Finite Group Databasebeta

← C22×#19104780 #35229337

Subgroups — C22×#19104780 #35229337

Order 20864.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#19104780C2148yesmaximal, direct product factor
2C22×#14579699C212yesmaximal, complemented
3C22×#14579698C214yesmaximal, complemented
4C23×C22⋊D163C214yesmaximal, complemented
5C22×D4×D163C214yesmaximal, complemented
6C23×D4×C163C211yesmaximal, characteristic, complemented
7C24×D163C2214yescomplemented
8#19104780C221256yesdirect product factor
9C2×#14579699C22148yescomplemented
10C2×#14579698C22196yescomplemented
11C22×Dic326C2212yescomplemented
12C22×D652C2212yescomplemented
13C22×C22⋊D163C2218yescomplemented
14C22×C22⋊D163C2214yescomplemented
15C22×C22⋊D163C22196yescomplemented
16C2×D4×D163C22196yescomplemented
17C23×Dic163C2214yescomplemented
18C22×C4×D163C2214yescomplemented
19C25×C163C2212yescomplemented
20C22×D4×C163C2214yescomplemented
21C22×D4×C163C22124yescomplemented
22C23×C4×C163C2211yescharacteristic, complemented
23C23×D163—416no—
24C23×D163C2314yes—
25C23×D163C23148yescomplemented
26#14579699C231128yescomplemented
27#14579698C231256yescomplemented
28C2×Dic326C23124yescomplemented
29C2×D652C23124yescomplemented
30C2×C22⋊D163C23196yescomplemented
31C2×C22⋊D163C23148yescomplemented
32C2×C22⋊D163C231256yescomplemented
33D4×D163C231256yescomplemented
34C22×Dic163C2314yes—
35C22×Dic163C23148yescomplemented
36C2×C4×D163C23148yescomplemented
37C24×C163—48no—
38C24×C163C2314yes—
39C24×C163C2311yescharacteristic
40C24×C163C23124yescomplemented
41C2×D4×C163C23148yescomplemented
42C2×D4×C163C23164yescomplemented
43C22×C4×C163C2312yes—
44C22×C4×C163C23112yescomplemented
45C22×D163—216no—
46C22×D163—496no—
47C22×D163C24124yes—
48C22×D163C24164yescomplemented
49Dic326C24132yescomplemented
50D652C24132yescomplemented
51C22⋊D163C241128yescomplemented
52C22⋊D163C24164yescomplemented
53C2×Dic163C24124yes—
54C2×Dic163C24164yescomplemented
55C4×D163C24164yescomplemented
56C23×C163—24no—
57C23×C163—216no—
58C23×C163—448no—
59C23×C163C2416yes—
60C23×C163C24124yes—
61C23×C163C2411yescharacteristic
62C23×C163C24132yescomplemented
63D4×C163C24164yescomplemented
64C2×C4×C163C24112yes—
65C2×C4×C163C24116yescomplemented
66C22×C163—212no—
67C22×C163—248no—
68C22×C163—432no—
69C22×C163H(5,𝔽2)14yescomplemented
70C22×C163C2514yes—
71C22×C163C25116yes—
72C22×C163C2513yes—
73C2×D163—248no—
74C2×D163—464no—
75C2×D163C25116yes—
76C4×C163C2518yes—
77Dic163C25116yes—
78C2×C163—216no—
79C2×C163—24no—
80C2×C163C2×H(5,𝔽2)16yescomplemented
81C2×C163C2611yescharacteristic
82D163—216no—
83C163C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1631nomaximal, Sylow
85C2×H(5,𝔽2)—16348no—
86C22×#41—1634no—
87C22×#41—1632no—
88C23×D4—1634no—
89C23×D4—1634no—
90C23×D4D16311yescharacteristic, complemented
91C25—1634no—
92C25C2×D16312yescomplemented
93H(5,𝔽2)—163256no—
94C2×#41—16396no—
95C2×#41—16348no—
96C22×Q8—1632no—
97C22×D4—1638no—
98C22×D4—1634no—
99C22×D4—16396no—
100C22×D4—16396no—
101C22×D4—1632no—
102C22×D4C2×D16314yescomplemented
103C22×D4C2×D163124yescomplemented
104C23×C4—1634no—
105C23×C4—1634no—
106C23×C4C2×D16311yescharacteristic, complemented
107C24—48no—
108C24—65216no—
109C24—1634no—
110C24—16348no—
111C24C22×D16314yes—
112C24C22×D16311yescharacteristic
113C24C22×D163124yescomplemented
114#41—163256no—
115#41—163128no—
116C2×Q8—16324no—
117C2×D4—16396no—
118C2×D4—16348no—
119C2×D4—163256no—
120C2×D4—163256no—
121C2×D4—16324no—
122C2×D4C22×D163148yescomplemented
123C2×D4C22×D163164yescomplemented
124C22×C4—16348no—
125C22×C4—1634no—
126C22×C4—16348no—
127C22×C4C22×D16312yes—
128C22×C4C22×D163112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—32616no—
133C23—65296no—
134C23—16324no—
135C23—16364no—
136C23C23×D163124yes—
137C23C23×D16316yes—
138C23C23×D16311yescharacteristic
139C23C23×D163132yescomplemented
140Q8—16332no—
141D4—163128no—
142D4—16364no—
143D4—16332no—
144D4C23×D163164yescomplemented
145C2×C4—16324no—
146C2×C4—16364no—
147C2×C4—16364no—
148C2×C4C23×D163112yes—
149C2×C4C23×D163116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—32648no—
154C22—65264no—
155C22—16316no—
156C22#1910478014yesdirect product factor
157C22C24×D163116yes—
158C22C24×D16314yes—
159C22C24×D16313yes—
160C4—16316no—
161C4C24×D16318yes—
162C2—24no—
163C2—216no—
164C2—32616no—
165C2C2×#1910478016yesdirect product factor
166C2C25×D16311yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC2×C163C26
FrattiniC2C25×D163
FittingC23×D4×C163C2
socleC23×C163C24
hypercenterC23×D4D163
hypocenterC163C22×H(5,𝔽2)
LCSG ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19104780C2direct (N × H)
2. C22×#14579699C2semidirect
3. C22×#14579698C2semidirect
4. C23×C22⋊D163C2semidirect
5. C22×D4×D163C2semidirect
6. C23×D4×C163C2semidirect
7. C24×D163C22semidirect
8. #19104780C22direct (N × H)
9. C2×#14579699C22semidirect
10. C2×#14579698C22semidirect
11. C22×Dic326C22semidirect
12. C22×D652C22semidirect
13. C22×C22⋊D163C22semidirect
14. C22×C22⋊D163C22semidirect
15. C22×C22⋊D163C22semidirect
16. C2×D4×D163C22semidirect
17. C23×Dic163C22semidirect
18. C22×C4×D163C22semidirect
19. C25×C163C22semidirect
20. C22×D4×C163C22semidirect
21. C22×D4×C163C22semidirect
22. C23×C4×C163C22semidirect
25. C23×D163C23semidirect
26. #14579699C23semidirect
27. #14579698C23semidirect
28. C2×Dic326C23semidirect
29. C2×D652C23semidirect
30. C2×C22⋊D163C23semidirect
31. C2×C22⋊D163C23semidirect
32. C2×C22⋊D163C23semidirect
33. D4×D163C23semidirect
35. C22×Dic163C23semidirect
36. C2×C4×D163C23semidirect
40. C24×C163C23semidirect
41. C2×D4×C163C23semidirect
42. C2×D4×C163C23semidirect
44. C22×C4×C163C23semidirect
48. C22×D163C24semidirect
49. Dic326C24semidirect
50. D652C24semidirect
51. C22⋊D163C24semidirect
52. C22⋊D163C24semidirect
54. C2×Dic163C24semidirect
55. C4×D163C24semidirect
62. C23×C163C24semidirect
63. D4×C163C24semidirect
65. C2×C4×C163C24semidirect
69. C22×C163H(5,𝔽2)semidirect
80. C2×C163C2×H(5,𝔽2)semidirect
83. C163C22×H(5,𝔽2)semidirect
90. C23×D4D163semidirect
92. C25C2×D163semidirect
102. C22×D4C2×D163semidirect
103. C22×D4C2×D163semidirect
106. C23×C4C2×D163semidirect
113. C24C22×D163semidirect
122. C2×D4C22×D163semidirect
123. C2×D4C22×D163semidirect
128. C22×C4C22×D163semidirect
139. C23C23×D163semidirect
144. D4C23×D163semidirect
149. C2×C4C23×D163semidirect
156. C22#19104780direct (N × H)
165. C2C2×#19104780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191047802
C22×#145796992
C22×#145796982
C23×C22⋊D1632
C22×D4×D1632
C23×D4×C1632
C22×H(5,𝔽2)163