Finite Group Databasebeta

← #35229346

Subgroups — #35229346

Order 20864.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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#23782501C2112yesmaximal, complemented
2#23782500C2116yesmaximal, complemented
3C2×#19104780C2118yesmaximal, complemented
4H(5,𝔽2)×D163C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C163C211yesmaximal, characteristic, complemented
6#19104783C2218yescomplemented
7#19104782C22148yescomplemented
8#19104781C22148yescomplemented
9#19104780C221144yescomplemented
10#19104780C22172yescomplemented
11C2×#14579699C22136yescomplemented
12C2×#14579698C22172yescomplemented
13#41×D163C22196yescomplemented
14C22×C22⋊D163C22124yescomplemented
15C2×D4×D163C22172yescomplemented
16H(5,𝔽2)×C163C22116yescomplemented
17C2×#41×C163C2216yescomplemented
18C22×D4×C163C2219yescomplemented
19C23×D163C23124yescomplemented
20#14579700C23132yescomplemented
21#14579699C231144yescomplemented
22#14579699C23148yescomplemented
23#14579698C231288yescomplemented
24C2×Dic326C23136yescomplemented
25Q8×D163C23132yescomplemented
26C2×D652C23136yescomplemented
27C2×C22⋊D163C231144yescomplemented
28C2×C22⋊D163C23172yescomplemented
29D4×D163C231288yescomplemented
30C22×Dic163C23124yescomplemented
31C2×C4×D163C23172yescomplemented
32C24×C163C2316yescomplemented
33#41×C163C23148yescomplemented
34C2×Q8×C163C2312yescomplemented
35C2×D4×C163C23172yescomplemented
36C2×D4×C163C23118yescomplemented
37C22×C4×C163C2319yescomplemented
38C22×D163—824no—
39C22×D163C24172yes—
40Dic326C24148yes—
41D652C24148yes—
42C22⋊D163C241144yes—
43C22⋊D163C24116yes—
44C2×Dic163C24172yes—
45C4×D163C24196yes—
46C23×C163—86no—
47C23×C163C2419yes—
48C23×C163C24124yes—
49Q8×C163C2418yes—
50D4×C163C24172yes—
51C2×C4×C163C2416yes—
52C2×C4×C163C24136yes—
53C22×C163—49no—
54C22×C163—424no—
55C22×C163C2511yescharacteristic
56C22×C163C25118yes—
57C2×D163—472no—
58C2×D163C25116yes—
59C4×C163C25112yes—
60Dic163C25116yes—
61C2×C163—218no—
62C2×C163—21no—
63C2×C163C2611yescharacteristic
64D163—216no—
65C163H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—1631nomaximal, Sylow
67#585—16316no—
68#585—16312no—
69C2×H(5,𝔽2)—16316no—
70C2×H(5,𝔽2)—16318no—
71C2×H(5,𝔽2)D16311yescharacteristic, complemented
722−1+4—16348no—
732−1+4—1638no—
74H(5,𝔽2)—16348no—
75H(5,𝔽2)—163144no—
76H(5,𝔽2)—16372no—
77H(5,𝔽2)C2×D163116yescomplemented
78C2×#41—16396no—
79C2×#41—16372no—
80C2×#41—16336no—
81C2×#41C2×D16316yescomplemented
82C22×D4—16324no—
83C22×D4—16372no—
84C22×D4C2×D16319yescomplemented
85C24—16324no—
86C24C22×D16316yescomplemented
87#41—16332no—
88#41—163288no—
89#41—163144no—
90#41—16348no—
91#41C22×D163148yescomplemented
92C2×Q8—16332no—
93C2×Q8—16336no—
94C2×Q8C22×D16312yescomplemented
95C2×D4—163144no—
96C2×D4—16372no—
97C2×D4—163288no—
98C2×D4—16336no—
99C2×D4C22×D163172yescomplemented
100C2×D4C22×D163118yescomplemented
101C22×C4—16324no—
102C22×C4—16372no—
103C22×C4C22×D16319yescomplemented
104C23—86no—
105C23—130424no—
106C23—16372no—
107C23C23×D163124yes—
108C23C23×D16319yes—
109Q8—16348no—
110Q8C23×D16318yes—
111D4—163144no—
112D4—16316no—
113D4—16348no—
114D4C23×D163172yes—
115C2×C4—16372no—
116C2×C4—16396no—
117C2×C4C23×D163136yes—
118C2×C4C23×D16316yes—
119C22—49no—
120C22—424no—
121C22—65272no—
122C22—16316no—
123C22C24×D163118yes—
124C22C24×D16311yescharacteristic
125C4—16316no—
126C4C24×D163112yes—
127C2—21no—
128C2—218no—
129C2—32616no—
130C2C25×D16311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D163
commutator, socle, cosocleC2×C163C26
FittingC2×H(5,𝔽2)×C163C2
hypercenterC2×H(5,𝔽2)D163
hypocenterC163H(7,𝔽2)
LCSG ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23782501C2semidirect
2. #23782500C2semidirect
3. C2×#19104780C2semidirect
4. H(5,𝔽2)×D163C2semidirect
5. C2×H(5,𝔽2)×C163C2semidirect
6. #19104783C22semidirect
7. #19104782C22semidirect
8. #19104781C22semidirect
9. #19104780C22semidirect
10. #19104780C22semidirect
11. C2×#14579699C22semidirect
12. C2×#14579698C22semidirect
13. #41×D163C22semidirect
14. C22×C22⋊D163C22semidirect
15. C2×D4×D163C22semidirect
16. H(5,𝔽2)×C163C22semidirect
17. C2×#41×C163C22semidirect
18. C22×D4×C163C22semidirect
19. C23×D163C23semidirect
20. #14579700C23semidirect
21. #14579699C23semidirect
22. #14579699C23semidirect
23. #14579698C23semidirect
24. C2×Dic326C23semidirect
25. Q8×D163C23semidirect
26. C2×D652C23semidirect
27. C2×C22⋊D163C23semidirect
28. C2×C22⋊D163C23semidirect
29. D4×D163C23semidirect
30. C22×Dic163C23semidirect
31. C2×C4×D163C23semidirect
32. C24×C163C23semidirect
33. #41×C163C23semidirect
34. C2×Q8×C163C23semidirect
35. C2×D4×C163C23semidirect
36. C2×D4×C163C23semidirect
37. C22×C4×C163C23semidirect
65. C163H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D163semidirect
77. H(5,𝔽2)C2×D163semidirect
81. C2×#41C2×D163semidirect
84. C22×D4C2×D163semidirect
86. C24C22×D163semidirect
91. #41C22×D163semidirect
94. C2×Q8C22×D163semidirect
99. C2×D4C22×D163semidirect
100. C2×D4C22×D163semidirect
103. C22×C4C22×D163semidirect

Maximal subgroups

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

subgroupindex
#237825012
#237825002
C2×#191047802
H(5,𝔽2)×D1632
C2×H(5,𝔽2)×C1632
H(7,𝔽2)163