Finite Group Databasebeta

← C22×#19104781 #35229338

Subgroups — C22×#19104781 #35229338

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×#19104781C2148yesmaximal, direct product factor
2C22×#14579700C212yesmaximal, complemented
3C22×#14579699C213yesmaximal, complemented
4C23×D652C213yesmaximal, complemented
5C22×D4×D163C216yesmaximal, complemented
6C22×#41×C163C211yesmaximal, characteristic, complemented
7C24×D163C2216yescomplemented
8#19104781C221256yesdirect product factor
9C2×#14579700C22148yescomplemented
10C2×#14579699C22172yescomplemented
11C22×Dic326C2211yescharacteristic, complemented
12C22×D652C2216yescomplemented
13C22×D652C2213yescomplemented
14C22×D652C22172yescomplemented
15C22×C22⋊D163C2216yescomplemented
16C2×D4×D163C221144yescomplemented
17C22×C4×D163C2216yescomplemented
18C2×#41×C163C22124yescomplemented
19C22×Q8×C163C2211yescharacteristic, complemented
20C22×D4×C163C2213yescomplemented
21C23×C4×C163C2213yescomplemented
22C23×D163—424no—
23C23×D163C2316yes—
24C23×D163C23172yescomplemented
25#14579700C231128yescomplemented
26#14579699C231192yescomplemented
27C2×Dic326C23112yescomplemented
28C2×D652C23172yescomplemented
29C2×D652C23136yescomplemented
30C2×D652C231192yescomplemented
31C2×C22⋊D163C23172yescomplemented
32D4×D163C231384yescomplemented
33C22×Dic163C2312yes—
34C2×C4×D163C23172yescomplemented
35C24×C163C2313yes—
36#41×C163C23164yescomplemented
37C2×Q8×C163C23112yescomplemented
38C2×D4×C163C23136yescomplemented
39C22×C4×C163C2313yes—
40C22×C4×C163C2311yescharacteristic
41C22×C4×C163C23136yescomplemented
42C22×D163—224no—
43C22×D163—4144no—
44C22×D163C24136yes—
45C22×D163C24196yescomplemented
46Dic326C24116yescomplemented
47D652C24196yescomplemented
48D652C24148yescomplemented
49C22⋊D163C24196yescomplemented
50C2×Dic163C24112yes—
51C4×D163C24196yescomplemented
52C23×C163—212no—
53C23×C163C24118yes—
54C23×C163C2411yescharacteristic
55Q8×C163C24116yescomplemented
56D4×C163C24148yescomplemented
57C2×C4×C163C2416yes—
58C2×C4×C163C24118yes—
59C2×C4×C163C24148yescomplemented
60C22×C163—236no—
61C22×C163H(5,𝔽2)14yescomplemented
62C22×C163C25112yes—
63C22×C163C2513yes—
64C2×D163—272no—
65C2×D163—496no—
66C2×D163C25124yes—
67C4×C163C2514yes—
68C4×C163C25112yes—
69Dic163C2518yes—
70C2×C163—212no—
71C2×C163C2×H(5,𝔽2)16yescomplemented
72C2×C163C2611yescharacteristic
73D163—224no—
74C163C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1631nomaximal, Sylow
76C2×H(5,𝔽2)—16348no—
77C22×#41—1632no—
78C22×#41—1633no—
79C22×#41D16311yescharacteristic, complemented
80C23×D4—1636no—
81C23×D4—1633no—
82C25—1636no—
83H(5,𝔽2)—163256no—
84C2×#41—16348no—
85C2×#41—16372no—
86C2×#41C2×D163124yescomplemented
87C22×Q8—1631no—
88C22×Q8C2×D16311yescharacteristic, complemented
89C22×D4—1636no—
90C22×D4—163144no—
91C22×D4—1636no—
92C22×D4—1633no—
93C22×D4—16372no—
94C22×D4C2×D16313yescomplemented
95C23×C4—1636no—
96C23×C4C2×D16313yescomplemented
97C24—65224no—
98C24—1636no—
99C24—16372no—
100C24C22×D16313yes—
101#41—163128no—
102#41—163192no—
103#41C22×D163164yescomplemented
104C2×Q8—16312no—
105C2×Q8C22×D163112yescomplemented
106C2×D4—16372no—
107C2×D4—163384no—
108C2×D4—16372no—
109C2×D4—16336no—
110C2×D4—163192no—
111C2×D4C22×D163136yescomplemented
112C22×C4—1632no—
113C22×C4—16372no—
114C22×C4C22×D16313yes—
115C22×C4C22×D16311yescharacteristic
116C22×C4C22×D163136yescomplemented
117C23—212no—
118C23—32624no—
119C23—652144no—
120C23—16336no—
121C23—16396no—
122C23C23×D163118yes—
123C23C23×D16311yescharacteristic
124Q8—16316no—
125Q8C23×D163116yescomplemented
126D4—16396no—
127D4—16396no—
128D4—16348no—
129D4C23×D163148yescomplemented
130C2×C4—16312no—
131C2×C4—16396no—
132C2×C4C23×D163118yes—
133C2×C4C23×D16316yes—
134C2×C4C23×D163148yescomplemented
135C22—236no—
136C22—32672no—
137C22—65296no—
138C22—16324no—
139C22#1910478114yesdirect product factor
140C22C24×D163112yes—
141C22C24×D16313yes—
142C4—1638no—
143C4C24×D163112yes—
144C4C24×D16314yes—
145C2—212no—
146C2—32624no—
147C2C2×#1910478116yesdirect product factor
148C2C25×D16311yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC2×C163C26
FrattiniC2C25×D163
FittingC22×#41×C163C2
socleC23×C163C24
hypercenterC22×#41D163
hypocenterC163C22×H(5,𝔽2)
LCSG ▷ C2×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2direct (N × H)
2. C22×#14579700C2semidirect
3. C22×#14579699C2semidirect
4. C23×D652C2semidirect
5. C22×D4×D163C2semidirect
6. C22×#41×C163C2semidirect
7. C24×D163C22semidirect
8. #19104781C22direct (N × H)
9. C2×#14579700C22semidirect
10. C2×#14579699C22semidirect
11. C22×Dic326C22semidirect
12. C22×D652C22semidirect
13. C22×D652C22semidirect
14. C22×D652C22semidirect
15. C22×C22⋊D163C22semidirect
16. C2×D4×D163C22semidirect
17. C22×C4×D163C22semidirect
18. C2×#41×C163C22semidirect
19. C22×Q8×C163C22semidirect
20. C22×D4×C163C22semidirect
21. C23×C4×C163C22semidirect
24. C23×D163C23semidirect
25. #14579700C23semidirect
26. #14579699C23semidirect
27. C2×Dic326C23semidirect
28. C2×D652C23semidirect
29. C2×D652C23semidirect
30. C2×D652C23semidirect
31. C2×C22⋊D163C23semidirect
32. D4×D163C23semidirect
34. C2×C4×D163C23semidirect
36. #41×C163C23semidirect
37. C2×Q8×C163C23semidirect
38. C2×D4×C163C23semidirect
41. C22×C4×C163C23semidirect
45. C22×D163C24semidirect
46. Dic326C24semidirect
47. D652C24semidirect
48. D652C24semidirect
49. C22⋊D163C24semidirect
51. C4×D163C24semidirect
55. Q8×C163C24semidirect
56. D4×C163C24semidirect
59. C2×C4×C163C24semidirect
61. C22×C163H(5,𝔽2)semidirect
71. C2×C163C2×H(5,𝔽2)semidirect
74. C163C22×H(5,𝔽2)semidirect
79. C22×#41D163semidirect
86. C2×#41C2×D163semidirect
88. C22×Q8C2×D163semidirect
94. C22×D4C2×D163semidirect
96. C23×C4C2×D163semidirect
103. #41C22×D163semidirect
105. C2×Q8C22×D163semidirect
111. C2×D4C22×D163semidirect
116. C22×C4C22×D163semidirect
125. Q8C23×D163semidirect
129. D4C23×D163semidirect
134. C2×C4C23×D163semidirect
139. C22#19104781direct (N × H)
147. C2C2×#19104781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191047812
C22×#145797002
C22×#145796992
C23×D6522
C22×D4×D1632
C22×#41×C1632
C22×H(5,𝔽2)163