Finite Group Databasebeta

← C22×#19128877 #36338939

Subgroups — C22×#19128877 #36338939

Order 21376.

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×#19128877C2148yesmaximal, direct product factor
2C22×#14581950C212yesmaximal, complemented
3C22×#14581949C214yesmaximal, complemented
4C23×C22⋊D167C214yesmaximal, complemented
5C22×D4×D167C214yesmaximal, complemented
6C23×D4×C167C211yesmaximal, characteristic, complemented
7C24×D167C2214yescomplemented
8#19128877C221256yesdirect product factor
9C2×#14581950C22148yescomplemented
10C2×#14581949C22196yescomplemented
11C22×Dic334C2212yescomplemented
12C22×D668C2212yescomplemented
13C22×C22⋊D167C2218yescomplemented
14C22×C22⋊D167C2214yescomplemented
15C22×C22⋊D167C22196yescomplemented
16C2×D4×D167C22196yescomplemented
17C23×Dic167C2214yescomplemented
18C22×C4×D167C2214yescomplemented
19C25×C167C2212yescomplemented
20C22×D4×C167C2214yescomplemented
21C22×D4×C167C22124yescomplemented
22C23×C4×C167C2211yescharacteristic, complemented
23C23×D167—416no—
24C23×D167C2314yes—
25C23×D167C23148yescomplemented
26#14581950C231128yescomplemented
27#14581949C231256yescomplemented
28C2×Dic334C23124yescomplemented
29C2×D668C23124yescomplemented
30C2×C22⋊D167C23196yescomplemented
31C2×C22⋊D167C23148yescomplemented
32C2×C22⋊D167C231256yescomplemented
33D4×D167C231256yescomplemented
34C22×Dic167C2314yes—
35C22×Dic167C23148yescomplemented
36C2×C4×D167C23148yescomplemented
37C24×C167—48no—
38C24×C167C2314yes—
39C24×C167C2311yescharacteristic
40C24×C167C23124yescomplemented
41C2×D4×C167C23148yescomplemented
42C2×D4×C167C23164yescomplemented
43C22×C4×C167C2312yes—
44C22×C4×C167C23112yescomplemented
45C22×D167—216no—
46C22×D167—496no—
47C22×D167C24124yes—
48C22×D167C24164yescomplemented
49Dic334C24132yescomplemented
50D668C24132yescomplemented
51C22⋊D167C241128yescomplemented
52C22⋊D167C24164yescomplemented
53C2×Dic167C24124yes—
54C2×Dic167C24164yescomplemented
55C4×D167C24164yescomplemented
56C23×C167—24no—
57C23×C167—216no—
58C23×C167—448no—
59C23×C167C2416yes—
60C23×C167C24124yes—
61C23×C167C2411yescharacteristic
62C23×C167C24132yescomplemented
63D4×C167C24164yescomplemented
64C2×C4×C167C24112yes—
65C2×C4×C167C24116yescomplemented
66C22×C167—212no—
67C22×C167—248no—
68C22×C167—432no—
69C22×C167H(5,𝔽2)14yescomplemented
70C22×C167C2514yes—
71C22×C167C25116yes—
72C22×C167C2513yes—
73C2×D167—248no—
74C2×D167—464no—
75C2×D167C25116yes—
76C4×C167C2518yes—
77Dic167C25116yes—
78C2×C167—216no—
79C2×C167—24no—
80C2×C167C2×H(5,𝔽2)16yescomplemented
81C2×C167C2611yescharacteristic
82D167—216no—
83C167C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1671nomaximal, Sylow
85C2×H(5,𝔽2)—16748no—
86C22×#41—1674no—
87C22×#41—1672no—
88C23×D4—1674no—
89C23×D4—1674no—
90C23×D4D16711yescharacteristic, complemented
91C25—1674no—
92C25C2×D16712yescomplemented
93H(5,𝔽2)—167256no—
94C2×#41—16796no—
95C2×#41—16748no—
96C22×Q8—1672no—
97C22×D4—1678no—
98C22×D4—1674no—
99C22×D4—16796no—
100C22×D4—16796no—
101C22×D4—1672no—
102C22×D4C2×D16714yescomplemented
103C22×D4C2×D167124yescomplemented
104C23×C4—1674no—
105C23×C4—1674no—
106C23×C4C2×D16711yescharacteristic, complemented
107C24—48no—
108C24—66816no—
109C24—1674no—
110C24—16748no—
111C24C22×D16714yes—
112C24C22×D16711yescharacteristic
113C24C22×D167124yescomplemented
114#41—167256no—
115#41—167128no—
116C2×Q8—16724no—
117C2×D4—16796no—
118C2×D4—16748no—
119C2×D4—167256no—
120C2×D4—167256no—
121C2×D4—16724no—
122C2×D4C22×D167148yescomplemented
123C2×D4C22×D167164yescomplemented
124C22×C4—16748no—
125C22×C4—1674no—
126C22×C4—16748no—
127C22×C4C22×D16712yes—
128C22×C4C22×D167112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—33416no—
133C23—66896no—
134C23—16724no—
135C23—16764no—
136C23C23×D167124yes—
137C23C23×D16716yes—
138C23C23×D16711yescharacteristic
139C23C23×D167132yescomplemented
140Q8—16732no—
141D4—167128no—
142D4—16764no—
143D4—16732no—
144D4C23×D167164yescomplemented
145C2×C4—16724no—
146C2×C4—16764no—
147C2×C4—16764no—
148C2×C4C23×D167112yes—
149C2×C4C23×D167116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—33448no—
154C22—66864no—
155C22—16716no—
156C22#1912887714yesdirect product factor
157C22C24×D167116yes—
158C22C24×D16714yes—
159C22C24×D16713yes—
160C4—16716no—
161C4C24×D16718yes—
162C2—24no—
163C2—216no—
164C2—33416no—
165C2C2×#1912887716yesdirect product factor
166C2C25×D16711yescharacteristic
subgroupG/H
centerC23C23×D167
commutator, cosocleC2×C167C26
FrattiniC2C25×D167
FittingC23×D4×C167C2
socleC23×C167C24
hypercenterC23×D4D167
hypocenterC167C22×H(5,𝔽2)
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19128877C2direct (N × H)
2. C22×#14581950C2semidirect
3. C22×#14581949C2semidirect
4. C23×C22⋊D167C2semidirect
5. C22×D4×D167C2semidirect
6. C23×D4×C167C2semidirect
7. C24×D167C22semidirect
8. #19128877C22direct (N × H)
9. C2×#14581950C22semidirect
10. C2×#14581949C22semidirect
11. C22×Dic334C22semidirect
12. C22×D668C22semidirect
13. C22×C22⋊D167C22semidirect
14. C22×C22⋊D167C22semidirect
15. C22×C22⋊D167C22semidirect
16. C2×D4×D167C22semidirect
17. C23×Dic167C22semidirect
18. C22×C4×D167C22semidirect
19. C25×C167C22semidirect
20. C22×D4×C167C22semidirect
21. C22×D4×C167C22semidirect
22. C23×C4×C167C22semidirect
25. C23×D167C23semidirect
26. #14581950C23semidirect
27. #14581949C23semidirect
28. C2×Dic334C23semidirect
29. C2×D668C23semidirect
30. C2×C22⋊D167C23semidirect
31. C2×C22⋊D167C23semidirect
32. C2×C22⋊D167C23semidirect
33. D4×D167C23semidirect
35. C22×Dic167C23semidirect
36. C2×C4×D167C23semidirect
40. C24×C167C23semidirect
41. C2×D4×C167C23semidirect
42. C2×D4×C167C23semidirect
44. C22×C4×C167C23semidirect
48. C22×D167C24semidirect
49. Dic334C24semidirect
50. D668C24semidirect
51. C22⋊D167C24semidirect
52. C22⋊D167C24semidirect
54. C2×Dic167C24semidirect
55. C4×D167C24semidirect
62. C23×C167C24semidirect
63. D4×C167C24semidirect
65. C2×C4×C167C24semidirect
69. C22×C167H(5,𝔽2)semidirect
80. C2×C167C2×H(5,𝔽2)semidirect
83. C167C22×H(5,𝔽2)semidirect
90. C23×D4D167semidirect
92. C25C2×D167semidirect
102. C22×D4C2×D167semidirect
103. C22×D4C2×D167semidirect
106. C23×C4C2×D167semidirect
113. C24C22×D167semidirect
122. C2×D4C22×D167semidirect
123. C2×D4C22×D167semidirect
128. C22×C4C22×D167semidirect
139. C23C23×D167semidirect
144. D4C23×D167semidirect
149. C2×C4C23×D167semidirect
156. C22#19128877direct (N × H)
165. C2C2×#19128877direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191288772
C22×#145819502
C22×#145819492
C23×C22⋊D1672
C22×D4×D1672
C23×D4×C1672
C22×H(5,𝔽2)167