Finite Group Databasebeta

← C2×#26042084 #37548474

Subgroups — C2×#26042084 #37548474

Order 23168.

1512 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 549 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×#19153862C212yesmaximal, complemented
2#26042084C2116yesmaximal, direct product factor
3C2×#19153844C216yesmaximal, complemented
4C2×#19153838C213yesmaximal, complemented
5C22×D1448C213yesmaximal, complemented
6C2×#131×C181C211yesmaximal, characteristic, complemented
7#19153862—28no—
8#19153862C22116yescomplemented
9C2×#15668450—22no—
10C2×#15668449C2216yescomplemented
11C22×D724C2216yescomplemented
12C2×D4×D181—26no—
13#19153844C22148yescomplemented
14#19153838C22124yescomplemented
15C2×Dic724C2211yescharacteristic, complemented
16C2×#15668438C2216yescomplemented
17C2×D1448C2213yescomplemented
18C2×D1448C2216yescomplemented
19C2×D1448C22124yescomplemented
20C2×#41×C181C2211yescharacteristic
21#131×C181C2218yescomplemented
22C2×M16×C181C2213yescomplemented
23C22×C8×C181C2213yescomplemented
24C23×D181—26no—
25#15668450—216no—
26#15668449—212no—
27#15668449C23124yescomplemented
28C2×Dic362C2312yes—
29C2×D724—26no—
30C2×D724—212no—
31C2×D724C2316yes—
32C2×D724C23124yescomplemented
33C2×C22⋊D181—26no—
34D4×D181—248no—
35C2×C4×D181—26no—
36Dic724C2314yescomplemented
37#15668438C23124yescomplemented
38D1448C23112yescomplemented
39D1448C23124yescomplemented
40#41×C181D414yes—
41#41×C181C2314yes—
42C2×Q8×C181D411yescharacteristic
43C2×D4×C181D413yes—
44C22×C4×C181C2313yes—
45M16×C181C23112yescomplemented
46C2×C8×C181C2311yescharacteristic
47C2×C8×C181C2313yes—
48C2×C8×C181C23112yescomplemented
49C22×D181—26no—
50C22×D181—224no—
51C22×D181—812no—
52Dic362—22no—
53Dic362C2414yes—
54D724—224no—
55D724—26no—
56D724C24112yes—
57C22⋊D181—224no—
58C2×Dic181—22no—
59C4×D181—224no—
60C23×C181C2×D413yes—
61Q8×C181C2×D414yes—
62D4×C181C2×D4112yes—
63C2×C4×C181C2×D413yes—
64C2×C4×C181C2×D416yes—
65C2×C4×C181C2411yescharacteristic
66C2×C4×C181C2416yes—
67C8×C181C2412yes—
68C8×C181C2416yes—
69C22×C181—26no—
70C22×C181C22×D416yes—
71C22×C181C22×D411yescharacteristic
72C2×D181—212no—
73C2×D181—412no—
74C2×D181—824no—
75C4×C181C22×D411yescharacteristic
76C4×C181C22×D416yes—
77C4×C181C2511yescharacteristic
78Dic181—24no—
79C2×C181—26no—
80C2×C181#57612yescomplemented
81C2×C181C23×D411yescharacteristic
82D181—412no—
83C181C2×#57611yescharacteristic, complemented, Sylow
84C2×#576—1811nomaximal, Sylow
85C2×H(5,𝔽2)—1812no—
86#576—18116no—
87C2×AGL(1,ℤ/8)—1816no—
88C2×#135—1813no—
89C22×D8—1813no—
90C2×#131D18111yescharacteristic, complemented
91H(5,𝔽2)—3628no—
92H(5,𝔽2)—18116no—
93C2×#41—3622no—
94C2×#41—1816no—
95C2×#41C2×D18111yescharacteristic
96C22×D4—3626no—
97C22×D4—1816no—
98AGL(1,ℤ/8)—18148no—
99#135—18124no—
100C2×Q16—1811no—
101C2×SD16—1816no—
102C2×D8—1813no—
103C2×D8—1816no—
104C2×D8—18124no—
105#131C2×D18118yescomplemented
106C2×M16C2×D18113yescomplemented
107C22×C8C2×D18113yescomplemented
108C24—3626no—
109#41—36216no—
110#41—36212no—
111#41—18124no—
112#41D72414yes—
113#41C22×D18114yes—
114C2×Q8—1812no—
115C2×Q8D72411yescharacteristic
116C2×D4—3626no—
117C2×D4—36248no—
118C2×D4—3626no—
119C2×D4—36212no—
120C2×D4—1816no—
121C2×D4—18124no—
122C2×D4D72413yes—
123C22×C4—3626no—
124C22×C4C22×D18113yes—
125Q16—1814no—
126SD16—18124no—
127D8—18112no—
128D8—18124no—
129M16C22×D181112yescomplemented
130C2×C8C22×D18111yescharacteristic
131C2×C8C22×D18113yes—
132C2×C8C22×D181112yescomplemented
133C23—36224no—
134C23—3626no—
135C23—144812no—
136C23C2×D72413yes—
137Q8—3622no—
138Q8—1814no—
139Q8C2×D72414yes—
140D4—36224no—
141D4—36224no—
142D4—3626no—
143D4—18112no—
144D4C2×D724112yes—
145C2×C4—3622no—
146C2×C4—36224no—
147C2×C4C2×D72416yes—
148C2×C4C2×D72413yes—
149C2×C4C23×D18116yes—
150C2×C4C23×D18111yescharacteristic
151C8C23×D18112yes—
152C8C23×D18116yes—
153C22—26no—
154C22—36212no—
155C22—144824no—
156C22—72412no—
157C22C22×D72416yes—
158C22C22×D72411yescharacteristic
159C4—3624no—
160C4C22×D72416yes—
161C4C22×D72411yescharacteristic
162C4C24×D18111yescharacteristic
163C2—26no—
164C2—72412no—
165C2#2604208412yesdirect product factor
166C2C23×D72411yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC4×C181C25
FrattiniC4C24×D181
FittingC2×#131×C181C2
socleC22×C181C22×D4
hypercenterC2×#131D181
hypocenterC181C2×#576
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. #26042084C2direct (N × H)
3. C2×#19153844C2semidirect
4. C2×#19153838C2semidirect
5. C22×D1448C2semidirect
6. C2×#131×C181C2semidirect
8. #19153862C22semidirect
10. C2×#15668449C22semidirect
11. C22×D724C22semidirect
13. #19153844C22semidirect
14. #19153838C22semidirect
15. C2×Dic724C22semidirect
16. C2×#15668438C22semidirect
17. C2×D1448C22semidirect
18. C2×D1448C22semidirect
19. C2×D1448C22semidirect
21. #131×C181C22semidirect
22. C2×M16×C181C22semidirect
23. C22×C8×C181C22semidirect
27. #15668449C23semidirect
32. C2×D724C23semidirect
36. Dic724C23semidirect
37. #15668438C23semidirect
38. D1448C23semidirect
39. D1448C23semidirect
45. M16×C181C23semidirect
48. C2×C8×C181C23semidirect
80. C2×C181#576semidirect
83. C181C2×#576semidirect
90. C2×#131D181semidirect
105. #131C2×D181semidirect
106. C2×M16C2×D181semidirect
107. C22×C8C2×D181semidirect
129. M16C22×D181semidirect
132. C2×C8C22×D181semidirect
165. C2#26042084direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191538622
#260420842
C2×#191538442
C2×#191538382
C22×D14482
C2×#131×C1812
C2×#576181