Finite Group Databasebeta

← C22×#16813667 #28405375

Subgroups — C22×#16813667 #28405375

Order 14464.

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×#16813667C2148yesmaximal, direct product factor
2C22×#14268647C212yesmaximal, complemented
3C22×#14268646C214yesmaximal, complemented
4C23×C22⋊D113C214yesmaximal, complemented
5C22×D4×D113C214yesmaximal, complemented
6C23×D4×C113C211yesmaximal, characteristic, complemented
7C24×D113C2214yescomplemented
8#16813667C221256yesdirect product factor
9C2×#14268647C22148yescomplemented
10C2×#14268646C22196yescomplemented
11C22×Dic226C2212yescomplemented
12C22×D452C2212yescomplemented
13C22×C22⋊D113C2218yescomplemented
14C22×C22⋊D113C2214yescomplemented
15C22×C22⋊D113C22196yescomplemented
16C2×D4×D113C22196yescomplemented
17C23×Dic113C2214yescomplemented
18C22×C4×D113C2214yescomplemented
19C25×C113C2212yescomplemented
20C22×D4×C113C2214yescomplemented
21C22×D4×C113C22124yescomplemented
22C23×C4×C113C2211yescharacteristic, complemented
23C23×D113—416no—
24C23×D113C2314yes—
25C23×D113C23148yescomplemented
26#14268647C231128yescomplemented
27#14268646C231256yescomplemented
28C2×Dic226C23124yescomplemented
29C2×D452C23124yescomplemented
30C2×C22⋊D113C23196yescomplemented
31C2×C22⋊D113C23148yescomplemented
32C2×C22⋊D113C231256yescomplemented
33D4×D113C231256yescomplemented
34C22×Dic113C2314yes—
35C22×Dic113C23148yescomplemented
36C2×C4×D113C23148yescomplemented
37C24×C113—48no—
38C24×C113C2314yes—
39C24×C113C2311yescharacteristic
40C24×C113C23124yescomplemented
41C2×D4×C113C23148yescomplemented
42C2×D4×C113C23164yescomplemented
43C22×C4×C113C2312yes—
44C22×C4×C113C23112yescomplemented
45C23×C113—24no—
46C23×C113—216no—
47C23×C113—448no—
48C23×C113C2416yes—
49C23×C113C24124yes—
50C23×C113C2411yescharacteristic
51C23×C113C24132yescomplemented
52C22×D113—216no—
53C22×D113—496no—
54C22×D113C24124yes—
55C22×D113C24164yescomplemented
56D4×C113C24164yescomplemented
57C2×C4×C113C24112yes—
58C2×C4×C113C24116yescomplemented
59C22⋊D113C241128yescomplemented
60C22⋊D113C24164yescomplemented
61C2×Dic113C24124yes—
62C2×Dic113C24164yescomplemented
63D452C24132yescomplemented
64C4×D113C24164yescomplemented
65Dic226C24132yescomplemented
66C22×C113—248no—
67C22×C113—212no—
68C22×C113—432no—
69C22×C113H(5,𝔽2)14yescomplemented
70C22×C113C25116yes—
71C22×C113C2514yes—
72C22×C113C2513yes—
73C2×D113—248no—
74C2×D113—464no—
75C2×D113C25116yes—
76C4×C113C2518yes—
77Dic113C25116yes—
78C2×C113—216no—
79C2×C113—24no—
80C2×C113C2×H(5,𝔽2)16yescomplemented
81C2×C113C2611yescharacteristic
82D113—216no—
83C22×H(5,𝔽2)—1131nomaximal, Sylow
84C113C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—11348no—
86C22×#41—1134no—
87C22×#41—1132no—
88C23×D4—1134no—
89C23×D4—1134no—
90C23×D4D11311yescharacteristic, complemented
91C25—1134no—
92C25C2×D11312yescomplemented
93H(5,𝔽2)—113256no—
94C2×#41—11396no—
95C2×#41—11348no—
96C22×Q8—1132no—
97C22×D4—1138no—
98C22×D4—1134no—
99C22×D4—11396no—
100C22×D4—11396no—
101C22×D4—1132no—
102C22×D4C2×D11314yescomplemented
103C22×D4C2×D113124yescomplemented
104C23×C4—1134no—
105C23×C4—1134no—
106C23×C4C2×D11311yescharacteristic, complemented
107C24—48no—
108C24—11348no—
109C24—1134no—
110C24—45216no—
111C24C22×D11314yes—
112C24C22×D11311yescharacteristic
113C24C22×D113124yescomplemented
114#41—113256no—
115#41—113128no—
116C2×Q8—11324no—
117C2×D4—11396no—
118C2×D4—11348no—
119C2×D4—113256no—
120C2×D4—113256no—
121C2×D4—11324no—
122C2×D4C22×D113148yescomplemented
123C2×D4C22×D113164yescomplemented
124C22×C4—11348no—
125C22×C4—1134no—
126C22×C4—11348no—
127C22×C4C22×D11312yes—
128C22×C4C22×D113112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—11364no—
133C23—11324no—
134C23—45296no—
135C23—22616no—
136C23C23×D113124yes—
137C23C23×D11316yes—
138C23C23×D11311yescharacteristic
139C23C23×D113132yescomplemented
140Q8—11332no—
141D4—113128no—
142D4—11364no—
143D4—11332no—
144D4C23×D113164yescomplemented
145C2×C4—11324no—
146C2×C4—11364no—
147C2×C4—11364no—
148C2×C4C23×D113112yes—
149C2×C4C23×D113116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—11316no—
154C22—45264no—
155C22—22648no—
156C22#1681366714yesdirect product factor
157C22C24×D113116yes—
158C22C24×D11314yes—
159C22C24×D11313yes—
160C4—11316no—
161C4C24×D11318yes—
162C2—24no—
163C2—216no—
164C2—22616no—
165C2C2×#1681366716yesdirect product factor
166C2C25×D11311yescharacteristic
subgroupG/H
centerC23C23×D113
commutator, cosocleC2×C113C26
FrattiniC2C25×D113
FittingC23×D4×C113C2
socleC23×C113C24
hypercenterC23×D4D113
hypocenterC113C22×H(5,𝔽2)
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16813667C2direct (N × H)
2. C22×#14268647C2semidirect
3. C22×#14268646C2semidirect
4. C23×C22⋊D113C2semidirect
5. C22×D4×D113C2semidirect
6. C23×D4×C113C2semidirect
7. C24×D113C22semidirect
8. #16813667C22direct (N × H)
9. C2×#14268647C22semidirect
10. C2×#14268646C22semidirect
11. C22×Dic226C22semidirect
12. C22×D452C22semidirect
13. C22×C22⋊D113C22semidirect
14. C22×C22⋊D113C22semidirect
15. C22×C22⋊D113C22semidirect
16. C2×D4×D113C22semidirect
17. C23×Dic113C22semidirect
18. C22×C4×D113C22semidirect
19. C25×C113C22semidirect
20. C22×D4×C113C22semidirect
21. C22×D4×C113C22semidirect
22. C23×C4×C113C22semidirect
25. C23×D113C23semidirect
26. #14268647C23semidirect
27. #14268646C23semidirect
28. C2×Dic226C23semidirect
29. C2×D452C23semidirect
30. C2×C22⋊D113C23semidirect
31. C2×C22⋊D113C23semidirect
32. C2×C22⋊D113C23semidirect
33. D4×D113C23semidirect
35. C22×Dic113C23semidirect
36. C2×C4×D113C23semidirect
40. C24×C113C23semidirect
41. C2×D4×C113C23semidirect
42. C2×D4×C113C23semidirect
44. C22×C4×C113C23semidirect
51. C23×C113C24semidirect
55. C22×D113C24semidirect
56. D4×C113C24semidirect
58. C2×C4×C113C24semidirect
59. C22⋊D113C24semidirect
60. C22⋊D113C24semidirect
62. C2×Dic113C24semidirect
63. D452C24semidirect
64. C4×D113C24semidirect
65. Dic226C24semidirect
69. C22×C113H(5,𝔽2)semidirect
80. C2×C113C2×H(5,𝔽2)semidirect
84. C113C22×H(5,𝔽2)semidirect
90. C23×D4D113semidirect
92. C25C2×D113semidirect
102. C22×D4C2×D113semidirect
103. C22×D4C2×D113semidirect
106. C23×C4C2×D113semidirect
113. C24C22×D113semidirect
122. C2×D4C22×D113semidirect
123. C2×D4C22×D113semidirect
128. C22×C4C22×D113semidirect
139. C23C23×D113semidirect
144. D4C23×D113semidirect
149. C2×C4C23×D113semidirect
156. C22#16813667direct (N × H)
165. C2C2×#16813667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136672
C22×#142686472
C22×#142686462
C23×C22⋊D1132
C22×D4×D1132
C23×D4×C1132
C22×H(5,𝔽2)113