Finite Group Databasebeta

← C22×#19153862 #37548523

Subgroups — C22×#19153862 #37548523

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×#19153862C2148yesmaximal, direct product factor
2C22×#15668450C212yesmaximal, complemented
3C22×#15668449C213yesmaximal, complemented
4C23×D724C213yesmaximal, complemented
5C22×D4×D181C216yesmaximal, complemented
6C22×#41×C181C211yesmaximal, characteristic, complemented
7C24×D181C2216yescomplemented
8#19153862C221256yesdirect product factor
9C2×#15668450C22148yescomplemented
10C2×#15668449C22172yescomplemented
11C22×Dic362C2211yescharacteristic, complemented
12C22×D724C2216yescomplemented
13C22×D724C2213yescomplemented
14C22×D724C22172yescomplemented
15C22×C22⋊D181C2216yescomplemented
16C2×D4×D181C221144yescomplemented
17C22×C4×D181C2216yescomplemented
18C2×#41×C181C22124yescomplemented
19C22×Q8×C181C2211yescharacteristic, complemented
20C22×D4×C181C2213yescomplemented
21C23×C4×C181C2213yescomplemented
22C23×D181—424no—
23C23×D181C2316yes—
24C23×D181C23172yescomplemented
25#15668450C231128yescomplemented
26#15668449C231192yescomplemented
27C2×Dic362C23112yescomplemented
28C2×D724C23172yescomplemented
29C2×D724C23136yescomplemented
30C2×D724C231192yescomplemented
31C2×C22⋊D181C23172yescomplemented
32D4×D181C231384yescomplemented
33C22×Dic181C2312yes—
34C2×C4×D181C23172yescomplemented
35C24×C181C2313yes—
36#41×C181C23164yescomplemented
37C2×Q8×C181C23112yescomplemented
38C2×D4×C181C23136yescomplemented
39C22×C4×C181C2311yescharacteristic
40C22×C4×C181C2313yes—
41C22×C4×C181C23136yescomplemented
42C22×D181—224no—
43C22×D181—4144no—
44C22×D181C24136yes—
45C22×D181C24196yescomplemented
46Dic362C24116yescomplemented
47D724C24196yescomplemented
48D724C24148yescomplemented
49C22⋊D181C24196yescomplemented
50C2×Dic181C24112yes—
51C4×D181C24196yescomplemented
52C23×C181—212no—
53C23×C181C24118yes—
54C23×C181C2411yescharacteristic
55Q8×C181C24116yescomplemented
56D4×C181C24148yescomplemented
57C2×C4×C181C2416yes—
58C2×C4×C181C24118yes—
59C2×C4×C181C24148yescomplemented
60C22×C181—236no—
61C22×C181H(5,𝔽2)14yescomplemented
62C22×C181C25112yes—
63C22×C181C2513yes—
64C2×D181—272no—
65C2×D181—496no—
66C2×D181C25124yes—
67C4×C181C2514yes—
68C4×C181C25112yes—
69Dic181C2518yes—
70C2×C181—212no—
71C2×C181C2×H(5,𝔽2)16yescomplemented
72C2×C181C2611yescharacteristic
73D181—224no—
74C181C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1811nomaximal, Sylow
76C2×H(5,𝔽2)—18148no—
77C22×#41—1812no—
78C22×#41—1813no—
79C22×#41D18111yescharacteristic, complemented
80C23×D4—1816no—
81C23×D4—1813no—
82C25—1816no—
83H(5,𝔽2)—181256no—
84C2×#41—18148no—
85C2×#41—18172no—
86C2×#41C2×D181124yescomplemented
87C22×Q8—1811no—
88C22×Q8C2×D18111yescharacteristic, complemented
89C22×D4—1816no—
90C22×D4—181144no—
91C22×D4—1816no—
92C22×D4—1813no—
93C22×D4—18172no—
94C22×D4C2×D18113yescomplemented
95C23×C4—1816no—
96C23×C4C2×D18113yescomplemented
97C24—1816no—
98C24—18172no—
99C24—72424no—
100C24C22×D18113yes—
101#41—181128no—
102#41—181192no—
103#41C22×D181164yescomplemented
104C2×Q8—18112no—
105C2×Q8C22×D181112yescomplemented
106C2×D4—18172no—
107C2×D4—181384no—
108C2×D4—18172no—
109C2×D4—18136no—
110C2×D4—181192no—
111C2×D4C22×D181136yescomplemented
112C22×C4—1812no—
113C22×C4—18172no—
114C22×C4C22×D18113yes—
115C22×C4C22×D18111yescharacteristic
116C22×C4C22×D181136yescomplemented
117C23—212no—
118C23—36224no—
119C23—18136no—
120C23—18196no—
121C23—724144no—
122C23C23×D181118yes—
123C23C23×D18111yescharacteristic
124Q8—18116no—
125Q8C23×D181116yescomplemented
126D4—18196no—
127D4—18196no—
128D4—18148no—
129D4C23×D181148yescomplemented
130C2×C4—18112no—
131C2×C4—18196no—
132C2×C4C23×D181118yes—
133C2×C4C23×D18116yes—
134C2×C4C23×D181148yescomplemented
135C22—236no—
136C22—36272no—
137C22—18124no—
138C22—72496no—
139C22#1915386214yesdirect product factor
140C22C24×D181112yes—
141C22C24×D18113yes—
142C4—1818no—
143C4C24×D181112yes—
144C4C24×D18114yes—
145C2—212no—
146C2—36224no—
147C2C2×#1915386216yesdirect product factor
148C2C25×D18111yescharacteristic
subgroupG/H
centerC23C23×D181
commutator, cosocleC2×C181C26
FrattiniC2C25×D181
FittingC22×#41×C181C2
socleC23×C181C24
hypercenterC22×#41D181
hypocenterC181C22×H(5,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2direct (N × H)
2. C22×#15668450C2semidirect
3. C22×#15668449C2semidirect
4. C23×D724C2semidirect
5. C22×D4×D181C2semidirect
6. C22×#41×C181C2semidirect
7. C24×D181C22semidirect
8. #19153862C22direct (N × H)
9. C2×#15668450C22semidirect
10. C2×#15668449C22semidirect
11. C22×Dic362C22semidirect
12. C22×D724C22semidirect
13. C22×D724C22semidirect
14. C22×D724C22semidirect
15. C22×C22⋊D181C22semidirect
16. C2×D4×D181C22semidirect
17. C22×C4×D181C22semidirect
18. C2×#41×C181C22semidirect
19. C22×Q8×C181C22semidirect
20. C22×D4×C181C22semidirect
21. C23×C4×C181C22semidirect
24. C23×D181C23semidirect
25. #15668450C23semidirect
26. #15668449C23semidirect
27. C2×Dic362C23semidirect
28. C2×D724C23semidirect
29. C2×D724C23semidirect
30. C2×D724C23semidirect
31. C2×C22⋊D181C23semidirect
32. D4×D181C23semidirect
34. C2×C4×D181C23semidirect
36. #41×C181C23semidirect
37. C2×Q8×C181C23semidirect
38. C2×D4×C181C23semidirect
41. C22×C4×C181C23semidirect
45. C22×D181C24semidirect
46. Dic362C24semidirect
47. D724C24semidirect
48. D724C24semidirect
49. C22⋊D181C24semidirect
51. C4×D181C24semidirect
55. Q8×C181C24semidirect
56. D4×C181C24semidirect
59. C2×C4×C181C24semidirect
61. C22×C181H(5,𝔽2)semidirect
71. C2×C181C2×H(5,𝔽2)semidirect
74. C181C22×H(5,𝔽2)semidirect
79. C22×#41D181semidirect
86. C2×#41C2×D181semidirect
88. C22×Q8C2×D181semidirect
94. C22×D4C2×D181semidirect
96. C23×C4C2×D181semidirect
103. #41C22×D181semidirect
105. C2×Q8C22×D181semidirect
111. C2×D4C22×D181semidirect
116. C22×C4C22×D181semidirect
125. Q8C23×D181semidirect
129. D4C23×D181semidirect
134. C2×C4C23×D181semidirect
139. C22#19153862direct (N × H)
147. C2C2×#19153862direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191538622
C22×#156684502
C22×#156684492
C23×D7242
C22×D4×D1812
C22×#41×C1812
C22×H(5,𝔽2)181