264no—
118C2×D4—3262no—
119C2×D4—6522no—
120C2×D4—6524no—
121Q16C22×D16312yes—
122SD16—22no—
123SD16—3264no—
124D8—3264no—
125D8—3262no—
126D8—1634no—
127D8C22×D16312yes—
128M16—1631no—
129M16—1632no—
130C2×C8—1632no—
131C2×C8C22×D16311yescharacteristic
132C16—1632no—
133C16C22×D16311yescharacteristic
134C16C22×D16311yescharacteristic
135C23—6524no—
136C23—6522no—
137Q8—22no—
138Q8—1631no—
139D4—22no—
140D4—42no—
141D4—3264no—
142D4—6524no—
143D4—6524no—
144D4—6522no—
145D4—1632no—
146D4—1631no—
147C2×C4—42no—
148C2×C4—6524no—
149C2×C4—1632no—
150C2×C4D4×D16311yescharacteristic
151C8—1632no—
152C8D4×D16311yescharacteristic
153C8C23×D16311yescharacteristic
154C22—42no—
155C22—26082no—
156C22—26084no—
157C22—6524no—
158C22—1632no—
159C22D8×D16311yescharacteristic
160C4—42no—
161C4—1632no—
162C4D8×D16311yescharacteristic
163C4C2×D4×D16311yescharacteristic
164C2—21no—
165C2—82no—
166C2—13044no—
167C2—3262no—
168C2C2×D8×D16311yescharacteristic
subgroupG/H
centerC2C2×D8×D163
commutator, cosocleC8×C163C24
FrattiniC8C23×D163
Fitting#508×C163C2
socleC2×C163C22×D8
hypercenter#508D163
hypocenterC163#3415
LCSG ▷ C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #23782467C2semidirect
2. #23782102C2semidirect
3. #23782100C2semidirect
4. #23782098C2semidirect
5. C2×D2608C2semidirect
6. D16×D163C2semidirect
7. #23782081C2semidirect
8. #508×C163C2semidirect
12. #19104758C22semidirect
13. C2×D1304C22semidirect
14. D8×D163C22semidirect
16. #19104678C22semidirect
17. D2608C22semidirect
18. D2608C22semidirect
19. #19104676C22semidirect
20. #19104675C22semidirect
21. #19104674C22semidirect
22. C16×D163C22semidirect
23. #135×C163C22semidirect
24. Q32×C163C22semidirect
25. SD32×C163C22semidirect
26. D16×C163C22semidirect
27. C2×C16×C163C22semidirect
84. C163#3415semidirect
90. #508D163semidirect
98. #135C2×D163semidirect
102. Q32C2×D163semidirect
104. SD32C2×D163semidirect
108. D16C2×D163semidirect
112. C2×C16C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824672
#237821022
#237821002
#237820982
C2×D26082
D16×D1632
#237820812
#508×C1632
#3415163