Finite Group Databasebeta

← C2×#26042116 #37548530

Subgroups — C2×#26042116 #37548530

Order 23168.

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#26042116C2132yesmaximal, direct product factor
2C2×#19153864C215yesmaximal, complemented
3C2×#19153862C2110yesmaximal, complemented
4C22×#15668450C215yesmaximal, complemented
5C2×#41×D181C2110yesmaximal, complemented
6C2×2−1+4×C181C211yesmaximal, characteristic, complemented
7#19153864C22180yescomplemented
8#19153862C221160yescomplemented
9C2×#15668450C22120yescomplemented
10C2×#15668450C22110yescomplemented
11C2×#15668450C22180yescomplemented
12C2×#15668449C22130yescomplemented
13C2×#15668448C22110yescomplemented
14#41×D181C221160yescomplemented
15C2×Q8×D181C22110yescomplemented
16C22×D724C22115yescomplemented
17C2×D4×D181C22130yescomplemented
18C22×C4×D181C22115yescomplemented
192−1+4×C181C22116yescomplemented
20C2×#41×C181C22110yescomplemented
21C22×Q8×C181C2215yescomplemented
22C23×D181C23115yes—
23#15668450C23180yescomplemented
24#15668450C231160yescomplemented
25#15668449C231240yescomplemented
26#15668448C23180yescomplemented
27C2×Dic362C23110yes—
28Q8×D181C23180yescomplemented
29C2×D724C23130yes—
30C2×D724C231120yescomplemented
31C2×C22⋊D181C23120yes—
32D4×D181C231240yescomplemented
33C22×Dic181C2315yes—
34C2×C4×D181C23110yes—
35C2×C4×D181C23130yes—
36C2×C4×D181C231120yescomplemented
37#41×C181C23180yescomplemented
38C2×Q8×C181C23110yes—
39C2×Q8×C181C23140yescomplemented
40C2×D4×C181C23110yes—
41C22×C4×C181C23115yes—
42C22×D181—430no—
43C22×D181C24110yes—
44C22×D181C24160yes—
45Dic362C24140yes—
46D724C241120yes—
47C22⋊D181C24180yes—
48C2×Dic181C2411yescharacteristic
49C2×Dic181C2415yes—
50C2×Dic181C24120yes—
51C4×D181C24140yes—
52C4×D181C241120yes—
53C23×C181C2415yes—
54Q8×C181C24140yes—
55D4×C181C24140yes—
56C2×C4×C181C24110yes—
57C2×C4×C181C24160yes—
58C22×C181—210no—
59C22×C181C25110yes—
60C22×C181C2511yescharacteristic
61C2×D181—220no—
62C2×D181—460no—
63C2×D181C25120yes—
64C4×C181C25120yes—
65Dic181C2512yes—
66Dic181C25110yes—
67C2×C181—210no—
68C2×C181#58512yescomplemented
69C2×C181C2611yescharacteristic
70D181—220no—
71C181C2×#58511yescharacteristic, complemented, Sylow
72C2×#585—1811nomaximal, Sylow
73#585—18132no—
74C2×2−1+4—1815no—
75C2×2−1+4D18111yescharacteristic, complemented
76C2×H(5,𝔽2)—18110no—
77C22×#41—18110no—
78C22×#41—1815no—
792−1+4—18180no—
802−1+4C2×D181116yescomplemented
81H(5,𝔽2)—181160no—
82C2×#41—181160no—
83C2×#41—18110no—
84C2×#41—18120no—
85C2×#41—18180no—
86C2×#41—18110no—
87C2×#41—18130no—
88C2×#41C2×D181110yescomplemented
89C22×Q8—18110no—
90C22×Q8C2×D18115yescomplemented
91C22×D4—18130no—
92C22×D4—18115no—
93C23×C4—18115no—
94C24—18115no—
95#41—18180no—
96#41—181160no—
97#41—18180no—
98#41—181240no—
99#41C22×D181180yescomplemented
100C2×Q8—18180no—
101C2×Q8—18110no—
102C2×Q8C22×D181110yes—
103C2×Q8C22×D181140yescomplemented
104C2×D4—18120no—
105C2×D4—181240no—
106C2×D4—18130no—
107C2×D4—181120no—
108C2×D4C22×D181110yes—
109C22×C4—1815no—
110C22×C4—18110no—
111C22×C4—18130no—
112C22×C4—181120no—
113C22×C4C22×D181115yes—
114C23—18110no—
115C23—18160no—
116C23—72430no—
117C23C23×D18115yes—
118Q8—18140no—
119Q8C23×D181140yes—
120D4—18180no—
121D4—181120no—
122D4C23×D181140yes—
123C2×C4—18120no—
124C2×C4—1811no—
125C2×C4—1815no—
126C2×C4—18140no—
127C2×C4—181120no—
128C2×C4C23×D181160yes—
129C2×C4C23×D181110yes—
130C22—210no—
131C22—36220no—
132C22—18120no—
133C22—72460no—
134C22C24×D181110yes—
135C22C24×D18111yescharacteristic
136C4—1812no—
137C4—18110no—
138C4C24×D181120yes—
139C2—210no—
140C2—36220no—
141C2#2604211612yesdirect product factor
142C2C25×D18111yescharacteristic
subgroupG/H
centerC22C24×D181
commutator, cosocleC2×C181C26
FrattiniC2C25×D181
FittingC2×2−1+4×C181C2
socleC22×C181C25
hypercenterC2×2−1+4D181
hypocenterC181C2×#585
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #26042116C2direct (N × H)
2. C2×#19153864C2semidirect
3. C2×#19153862C2semidirect
4. C22×#15668450C2semidirect
5. C2×#41×D181C2semidirect
6. C2×2−1+4×C181C2semidirect
7. #19153864C22semidirect
8. #19153862C22semidirect
9. C2×#15668450C22semidirect
10. C2×#15668450C22semidirect
11. C2×#15668450C22semidirect
12. C2×#15668449C22semidirect
13. C2×#15668448C22semidirect
14. #41×D181C22semidirect
15. C2×Q8×D181C22semidirect
16. C22×D724C22semidirect
17. C2×D4×D181C22semidirect
18. C22×C4×D181C22semidirect
19. 2−1+4×C181C22semidirect
20. C2×#41×C181C22semidirect
21. C22×Q8×C181C22semidirect
23. #15668450C23semidirect
24. #15668450C23semidirect
25. #15668449C23semidirect
26. #15668448C23semidirect
28. Q8×D181C23semidirect
30. C2×D724C23semidirect
32. D4×D181C23semidirect
36. C2×C4×D181C23semidirect
37. #41×C181C23semidirect
39. C2×Q8×C181C23semidirect
68. C2×C181#585semidirect
71. C181C2×#585semidirect
75. C2×2−1+4D181semidirect
80. 2−1+4C2×D181semidirect
88. C2×#41C2×D181semidirect
90. C22×Q8C2×D181semidirect
99. #41C22×D181semidirect
103. C2×Q8C22×D181semidirect
141. C2#26042116direct (N × H)

Maximal subgroups

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

subgroupindex
#260421162
C2×#191538642
C2×#191538622
C22×#156684502
C2×#41×D1812
C2×2−1+4×C1812
C2×#585181