Finite Group Databasebeta

← #27164583

Subgroups — #27164583

Order 12736.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#15691885C211yesmaximal, characteristic, complemented
2C22×C22⋊D199C211yesmaximal, characteristic, complemented
3#20263951C212yesmaximal, complemented
4#20263941C212yesmaximal, complemented
5#20263928C212yesmaximal, complemented
6#20263927C212yesmaximal, complemented
7#20263925C211yesmaximal, characteristic, complemented
8#20263924C211yesmaximal, characteristic, complemented
9#20263923C211yesmaximal, characteristic, complemented
10#20263911C211yesmaximal, characteristic, complemented
11#120×C199C211yesmaximal, characteristic, complemented
12C23×D199C2211yescharacteristic
13#15691885—24no—
14C2×Dic398C2211yescharacteristic, complemented
15C2×D796C2212yescomplemented
16C2×C22⋊D199—24no—
17C2×C22⋊D199C2212yes—
18C2×C22⋊D199C2212yes—
19C2×C22⋊D199C2211yescharacteristic
20C2×C22⋊D199C2211yescharacteristic, complemented
21C2×C22⋊D199C2212yescomplemented
22C2×C22⋊D199C2212yescomplemented
23C22×Dic199C2211yescharacteristic
24C22×Dic199C2211yescharacteristic, complemented
25C2×C4×D199C2211yescharacteristic, complemented
26Dic199⋊C4C2212yescomplemented
27C22⋊Dic199C2211yescharacteristic
28C22⋊Dic199C2212yescomplemented
29(C2×D199)⋊C4C2212yescomplemented
30(C2×D199)⋊C4C2212yescomplemented
31C4×Dic199C2212yescomplemented
32C24×C199C2211yescharacteristic
33C2×D4×C199C2211yescharacteristic
34C2×D4×C199C2212yescomplemented
35C22⋊C4×C199C2211yescharacteristic
36C22⋊C4×C199C2212yescomplemented
37C22×D199—22no—
38C22×D199—42no—
39C22×D199C2312yes—
40C22×D199C2311yescharacteristic
41Dic398—22no—
42D796—42no—
43C22⋊D199—22no—
44C22⋊D199—24no—
45C22⋊D199—24no—
46C22⋊D199—22no—
47C22⋊D199—42no—
48C22⋊D199—42no—
49C2×Dic199—22no—
50C2×Dic199D414yescomplemented
51C2×Dic199C2312yes—
52C2×Dic199C2311yescharacteristic
53C2×Dic199C2311yescharacteristic
54C2×Dic199C2311yescharacteristic
55C4×D199—22no—
56C23×C199—22no—
57C23×C199—42no—
58C23×C199C2312yes—
59C23×C199C2311yescharacteristic
60C23×C199C2311yescharacteristic
61D4×C199—22no—
62D4×C199—42no—
63C2×C4×C199C2312yes—
64C2×C4×C199C2311yescharacteristic
65C22×C199—22no—
66C22×C199—21no—
67C22×C199—22no—
68C22×C199—22no—
69C22×C199—22no—
70C22×C199—22no—
71C22×C199—44no—
72C22×C199C2×D412yes—
73C22×C199C2411yescharacteristic
74C2×D199—21no—
75C2×D199—22no—
76C2×D199—22no—
77C2×D199—22no—
78C2×D199—22no—
79C2×D199—44no—
80C4×C199—21no—
81C4×C199—22no—
82Dic199—21no—
83Dic199—22no—
84Dic199C2×D412yes—
85Dic199C2×D412yes—
86C2×C199—22no—
87C2×C199—41no—
88C2×C199—42no—
89C2×C199C22×D411yescharacteristic
90C2×C199H(5,𝔽2)12yes—
91D199—41no—
92D199—42no—
93C199#53511yescharacteristic, complemented, Sylow
94#535—1991nomaximal, Sylow
95C2×#41—1991no—
96C22×D4—1991no—
97#127—1992no—
98#124—1992no—
99#121—1992no—
100#121—1992no—
101#120—1991no—
102#120—1991no—
103#120—1991no—
104#120D19911yescharacteristic, complemented
105#117—1991no—
106C24—1991no—
107C24C2×D19911yescharacteristic
108#41—3984no—
109C2×Q8—1991no—
110C2×D4—3984no—
111C2×D4—1992no—
112C2×D4—1992no—
113C2×D4—1991no—
114C2×D4—1992no—
115C2×D4—1992no—
116C2×D4—1991no—
117C2×D4—1992no—
118C2×D4C2×D19911yescharacteristic
119C2×D4C2×D19912yescomplemented
120C22×C4—1991no—
121C22×C4—1991no—
122C22×C4—1991no—
123C4⋊C4—1992no—
124C22⋊C4—1991no—
125C22⋊C4—1992no—
126C22⋊C4—1992no—
127C22⋊C4—1992no—
128C22⋊C4C2×D19911yescharacteristic
129C22⋊C4C2×D19912yescomplemented
130C42—1992no—
131C23—22no—
132C23—42no—
133C23—7962no—
134C23—3982no—
135C23—1992no—
136C23—1991no—
137C23C22×D19912yes—
138C23C22×D19911yescharacteristic
139C23C22×D19911yescharacteristic
140Q8—3982no—
141D4—22no—
142D4—42no—
143D4—7962no—
144D4—7962no—
145D4—7962no—
146D4—3982no—
147D4—3984no—
148D4—3984no—
149D4—3982no—
150C2×C4—3982no—
151C2×C4—3982no—
152C2×C4—1992no—
153C2×C4—1991no—
154C2×C4—1994no—
155C2×C4—1991no—
156C2×C4—1991no—
157C2×C4C22×D19912yes—
158C2×C4C22×D19911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—7964no—
167C22—3981no—
168C22—3982no—
169C22—3982no—
170C22—3982no—
171C22—3982no—
172C22D4×D19912yes—
173C22C23×D19911yescharacteristic
174C4—21no—
175C4—22no—
176C4—3981no—
177C4—3982no—
178C4—1992no—
179C4—1992no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—7962no—
184C2—7961no—
185C2C2×D4×D19911yescharacteristic
186C2#2026400012yes—
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#120×C199C2
hypercenter#120D199
hypocenterC199#535
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. C22×C22⋊D199C2semidirect
3. #20263951C2semidirect
4. #20263941C2semidirect
5. #20263928C2semidirect
6. #20263927C2semidirect
7. #20263925C2semidirect
8. #20263924C2semidirect
9. #20263923C2semidirect
10. #20263911C2semidirect
11. #120×C199C2semidirect
14. C2×Dic398C22semidirect
15. C2×D796C22semidirect
20. C2×C22⋊D199C22semidirect
21. C2×C22⋊D199C22semidirect
22. C2×C22⋊D199C22semidirect
24. C22×Dic199C22semidirect
25. C2×C4×D199C22semidirect
26. Dic199⋊C4C22semidirect
28. C22⋊Dic199C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. C4×Dic199C22semidirect
34. C2×D4×C199C22semidirect
36. C22⋊C4×C199C22semidirect
50. C2×Dic199D4semidirect
93. C199#535semidirect
104. #120D199semidirect
119. C2×D4C2×D199semidirect
129. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
C22×C22⋊D1992
#202639512
#202639412
#202639282
#202639272
#202639252
#202639242
#202639232
#202639112
#120×C1992
#535199