Finite Group Databasebeta

← C22×#19153861 #37548522

Subgroups — C22×#19153861 #37548522

Order 23168.

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×#19153861C2148yesmaximal, direct product factor
2C22×#15668449C212yesmaximal, complemented
3C22×#15668448C214yesmaximal, complemented
4C23×C22⋊D181C214yesmaximal, complemented
5C22×D4×D181C214yesmaximal, complemented
6C23×D4×C181C211yesmaximal, characteristic, complemented
7C24×D181C2214yescomplemented
8#19153861C221256yesdirect product factor
9C2×#15668449C22148yescomplemented
10C2×#15668448C22196yescomplemented
11C22×Dic362C2212yescomplemented
12C22×D724C2212yescomplemented
13C22×C22⋊D181C2218yescomplemented
14C22×C22⋊D181C2214yescomplemented
15C22×C22⋊D181C22196yescomplemented
16C2×D4×D181C22196yescomplemented
17C23×Dic181C2214yescomplemented
18C22×C4×D181C2214yescomplemented
19C25×C181C2212yescomplemented
20C22×D4×C181C2214yescomplemented
21C22×D4×C181C22124yescomplemented
22C23×C4×C181C2211yescharacteristic, complemented
23C23×D181—416no—
24C23×D181C2314yes—
25C23×D181C23148yescomplemented
26#15668449C231128yescomplemented
27#15668448C231256yescomplemented
28C2×Dic362C23124yescomplemented
29C2×D724C23124yescomplemented
30C2×C22⋊D181C23196yescomplemented
31C2×C22⋊D181C23148yescomplemented
32C2×C22⋊D181C231256yescomplemented
33D4×D181C231256yescomplemented
34C22×Dic181C2314yes—
35C22×Dic181C23148yescomplemented
36C2×C4×D181C23148yescomplemented
37C24×C181—48no—
38C24×C181C2311yescharacteristic
39C24×C181C2314yes—
40C24×C181C23124yescomplemented
41C2×D4×C181C23148yescomplemented
42C2×D4×C181C23164yescomplemented
43C22×C4×C181C2312yes—
44C22×C4×C181C23112yescomplemented
45C22×D181—216no—
46C22×D181—496no—
47C22×D181C24124yes—
48C22×D181C24164yescomplemented
49Dic362C24132yescomplemented
50D724C24132yescomplemented
51C22⋊D181C241128yescomplemented
52C22⋊D181C24164yescomplemented
53C2×Dic181C24124yes—
54C2×Dic181C24164yescomplemented
55C4×D181C24164yescomplemented
56C23×C181—24no—
57C23×C181—216no—
58C23×C181—448no—
59C23×C181C2416yes—
60C23×C181C24124yes—
61C23×C181C2411yescharacteristic
62C23×C181C24132yescomplemented
63D4×C181C24164yescomplemented
64C2×C4×C181C24112yes—
65C2×C4×C181C24116yescomplemented
66C22×C181—212no—
67C22×C181—248no—
68C22×C181—432no—
69C22×C181H(5,𝔽2)14yescomplemented
70C22×C181C2514yes—
71C22×C181C25116yes—
72C22×C181C2513yes—
73C2×D181—248no—
74C2×D181—464no—
75C2×D181C25116yes—
76C4×C181C2518yes—
77Dic181C25116yes—
78C2×C181—216no—
79C2×C181—24no—
80C2×C181C2×H(5,𝔽2)16yescomplemented
81C2×C181C2611yescharacteristic
82D181—216no—
83C181C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1811nomaximal, Sylow
85C2×H(5,𝔽2)—18148no—
86C22×#41—1814no—
87C22×#41—1812no—
88C23×D4—1814no—
89C23×D4—1814no—
90C23×D4D18111yescharacteristic, complemented
91C25—1814no—
92C25C2×D18112yescomplemented
93H(5,𝔽2)—181256no—
94C2×#41—18196no—
95C2×#41—18148no—
96C22×Q8—1812no—
97C22×D4—1818no—
98C22×D4—1814no—
99C22×D4—18196no—
100C22×D4—18196no—
101C22×D4—1812no—
102C22×D4C2×D18114yescomplemented
103C22×D4C2×D181124yescomplemented
104C23×C4—1814no—
105C23×C4—1814no—
106C23×C4C2×D18111yescharacteristic, complemented
107C24—48no—
108C24—1814no—
109C24—18148no—
110C24—72416no—
111C24C22×D18114yes—
112C24C22×D18111yescharacteristic
113C24C22×D181124yescomplemented
114#41—181256no—
115#41—181128no—
116C2×Q8—18124no—
117C2×D4—18196no—
118C2×D4—18148no—
119C2×D4—181256no—
120C2×D4—181256no—
121C2×D4—18124no—
122C2×D4C22×D181148yescomplemented
123C2×D4C22×D181164yescomplemented
124C22×C4—18148no—
125C22×C4—1814no—
126C22×C4—18148no—
127C22×C4C22×D18112yes—
128C22×C4C22×D181112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—36216no—
133C23—18124no—
134C23—18164no—
135C23—72496no—
136C23C23×D181124yes—
137C23C23×D18116yes—
138C23C23×D18111yescharacteristic
139C23C23×D181132yescomplemented
140Q8—18132no—
141D4—181128no—
142D4—18164no—
143D4—18132no—
144D4C23×D181164yescomplemented
145C2×C4—18124no—
146C2×C4—18164no—
147C2×C4—18164no—
148C2×C4C23×D181112yes—
149C2×C4C23×D181116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—36248no—
154C22—18116no—
155C22—72464no—
156C22#1915386114yesdirect product factor
157C22C24×D181116yes—
158C22C24×D18114yes—
159C22C24×D18113yes—
160C4—18116no—
161C4C24×D18118yes—
162C2—24no—
163C2—216no—
164C2—36216no—
165C2C2×#1915386116yesdirect product factor
166C2C25×D18111yescharacteristic
subgroupG/H
centerC23C23×D181
commutator, cosocleC2×C181C26
FrattiniC2C25×D181
FittingC23×D4×C181C2
socleC23×C181C24
hypercenterC23×D4D181
hypocenterC181C22×H(5,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19153861C2direct (N × H)
2. C22×#15668449C2semidirect
3. C22×#15668448C2semidirect
4. C23×C22⋊D181C2semidirect
5. C22×D4×D181C2semidirect
6. C23×D4×C181C2semidirect
7. C24×D181C22semidirect
8. #19153861C22direct (N × H)
9. C2×#15668449C22semidirect
10. C2×#15668448C22semidirect
11. C22×Dic362C22semidirect
12. C22×D724C22semidirect
13. C22×C22⋊D181C22semidirect
14. C22×C22⋊D181C22semidirect
15. C22×C22⋊D181C22semidirect
16. C2×D4×D181C22semidirect
17. C23×Dic181C22semidirect
18. C22×C4×D181C22semidirect
19. C25×C181C22semidirect
20. C22×D4×C181C22semidirect
21. C22×D4×C181C22semidirect
22. C23×C4×C181C22semidirect
25. C23×D181C23semidirect
26. #15668449C23semidirect
27. #15668448C23semidirect
28. C2×Dic362C23semidirect
29. C2×D724C23semidirect
30. C2×C22⋊D181C23semidirect
31. C2×C22⋊D181C23semidirect
32. C2×C22⋊D181C23semidirect
33. D4×D181C23semidirect
35. C22×Dic181C23semidirect
36. C2×C4×D181C23semidirect
40. C24×C181C23semidirect
41. C2×D4×C181C23semidirect
42. C2×D4×C181C23semidirect
44. C22×C4×C181C23semidirect
48. C22×D181C24semidirect
49. Dic362C24semidirect
50. D724C24semidirect
51. C22⋊D181C24semidirect
52. C22⋊D181C24semidirect
54. C2×Dic181C24semidirect
55. C4×D181C24semidirect
62. C23×C181C24semidirect
63. D4×C181C24semidirect
65. C2×C4×C181C24semidirect
69. C22×C181H(5,𝔽2)semidirect
80. C2×C181C2×H(5,𝔽2)semidirect
83. C181C22×H(5,𝔽2)semidirect
90. C23×D4D181semidirect
92. C25C2×D181semidirect
102. C22×D4C2×D181semidirect
103. C22×D4C2×D181semidirect
106. C23×C4C2×D181semidirect
113. C24C22×D181semidirect
122. C2×D4C22×D181semidirect
123. C2×D4C22×D181semidirect
128. C22×C4C22×D181semidirect
139. C23C23×D181semidirect
144. D4C23×D181semidirect
149. C2×C4C23×D181semidirect
156. C22#19153861direct (N × H)
165. C2C2×#19153861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191538612
C22×#156684492
C22×#156684482
C23×C22⋊D1812
C22×D4×D1812
C23×D4×C1812
C22×H(5,𝔽2)181