Finite Group Databasebeta

← C22×#15692300 #27185933

Subgroups — C22×#15692300 #27185933

Order 12928.

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×#15692300C2148yesmaximal, direct product factor
2C22×#13108074C212yesmaximal, complemented
3C22×#13108073C214yesmaximal, complemented
4C23×C22⋊D101C214yesmaximal, complemented
5C22×D4×D101C214yesmaximal, complemented
6C23×D4×C101C211yesmaximal, characteristic, complemented
7C24×D101C2214yescomplemented
8#15692300C221256yesdirect product factor
9C2×#13108074C22148yescomplemented
10C2×#13108073C22196yescomplemented
11C22×Dic202C2212yescomplemented
12C22×D404C2212yescomplemented
13C22×C22⋊D101C2218yescomplemented
14C22×C22⋊D101C2214yescomplemented
15C22×C22⋊D101C22196yescomplemented
16C2×D4×D101C22196yescomplemented
17C23×Dic101C2214yescomplemented
18C22×C4×D101C2214yescomplemented
19C25×C101C2212yescomplemented
20C22×D4×C101C2214yescomplemented
21C22×D4×C101C22124yescomplemented
22C23×C4×C101C2211yescharacteristic, complemented
23C23×D101—416no—
24C23×D101C2314yes—
25C23×D101C23148yescomplemented
26#13108074C231128yescomplemented
27#13108073C231256yescomplemented
28C2×Dic202C23124yescomplemented
29C2×D404C23124yescomplemented
30C2×C22⋊D101C23196yescomplemented
31C2×C22⋊D101C23148yescomplemented
32C2×C22⋊D101C231256yescomplemented
33D4×D101C231256yescomplemented
34C22×Dic101C2314yes—
35C22×Dic101C23148yescomplemented
36C2×C4×D101C23148yescomplemented
37C24×C101—48no—
38C24×C101C2314yes—
39C24×C101C2311yescharacteristic
40C24×C101C23124yescomplemented
41C2×D4×C101C23148yescomplemented
42C2×D4×C101C23164yescomplemented
43C22×C4×C101C2312yes—
44C22×C4×C101C23112yescomplemented
45C23×C101—24no—
46C23×C101—216no—
47C23×C101—448no—
48C23×C101C2416yes—
49C23×C101C24124yes—
50C23×C101C2411yescharacteristic
51C23×C101C24132yescomplemented
52C22×D101—216no—
53C22×D101—496no—
54C22×D101C24124yes—
55C22×D101C24164yescomplemented
56D4×C101C24164yescomplemented
57C2×C4×C101C24112yes—
58C2×C4×C101C24116yescomplemented
59C22⋊D101C241128yescomplemented
60C22⋊D101C24164yescomplemented
61C2×Dic101C24124yes—
62C2×Dic101C24164yescomplemented
63D404C24132yescomplemented
64C4×D101C24164yescomplemented
65Dic202C24132yescomplemented
66C22×C101—248no—
67C22×C101—212no—
68C22×C101—432no—
69C22×C101H(5,𝔽2)14yescomplemented
70C22×C101C25116yes—
71C22×C101C2514yes—
72C22×C101C2513yes—
73C2×D101—248no—
74C2×D101—464no—
75C2×D101C25116yes—
76C4×C101C2518yes—
77Dic101C25116yes—
78C2×C101—216no—
79C2×C101—24no—
80C2×C101C2×H(5,𝔽2)16yescomplemented
81C2×C101C2611yescharacteristic
82D101—216no—
83C22×H(5,𝔽2)—1011nomaximal, Sylow
84C101C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—10148no—
86C22×#41—1014no—
87C22×#41—1012no—
88C23×D4—1014no—
89C23×D4—1014no—
90C23×D4D10111yescharacteristic, complemented
91C25—1014no—
92C25C2×D10112yescomplemented
93H(5,𝔽2)—101256no—
94C2×#41—10196no—
95C2×#41—10148no—
96C22×Q8—1012no—
97C22×D4—1018no—
98C22×D4—1014no—
99C22×D4—10196no—
100C22×D4—10196no—
101C22×D4—1012no—
102C22×D4C2×D10114yescomplemented
103C22×D4C2×D101124yescomplemented
104C23×C4—1014no—
105C23×C4—1014no—
106C23×C4C2×D10111yescharacteristic, complemented
107C24—48no—
108C24—10148no—
109C24—1014no—
110C24—40416no—
111C24C22×D10114yes—
112C24C22×D10111yescharacteristic
113C24C22×D101124yescomplemented
114#41—101256no—
115#41—101128no—
116C2×Q8—10124no—
117C2×D4—10196no—
118C2×D4—10148no—
119C2×D4—101256no—
120C2×D4—101256no—
121C2×D4—10124no—
122C2×D4C22×D101148yescomplemented
123C2×D4C22×D101164yescomplemented
124C22×C4—10148no—
125C22×C4—1014no—
126C22×C4—10148no—
127C22×C4C22×D10112yes—
128C22×C4C22×D101112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—10164no—
133C23—10124no—
134C23—40496no—
135C23—20216no—
136C23C23×D101124yes—
137C23C23×D10116yes—
138C23C23×D10111yescharacteristic
139C23C23×D101132yescomplemented
140Q8—10132no—
141D4—101128no—
142D4—10164no—
143D4—10132no—
144D4C23×D101164yescomplemented
145C2×C4—10124no—
146C2×C4—10164no—
147C2×C4—10164no—
148C2×C4C23×D101112yes—
149C2×C4C23×D101116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—10116no—
154C22—40464no—
155C22—20248no—
156C22#1569230014yesdirect product factor
157C22C24×D101116yes—
158C22C24×D10114yes—
159C22C24×D10113yes—
160C4—10116no—
161C4C24×D10118yes—
162C2—24no—
163C2—216no—
164C2—20216no—
165C2C2×#1569230016yesdirect product factor
166C2C25×D10111yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC23×D4×C101C2
socleC23×C101C24
hypercenterC23×D4D101
hypocenterC101C22×H(5,𝔽2)
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15692300C2direct (N × H)
2. C22×#13108074C2semidirect
3. C22×#13108073C2semidirect
4. C23×C22⋊D101C2semidirect
5. C22×D4×D101C2semidirect
6. C23×D4×C101C2semidirect
7. C24×D101C22semidirect
8. #15692300C22direct (N × H)
9. C2×#13108074C22semidirect
10. C2×#13108073C22semidirect
11. C22×Dic202C22semidirect
12. C22×D404C22semidirect
13. C22×C22⋊D101C22semidirect
14. C22×C22⋊D101C22semidirect
15. C22×C22⋊D101C22semidirect
16. C2×D4×D101C22semidirect
17. C23×Dic101C22semidirect
18. C22×C4×D101C22semidirect
19. C25×C101C22semidirect
20. C22×D4×C101C22semidirect
21. C22×D4×C101C22semidirect
22. C23×C4×C101C22semidirect
25. C23×D101C23semidirect
26. #13108074C23semidirect
27. #13108073C23semidirect
28. C2×Dic202C23semidirect
29. C2×D404C23semidirect
30. C2×C22⋊D101C23semidirect
31. C2×C22⋊D101C23semidirect
32. C2×C22⋊D101C23semidirect
33. D4×D101C23semidirect
35. C22×Dic101C23semidirect
36. C2×C4×D101C23semidirect
40. C24×C101C23semidirect
41. C2×D4×C101C23semidirect
42. C2×D4×C101C23semidirect
44. C22×C4×C101C23semidirect
51. C23×C101C24semidirect
55. C22×D101C24semidirect
56. D4×C101C24semidirect
58. C2×C4×C101C24semidirect
59. C22⋊D101C24semidirect
60. C22⋊D101C24semidirect
62. C2×Dic101C24semidirect
63. D404C24semidirect
64. C4×D101C24semidirect
65. Dic202C24semidirect
69. C22×C101H(5,𝔽2)semidirect
80. C2×C101C2×H(5,𝔽2)semidirect
84. C101C22×H(5,𝔽2)semidirect
90. C23×D4D101semidirect
92. C25C2×D101semidirect
102. C22×D4C2×D101semidirect
103. C22×D4C2×D101semidirect
106. C23×C4C2×D101semidirect
113. C24C22×D101semidirect
122. C2×D4C22×D101semidirect
123. C2×D4C22×D101semidirect
128. C22×C4C22×D101semidirect
139. C23C23×D101semidirect
144. D4C23×D101semidirect
149. C2×C4C23×D101semidirect
156. C22#15692300direct (N × H)
165. C2C2×#15692300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923002
C22×#131080742
C22×#131080732
C23×C22⋊D1012
C22×D4×D1012
C23×D4×C1012
C22×H(5,𝔽2)101