Finite Group Databasebeta

← C22×#15690855 #27156537

Subgroups — C22×#15690855 #27156537

Order 12416.

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×#15690855C2148yesmaximal, direct product factor
2C22×#13095511C212yesmaximal, complemented
3C22×#13095510C214yesmaximal, complemented
4C23×C22⋊D97C214yesmaximal, complemented
5C22×D4×D97C214yesmaximal, complemented
6C23×D4×C97C211yesmaximal, characteristic, complemented
7C24×D97C2214yescomplemented
8#15690855C221256yesdirect product factor
9C2×#13095511C22148yescomplemented
10C2×#13095510C22196yescomplemented
11C22×Dic194C2212yescomplemented
12C22×D388C2212yescomplemented
13C22×C22⋊D97C2218yescomplemented
14C22×C22⋊D97C2214yescomplemented
15C22×C22⋊D97C22196yescomplemented
16C2×D4×D97C22196yescomplemented
17C23×Dic97C2214yescomplemented
18C22×C4×D97C2214yescomplemented
19C25×C97C2212yescomplemented
20C22×D4×C97C2214yescomplemented
21C22×D4×C97C22124yescomplemented
22C23×C4×C97C2211yescharacteristic, complemented
23C23×D97—416no—
24C23×D97C2314yes—
25C23×D97C23148yescomplemented
26#13095511C231128yescomplemented
27#13095510C231256yescomplemented
28C2×Dic194C23124yescomplemented
29C2×D388C23124yescomplemented
30C2×C22⋊D97C23196yescomplemented
31C2×C22⋊D97C23148yescomplemented
32C2×C22⋊D97C231256yescomplemented
33D4×D97C231256yescomplemented
34C22×Dic97C2314yes—
35C22×Dic97C23148yescomplemented
36C2×C4×D97C23148yescomplemented
37C24×C97—48no—
38C24×C97C2314yes—
39C24×C97C2311yescharacteristic
40C24×C97C23124yescomplemented
41C2×D4×C97C23148yescomplemented
42C2×D4×C97C23164yescomplemented
43C22×C4×C97C2312yes—
44C22×C4×C97C23112yescomplemented
45C23×C97—24no—
46C23×C97—216no—
47C23×C97—448no—
48C23×C97C2416yes—
49C23×C97C24124yes—
50C23×C97C2411yescharacteristic
51C23×C97C24132yescomplemented
52C22×D97—216no—
53C22×D97—496no—
54C22×D97C24124yes—
55C22×D97C24164yescomplemented
56D4×C97C24164yescomplemented
57C2×C4×C97C24112yes—
58C2×C4×C97C24116yescomplemented
59C22⋊D97C241128yescomplemented
60C22⋊D97C24164yescomplemented
61C2×Dic97C24124yes—
62C2×Dic97C24164yescomplemented
63D388C24132yescomplemented
64C4×D97C24164yescomplemented
65Dic194C24132yescomplemented
66C22×C97—248no—
67C22×C97—212no—
68C22×C97—432no—
69C22×C97H(5,𝔽2)14yescomplemented
70C22×C97C25116yes—
71C22×C97C2514yes—
72C22×C97C2513yes—
73C2×D97—248no—
74C2×D97—464no—
75C2×D97C25116yes—
76C4×C97C2518yes—
77Dic97C25116yes—
78C2×C97—216no—
79C2×C97—24no—
80C2×C97C2×H(5,𝔽2)16yescomplemented
81C2×C97C2611yescharacteristic
82D97—216no—
83C22×H(5,𝔽2)—971nomaximal, Sylow
84C97C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—9748no—
86C22×#41—974no—
87C22×#41—972no—
88C23×D4—974no—
89C23×D4—974no—
90C23×D4D9711yescharacteristic, complemented
91C25—974no—
92C25C2×D9712yescomplemented
93H(5,𝔽2)—97256no—
94C2×#41—9796no—
95C2×#41—9748no—
96C22×Q8—972no—
97C22×D4—978no—
98C22×D4—974no—
99C22×D4—9796no—
100C22×D4—9796no—
101C22×D4—972no—
102C22×D4C2×D9714yescomplemented
103C22×D4C2×D97124yescomplemented
104C23×C4—974no—
105C23×C4—974no—
106C23×C4C2×D9711yescharacteristic, complemented
107C24—48no—
108C24—9748no—
109C24—974no—
110C24—38816no—
111C24C22×D9714yes—
112C24C22×D9711yescharacteristic
113C24C22×D97124yescomplemented
114#41—97256no—
115#41—97128no—
116C2×Q8—9724no—
117C2×D4—9796no—
118C2×D4—9748no—
119C2×D4—97256no—
120C2×D4—97256no—
121C2×D4—9724no—
122C2×D4C22×D97148yescomplemented
123C2×D4C22×D97164yescomplemented
124C22×C4—9748no—
125C22×C4—974no—
126C22×C4—9748no—
127C22×C4C22×D9712yes—
128C22×C4C22×D97112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—9764no—
133C23—9724no—
134C23—38896no—
135C23—19416no—
136C23C23×D97124yes—
137C23C23×D9716yes—
138C23C23×D9711yescharacteristic
139C23C23×D97132yescomplemented
140Q8—9732no—
141D4—97128no—
142D4—9764no—
143D4—9732no—
144D4C23×D97164yescomplemented
145C2×C4—9724no—
146C2×C4—9764no—
147C2×C4—9764no—
148C2×C4C23×D97112yes—
149C2×C4C23×D97116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—9716no—
154C22—38864no—
155C22—19448no—
156C22#1569085514yesdirect product factor
157C22C24×D97116yes—
158C22C24×D9714yes—
159C22C24×D9713yes—
160C4—9716no—
161C4C24×D9718yes—
162C2—24no—
163C2—216no—
164C2—19416no—
165C2C2×#1569085516yesdirect product factor
166C2C25×D9711yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC23×D4×C97C2
socleC23×C97C24
hypercenterC23×D4D97
hypocenterC97C22×H(5,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690855C2direct (N × H)
2. C22×#13095511C2semidirect
3. C22×#13095510C2semidirect
4. C23×C22⋊D97C2semidirect
5. C22×D4×D97C2semidirect
6. C23×D4×C97C2semidirect
7. C24×D97C22semidirect
8. #15690855C22direct (N × H)
9. C2×#13095511C22semidirect
10. C2×#13095510C22semidirect
11. C22×Dic194C22semidirect
12. C22×D388C22semidirect
13. C22×C22⋊D97C22semidirect
14. C22×C22⋊D97C22semidirect
15. C22×C22⋊D97C22semidirect
16. C2×D4×D97C22semidirect
17. C23×Dic97C22semidirect
18. C22×C4×D97C22semidirect
19. C25×C97C22semidirect
20. C22×D4×C97C22semidirect
21. C22×D4×C97C22semidirect
22. C23×C4×C97C22semidirect
25. C23×D97C23semidirect
26. #13095511C23semidirect
27. #13095510C23semidirect
28. C2×Dic194C23semidirect
29. C2×D388C23semidirect
30. C2×C22⋊D97C23semidirect
31. C2×C22⋊D97C23semidirect
32. C2×C22⋊D97C23semidirect
33. D4×D97C23semidirect
35. C22×Dic97C23semidirect
36. C2×C4×D97C23semidirect
40. C24×C97C23semidirect
41. C2×D4×C97C23semidirect
42. C2×D4×C97C23semidirect
44. C22×C4×C97C23semidirect
51. C23×C97C24semidirect
55. C22×D97C24semidirect
56. D4×C97C24semidirect
58. C2×C4×C97C24semidirect
59. C22⋊D97C24semidirect
60. C22⋊D97C24semidirect
62. C2×Dic97C24semidirect
63. D388C24semidirect
64. C4×D97C24semidirect
65. Dic194C24semidirect
69. C22×C97H(5,𝔽2)semidirect
80. C2×C97C2×H(5,𝔽2)semidirect
84. C97C22×H(5,𝔽2)semidirect
90. C23×D4D97semidirect
92. C25C2×D97semidirect
102. C22×D4C2×D97semidirect
103. C22×D4C2×D97semidirect
106. C23×C4C2×D97semidirect
113. C24C22×D97semidirect
122. C2×D4C22×D97semidirect
123. C2×D4C22×D97semidirect
128. C22×C4C22×D97semidirect
139. C23C23×D97semidirect
144. D4C23×D97semidirect
149. C2×C4C23×D97semidirect
156. C22#15690855direct (N × H)
165. C2C2×#15690855direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908552
C22×#130955112
C22×#130955102
C23×C22⋊D972
C22×D4×D972
C23×D4×C972
C22×H(5,𝔽2)97