Finite Group Databasebeta

← #27161644

Subgroups — #27161644

Order 12608.

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×#15691357C211yesmaximal, characteristic, complemented
2C22×C22⋊D197C211yesmaximal, characteristic, complemented
3#20263322C212yesmaximal, complemented
4#20263312C212yesmaximal, complemented
5#20263299C212yesmaximal, complemented
6#20263298C212yesmaximal, complemented
7#20263296C211yesmaximal, characteristic, complemented
8#20263295C211yesmaximal, characteristic, complemented
9#20263294C211yesmaximal, characteristic, complemented
10#20263282C211yesmaximal, characteristic, complemented
11#120×C197C211yesmaximal, characteristic, complemented
12C23×D197C2211yescharacteristic
13#15691357—24no—
14C2×Dic394C2211yescharacteristic, complemented
15C2×D788C2212yescomplemented
16C2×C22⋊D197—24no—
17C2×C22⋊D197C2211yescharacteristic
18C2×C22⋊D197C2212yes—
19C2×C22⋊D197C2212yes—
20C2×C22⋊D197C2211yescharacteristic, complemented
21C2×C22⋊D197C2212yescomplemented
22C2×C22⋊D197C2212yescomplemented
23C22×Dic197C2211yescharacteristic
24C22×Dic197C2211yescharacteristic, complemented
25C2×C4×D197C2211yescharacteristic, complemented
26#15691337C2212yescomplemented
27C22⋊Dic197C2211yescharacteristic
28C22⋊Dic197C2212yescomplemented
29#15691335C2212yescomplemented
30#15691335C2212yescomplemented
31C4×Dic197C2212yescomplemented
32C24×C197C2211yescharacteristic
33C2×D4×C197C2211yescharacteristic
34C2×D4×C197C2212yescomplemented
35C22⋊C4×C197C2211yescharacteristic
36C22⋊C4×C197C2212yescomplemented
37C22×D197—22no—
38C22×D197—42no—
39C22×D197C2312yes—
40C22×D197C2311yescharacteristic
41Dic394—22no—
42D788—42no—
43C22⋊D197—22no—
44C22⋊D197—24no—
45C22⋊D197—24no—
46C22⋊D197—22no—
47C22⋊D197—42no—
48C22⋊D197—42no—
49C2×Dic197—22no—
50C2×Dic197D414yescomplemented
51C2×Dic197C2312yes—
52C2×Dic197C2311yescharacteristic
53C2×Dic197C2311yescharacteristic
54C2×Dic197C2311yescharacteristic
55C4×D197—22no—
56C23×C197—22no—
57C23×C197—42no—
58C23×C197C2312yes—
59C23×C197C2311yescharacteristic
60C23×C197C2311yescharacteristic
61D4×C197—22no—
62D4×C197—42no—
63C2×C4×C197C2312yes—
64C2×C4×C197C2311yescharacteristic
65C22×C197—22no—
66C22×C197—21no—
67C22×C197—22no—
68C22×C197—22no—
69C22×C197—22no—
70C22×C197—22no—
71C22×C197—44no—
72C22×C197C2×D412yes—
73C22×C197C2411yescharacteristic
74C2×D197—21no—
75C2×D197—22no—
76C2×D197—22no—
77C2×D197—22no—
78C2×D197—22no—
79C2×D197—44no—
80C4×C197—21no—
81C4×C197—22no—
82Dic197—21no—
83Dic197—22no—
84Dic197C2×D412yes—
85Dic197C2×D412yes—
86C2×C197—22no—
87C2×C197—41no—
88C2×C197—42no—
89C2×C197C22×D411yescharacteristic
90C2×C197H(5,𝔽2)12yes—
91D197—41no—
92D197—42no—
93C197#53511yescharacteristic, complemented, Sylow
94#535—1971nomaximal, Sylow
95C2×#41—1971no—
96C22×D4—1971no—
97#127—1972no—
98#124—1972no—
99#121—1972no—
100#121—1972no—
101#120—1971no—
102#120—1971no—
103#120—1971no—
104#120D19711yescharacteristic, complemented
105#117—1971no—
106C24—1971no—
107C24C2×D19711yescharacteristic
108#41—3944no—
109C2×Q8—1971no—
110C2×D4—3944no—
111C2×D4—1972no—
112C2×D4—1972no—
113C2×D4—1971no—
114C2×D4—1972no—
115C2×D4—1972no—
116C2×D4—1971no—
117C2×D4—1972no—
118C2×D4C2×D19711yescharacteristic
119C2×D4C2×D19712yescomplemented
120C22×C4—1971no—
121C22×C4—1971no—
122C22×C4—1971no—
123C4⋊C4—1972no—
124C22⋊C4—1971no—
125C22⋊C4—1972no—
126C22⋊C4—1972no—
127C22⋊C4—1972no—
128C22⋊C4C2×D19711yescharacteristic
129C22⋊C4C2×D19712yescomplemented
130C42—1972no—
131C23—22no—
132C23—42no—
133C23—7882no—
134C23—3942no—
135C23—1972no—
136C23—1971no—
137C23C22×D19712yes—
138C23C22×D19711yescharacteristic
139C23C22×D19711yescharacteristic
140Q8—3942no—
141D4—22no—
142D4—42no—
143D4—7882no—
144D4—7882no—
145D4—7882no—
146D4—3942no—
147D4—3944no—
148D4—3944no—
149D4—3942no—
150C2×C4—3942no—
151C2×C4—3942no—
152C2×C4—1972no—
153C2×C4—1971no—
154C2×C4—1974no—
155C2×C4—1971no—
156C2×C4—1971no—
157C2×C4C22×D19712yes—
158C2×C4C22×D19711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—7884no—
167C22—3941no—
168C22—3942no—
169C22—3942no—
170C22—3942no—
171C22—3942no—
172C22D4×D19712yes—
173C22C23×D19711yescharacteristic
174C4—21no—
175C4—22no—
176C4—3941no—
177C4—3942no—
178C4—1972no—
179C4—1972no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—7882no—
184C2—7881no—
185C2C2×D4×D19711yescharacteristic
186C2#2026337112yes—
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#120×C197C2
hypercenter#120D197
hypocenterC197#535
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#15691357C2semidirect
2. C22×C22⋊D197C2semidirect
3. #20263322C2semidirect
4. #20263312C2semidirect
5. #20263299C2semidirect
6. #20263298C2semidirect
7. #20263296C2semidirect
8. #20263295C2semidirect
9. #20263294C2semidirect
10. #20263282C2semidirect
11. #120×C197C2semidirect
14. C2×Dic394C22semidirect
15. C2×D788C22semidirect
20. C2×C22⋊D197C22semidirect
21. C2×C22⋊D197C22semidirect
22. C2×C22⋊D197C22semidirect
24. C22×Dic197C22semidirect
25. C2×C4×D197C22semidirect
26. #15691337C22semidirect
28. C22⋊Dic197C22semidirect
29. #15691335C22semidirect
30. #15691335C22semidirect
31. C4×Dic197C22semidirect
34. C2×D4×C197C22semidirect
36. C22⋊C4×C197C22semidirect
50. C2×Dic197D4semidirect
93. C197#535semidirect
104. #120D197semidirect
119. C2×D4C2×D197semidirect
129. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#156913572
C22×C22⋊D1972
#202633222
#202633122
#202632992
#202632982
#202632962
#202632952
#202632942
#202632822
#120×C1972
#535197