Finite Group Databasebeta

← C2×#23782502 #35229345

Subgroups — C2×#23782502 #35229345

Order 20864.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#23782502C2132yesmaximal, direct product factor
2C2×#19104783C215yesmaximal, complemented
3C2×#19104781C2110yesmaximal, complemented
4C22×#14579700C215yesmaximal, complemented
5C2×#41×D163C2110yesmaximal, complemented
6C2×2−1+4×C163C211yesmaximal, characteristic, complemented
7#19104783C22180yescomplemented
8#19104781C221160yescomplemented
9C2×#14579700C22110yescomplemented
10C2×#14579700C22120yescomplemented
11C2×#14579700C22180yescomplemented
12C2×#14579699C22130yescomplemented
13C2×#14579698C22110yescomplemented
14#41×D163C221160yescomplemented
15C2×Q8×D163C22110yescomplemented
16C22×D652C22115yescomplemented
17C2×D4×D163C22130yescomplemented
18C22×C4×D163C22115yescomplemented
192−1+4×C163C22116yescomplemented
20C2×#41×C163C22110yescomplemented
21C22×Q8×C163C2215yescomplemented
22C23×D163C23115yes—
23#14579700C23180yescomplemented
24#14579700C231160yescomplemented
25#14579699C231240yescomplemented
26#14579698C23180yescomplemented
27C2×Dic326C23110yes—
28Q8×D163C23180yescomplemented
29C2×D652C23130yes—
30C2×D652C231120yescomplemented
31C2×C22⋊D163C23120yes—
32D4×D163C231240yescomplemented
33C22×Dic163C2315yes—
34C2×C4×D163C23110yes—
35C2×C4×D163C23130yes—
36C2×C4×D163C231120yescomplemented
37#41×C163C23180yescomplemented
38C2×Q8×C163C23110yes—
39C2×Q8×C163C23140yescomplemented
40C2×D4×C163C23110yes—
41C22×C4×C163C23115yes—
42C22×D163—430no—
43C22×D163C24110yes—
44C22×D163C24160yes—
45Dic326C24140yes—
46D652C241120yes—
47C22⋊D163C24180yes—
48C2×Dic163C2411yescharacteristic
49C2×Dic163C2415yes—
50C2×Dic163C24120yes—
51C4×D163C24140yes—
52C4×D163C241120yes—
53C23×C163C2415yes—
54Q8×C163C24140yes—
55D4×C163C24140yes—
56C2×C4×C163C24110yes—
57C2×C4×C163C24160yes—
58C22×C163—210no—
59C22×C163C25110yes—
60C22×C163C2511yescharacteristic
61C2×D163—220no—
62C2×D163—460no—
63C2×D163C25120yes—
64C4×C163C25120yes—
65Dic163C2512yes—
66Dic163C25110yes—
67C2×C163—210no—
68C2×C163#58512yescomplemented
69C2×C163C2611yescharacteristic
70D163—220no—
71C163C2×#58511yescharacteristic, complemented, Sylow
72C2×#585—1631nomaximal, Sylow
73#585—16332no—
74C2×2−1+4—1635no—
75C2×2−1+4D16311yescharacteristic, complemented
76C2×H(5,𝔽2)—16310no—
77C22×#41—16310no—
78C22×#41—1635no—
792−1+4—16380no—
802−1+4C2×D163116yescomplemented
81H(5,𝔽2)—163160no—
82C2×#41—163160no—
83C2×#41—16310no—
84C2×#41—16320no—
85C2×#41—16380no—
86C2×#41—16310no—
87C2×#41—16330no—
88C2×#41C2×D163110yescomplemented
89C22×Q8—16310no—
90C22×Q8C2×D16315yescomplemented
91C22×D4—16330no—
92C22×D4—16315no—
93C23×C4—16315no—
94C24—16315no—
95#41—16380no—
96#41—163160no—
97#41—16380no—
98#41—163240no—
99#41C22×D163180yescomplemented
100C2×Q8—16380no—
101C2×Q8—16310no—
102C2×Q8C22×D163110yes—
103C2×Q8C22×D163140yescomplemented
104C2×D4—16320no—
105C2×D4—163240no—
106C2×D4—16330no—
107C2×D4—163120no—
108C2×D4C22×D163110yes—
109C22×C4—1635no—
110C22×C4—16310no—
111C22×C4—16330no—
112C22×C4—163120no—
113C22×C4C22×D163115yes—
114C23—65230no—
115C23—16310no—
116C23—16360no—
117C23C23×D16315yes—
118Q8—16340no—
119Q8C23×D163140yes—
120D4—16380no—
121D4—163120no—
122D4C23×D163140yes—
123C2×C4—16320no—
124C2×C4—1631no—
125C2×C4—1635no—
126C2×C4—16340no—
127C2×C4—163120no—
128C2×C4C23×D163160yes—
129C2×C4C23×D163110yes—
130C22—210no—
131C22—32620no—
132C22—65260no—
133C22—16320no—
134C22C24×D163110yes—
135C22C24×D16311yescharacteristic
136C4—1632no—
137C4—16310no—
138C4C24×D163120yes—
139C2—210no—
140C2—32620no—
141C2#2378250212yesdirect product factor
142C2C25×D16311yescharacteristic
subgroupG/H
centerC22C24×D163
commutator, cosocleC2×C163C26
FrattiniC2C25×D163
FittingC2×2−1+4×C163C2
socleC22×C163C25
hypercenterC2×2−1+4D163
hypocenterC163C2×#585
LCSG ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #23782502C2direct (N × H)
2. C2×#19104783C2semidirect
3. C2×#19104781C2semidirect
4. C22×#14579700C2semidirect
5. C2×#41×D163C2semidirect
6. C2×2−1+4×C163C2semidirect
7. #19104783C22semidirect
8. #19104781C22semidirect
9. C2×#14579700C22semidirect
10. C2×#14579700C22semidirect
11. C2×#14579700C22semidirect
12. C2×#14579699C22semidirect
13. C2×#14579698C22semidirect
14. #41×D163C22semidirect
15. C2×Q8×D163C22semidirect
16. C22×D652C22semidirect
17. C2×D4×D163C22semidirect
18. C22×C4×D163C22semidirect
19. 2−1+4×C163C22semidirect
20. C2×#41×C163C22semidirect
21. C22×Q8×C163C22semidirect
23. #14579700C23semidirect
24. #14579700C23semidirect
25. #14579699C23semidirect
26. #14579698C23semidirect
28. Q8×D163C23semidirect
30. C2×D652C23semidirect
32. D4×D163C23semidirect
36. C2×C4×D163C23semidirect
37. #41×C163C23semidirect
39. C2×Q8×C163C23semidirect
68. C2×C163#585semidirect
71. C163C2×#585semidirect
75. C2×2−1+4D163semidirect
80. 2−1+4C2×D163semidirect
88. C2×#41C2×D163semidirect
90. C22×Q8C2×D163semidirect
99. #41C22×D163semidirect
103. C2×Q8C22×D163semidirect
141. C2#23782502direct (N × H)

Maximal subgroups

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

subgroupindex
#237825022
C2×#191047832
C2×#191047812
C22×#145797002
C2×#41×D1632
C2×2−1+4×C1632
C2×#585163