Finite Group Databasebeta

← #27164652

Subgroups — #27164652

Order 12736.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×D796C211yesmaximal, characteristic, complemented
2C2×D4×D199C212yesmaximal, complemented
3C2×D4×D199C211yesmaximal, characteristic, complemented
4#20263951C211yesmaximal, characteristic, complemented
5#20263930C211yesmaximal, characteristic, complemented
6#20263928C212yesmaximal, complemented
7#20263925C212yesmaximal, complemented
8#20263923C212yesmaximal, complemented
9#20263917C211yesmaximal, characteristic, complemented
10C4×C22⋊D199C211yesmaximal, characteristic, complemented
11#121×C199C211yesmaximal, characteristic, complemented
12C23×D199C2212yescomplemented
13C23×D199C2212yescomplemented
14C2×D796—24no—
15C2×D796C2211yescharacteristic
16C2×D796C2211yescharacteristic, complemented
17C2×D796C2212yescomplemented
18C2×D796C2212yescomplemented
19C2×C22⋊D199C2211yescharacteristic
20C2×C22⋊D199C2212yescomplemented
21C2×C22⋊D199C2212yescomplemented
22C2×C22⋊D199C2212yescomplemented
23D4×D199—24no—
24D4×D199—28no—
25C2×C4×D199C2211yescharacteristic, complemented
26C2×C4×D199C2212yescomplemented
27Dic199⋊C4C2211yescharacteristic, complemented
28C22⋊Dic199C2211yescharacteristic, complemented
29(C2×D199)⋊C4C2211yescharacteristic, complemented
30(C2×D199)⋊C4C2212yescomplemented
31(C2×D199)⋊C4C2212yescomplemented
32C4×Dic199C2211yescharacteristic, complemented
33C2×D4×C199C2211yescharacteristic, complemented
34C2×D4×C199C2212yescomplemented
35C22×C4×C199C2211yescharacteristic, complemented
36C4⋊C4×C199C2211yescharacteristic, complemented
37C22⋊C4×C199C2212yescomplemented
38C22×D199—24no—
39C22×D199—24no—
40C22×D199—24no—
41C22×D199—24no—
42C22×D199—42no—
43C22×D199—42no—
44C22×D199C2312yes—
45C22×D199C2311yescharacteristic
46C22×D199C2312yes—
47D796—24no—
48D796—24no—
49D796—22no—
50D796D414yescomplemented
51C22⋊D199—24no—
52C22⋊D199—24no—
53C22⋊D199—42no—
54C22⋊D199D414yescomplemented
55C2×Dic199C2311yescharacteristic
56C2×Dic199C2311yescharacteristic
57C2×Dic199C2311yescharacteristic
58C4×D199—22no—
59C4×D199—24no—
60C23×C199C2312yes—
61C23×C199C2311yescharacteristic
62D4×C199—24no—
63D4×C199—22no—
64C2×C4×C199—22no—
65C2×C4×C199C2311yescharacteristic
66C2×C4×C199C2312yes—
67C2×C4×C199C2311yescharacteristic
68C22×C199—21no—
69C22×C199—21no—
70C22×C199—22no—
71C22×C199—22no—
72C22×C199—22no—
73C22×C199C2×D412yescomplemented
74C22×C199C2411yescharacteristic
75C2×D199—21no—
76C2×D199—22no—
77C2×D199—21no—
78C2×D199—22no—
79C2×D199—22no—
80C2×D199—22no—
81C2×D199—22no—
82C2×D199—48no—
83C2×D199—48no—
84C2×D199C2×D414yescomplemented
85C2×D199C2×D412yescomplemented
86C4×C199—21no—
87C4×C199—22no—
88C4×C199C2×D412yescomplemented
89Dic199—21no—
90Dic199—21no—
91Dic199C2×D412yescomplemented
92C2×C199—22no—
93C2×C199—42no—
94C2×C199C22×D411yescharacteristic
95C2×C199C22×D411yescharacteristic
96C2×C199H(5,𝔽2)11yescharacteristic
97D199—22no—
98D199—24no—
99D199—42no—
100C199D4211yescharacteristic, complemented, Sylow
101D42—1991nomaximal, Sylow
102C22×D4—1991no—
103C22×D4—1992no—
104C22×D4—1991no—
105#127—1991no—
106#121—1991no—
107#121—1992no—
108#121D19911yescharacteristic, complemented
109#120—1992no—
110#120—1992no—
111C4×D4—1991no—
112C4×D4—1991no—
113C24—1992no—
114C24—1992no—
115C2×D4—3984no—
116C2×D4—3988no—
117C2×D4—3984no—
118C2×D4—1992no—
119C2×D4—1992no—
120C2×D4—1992no—
121C2×D4—1991no—
122C2×D4—1992no—
123C2×D4—1992no—
124C2×D4—1991no—
125C2×D4—1991no—
126C2×D4C2×D19912yescomplemented
127C2×D4C2×D19911yescharacteristic, complemented
128C22×C4—1992no—
129C22×C4—1991no—
130C22×C4C2×D19911yescharacteristic, complemented
131C4⋊C4—1991no—
132C4⋊C4C2×D19911yescharacteristic, complemented
133C22⋊C4—1991no—
134C22⋊C4—1992no—
135C22⋊C4—1991no—
136C22⋊C4—1992no—
137C22⋊C4C2×D19912yescomplemented
138C42—1991no—
139C23—7962no—
140C23—7962no—
141C23—3984no—
142C23—3984no—
143C23—3984no—
144C23—3984no—
145C23—1992no—
146C23—1992no—
147C23—1991no—
148C23C22×D19912yes—
149C23C22×D19911yescharacteristic
150D4—24no—
151D4—22no—
152D4—7962no—
153D4—3984no—
154D4—3984no—
155D4—3984no—
156D4—3984no—
157D4—3982no—
158D4—1994no—
159D4—1994no—
160C2×C4—22no—
161C2×C4—3984no—
162C2×C4—3982no—
163C2×C4—1991no—
164C2×C4—1991no—
165C2×C4—1991no—
166C2×C4C22×D19912yes—
167C2×C4C22×D19911yescharacteristic
168C2×C4C22×D19911yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—7968no—
175C22—7968no—
176C22—3982no—
177C22—3982no—
178C22—3982no—
179C22—3982no—
180C22—3981no—
181C22—3982no—
182C22—3981no—
183C22—1992no—
184C22—1994no—
185C22D4×D19912yescomplemented
186C22C23×D19911yescharacteristic
187C4—21no—
188C4—22no—
189C4—3981no—
190C4—3981no—
191C4—1992no—
192C4D4×D19912yescomplemented
193C2—22no—
194C2—42no—
195C2—7962no—
196C2—3984no—
197C2—3982no—
198C2C2×D4×D19911yescharacteristic
199C2C2×D4×D19911yescharacteristic
200C2#2026400111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#121×C199C2
hypercenter#121D199
hypocenterC199D42
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D796C2semidirect
2. C2×D4×D199C2semidirect
3. C2×D4×D199C2semidirect
4. #20263951C2semidirect
5. #20263930C2semidirect
6. #20263928C2semidirect
7. #20263925C2semidirect
8. #20263923C2semidirect
9. #20263917C2semidirect
10. C4×C22⋊D199C2semidirect
11. #121×C199C2semidirect
12. C23×D199C22semidirect
13. C23×D199C22semidirect
16. C2×D796C22semidirect
17. C2×D796C22semidirect
18. C2×D796C22semidirect
20. C2×C22⋊D199C22semidirect
21. C2×C22⋊D199C22semidirect
22. C2×C22⋊D199C22semidirect
25. C2×C4×D199C22semidirect
26. C2×C4×D199C22semidirect
27. Dic199⋊C4C22semidirect
28. C22⋊Dic199C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. (C2×D199)⋊C4C22semidirect
32. C4×Dic199C22semidirect
33. C2×D4×C199C22semidirect
34. C2×D4×C199C22semidirect
35. C22×C4×C199C22semidirect
36. C4⋊C4×C199C22semidirect
37. C22⋊C4×C199C22semidirect
50. D796D4semidirect
54. C22⋊D199D4semidirect
73. C22×C199C2×D4semidirect
84. C2×D199C2×D4semidirect
85. C2×D199C2×D4semidirect
88. C4×C199C2×D4semidirect
91. Dic199C2×D4semidirect
100. C199D42semidirect
108. #121D199semidirect
126. C2×D4C2×D199semidirect
127. C2×D4C2×D199semidirect
130. C22×C4C2×D199semidirect
132. C4⋊C4C2×D199semidirect
137. C22⋊C4C2×D199semidirect
185. C22D4×D199semidirect
192. C4D4×D199semidirect

Maximal subgroups

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

subgroupindex
C22×D7962
C2×D4×D1992
#202639512
#202639302
#202639282
#202639252
#202639232
#202639172
C4×C22⋊D1992
#121×C1992
D42199