Finite Group Databasebeta

← #27161639

Subgroups — #27161639

Order 12608.

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

normal subgroups only

This group has 346 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
1C22×C22⋊D197C211yesmaximal, characteristic, complemented
2C2×D4×D197C211yesmaximal, characteristic, complemented
3#20263310C211yesmaximal, characteristic, complemented
4#20263308C212yesmaximal, complemented
5#20263307C211yesmaximal, characteristic, complemented
6#20263299C212yesmaximal, complemented
7#20263297C212yesmaximal, complemented
8#20263295C211yesmaximal, characteristic, complemented
9#20263294C212yesmaximal, complemented
10C22⋊C4×D197C211yesmaximal, characteristic, complemented
11#120×C197C211yesmaximal, characteristic, complemented
12C23×D197C2211yescharacteristic
13C23×D197C2211yescharacteristic, complemented
14C2×D788C2211yescharacteristic, complemented
15C2×C22⋊D197—24no—
16C2×C22⋊D197C2212yes—
17C2×C22⋊D197C2211yescharacteristic
18C2×C22⋊D197C2211yescharacteristic, complemented
19C2×C22⋊D197C2212yescomplemented
20C2×C22⋊D197C2212yescomplemented
21D4×D197—24no—
22C22×Dic197C2211yescharacteristic, complemented
23C2×C4×D197C2211yescharacteristic, complemented
24C2×C4×D197C2212yescomplemented
25C4⋊Dic197C2212yescomplemented
26#15691337C2212yescomplemented
27C22⋊Dic197C2211yescharacteristic, complemented
28C22⋊Dic197C2212yescomplemented
29C22⋊Dic197C2212yescomplemented
30#15691335C2212yescomplemented
31#15691335C2212yescomplemented
32C24×C197C2211yescharacteristic, complemented
33C2×D4×C197C2211yescharacteristic
34C2×D4×C197C2212yescomplemented
35C22⋊C4×C197C2211yescharacteristic, complemented
36C22⋊C4×C197C2212yescomplemented
37C22×D197—22no—
38C22×D197—42no—
39C22×D197—42no—
40C22×D197D414yescomplemented
41C22×D197C2311yescharacteristic
42C22×D197C2311yescharacteristic
43C22×D197C2311yescharacteristic
44D788—22no—
45C22⋊D197—22no—
46C22⋊D197—22no—
47C22⋊D197—24no—
48C22⋊D197—24no—
49C22⋊D197—42no—
50C2×Dic197—22no—
51C2×Dic197C2312yes—
52C2×Dic197C2312yes—
53C2×Dic197C2311yescharacteristic
54C4×D197—22no—
55C4×D197—42no—
56C23×C197—22no—
57C23×C197—42no—
58C23×C197C2312yes—
59C23×C197C2311yescharacteristic
60C23×C197C2311yescharacteristic
61D4×C197—22no—
62D4×C197—42no—
63C2×C4×C197C2311yescharacteristic
64C2×C4×C197C2312yes—
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—22no—
75C2×D197—22no—
76C2×D197—21no—
77C2×D197—22no—
78C2×D197—44no—
79C2×D197—44no—
80C2×D197C2×D412yes—
81C2×D197C2×D412yes—
82C4×C197—21no—
83C4×C197—22no—
84Dic197—21no—
85Dic197—22no—
86Dic197—22no—
87C2×C197—22no—
88C2×C197—41no—
89C2×C197—42no—
90C2×C197C22×D411yescharacteristic
91C2×C197H(5,𝔽2)12yes—
92D197—22no—
93D197—22no—
94D197—41no—
95C197#53411yescharacteristic, complemented, Sylow
96#534—1971nomaximal, Sylow
97C22×D4—1971no—
98C22×D4—1971no—
99#123—1971no—
100#123—1972no—
101#123—1971no—
102#121—1972no—
103#121—1972no—
104#120—1971no—
105#120—1972no—
106#120D19711yescharacteristic, complemented
107C2×C22⋊C4—1971no—
108C24—1971no—
109C24—1971no—
110C24C2×D19711yescharacteristic, complemented
111C2×D4—3944no—
112C2×D4—3944no—
113C2×D4—1972no—
114C2×D4—1972no—
115C2×D4—1971no—
116C2×D4—1971no—
117C2×D4—1972no—
118C2×D4—1971no—
119C2×D4C2×D19711yescharacteristic
120C2×D4C2×D19712yescomplemented
121C22×C4—1972no—
122C22×C4—1971no—
123C22×C4—1971no—
124C4⋊C4—1972no—
125C4⋊C4—1972no—
126C22⋊C4—1971no—
127C22⋊C4—1972no—
128C22⋊C4—1972no—
129C22⋊C4—1972no—
130C22⋊C4—1972no—
131C22⋊C4C2×D19711yescharacteristic, complemented
132C22⋊C4C2×D19712yescomplemented
133C23—22no—
134C23—42no—
135C23—7882no—
136C23—7882no—
137C23—3942no—
138C23—1974no—
139C23—1971no—
140C23—1971no—
141C23—1971no—
142C23C22×D19711yescharacteristic
143C23C22×D19712yes—
144C23C22×D19711yescharacteristic
145D4—22no—
146D4—42no—
147D4—7882no—
148D4—3942no—
149D4—3942no—
150D4—3944no—
151D4—3944no—
152D4—3942no—
153C2×C4—7882no—
154C2×C4—3942no—
155C2×C4—3942no—
156C2×C4—1972no—
157C2×C4—1971no—
158C2×C4—1972no—
159C2×C4C22×D19712yes—
160C2×C4C22×D19711yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—7884no—
169C22—7884no—
170C22—3941no—
171C22—3942no—
172C22—3942no—
173C22—3942no—
174C22—1972no—
175C22—1972no—
176C22D4×D19712yes—
177C22C23×D19711yescharacteristic
178C4—21no—
179C4—22no—
180C4—3941no—
181C4—3942no—
182C4—3942no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—7881no—
187C2—3942no—
188C2—3942no—
189C2C2×D4×D19711yescharacteristic
190C2#2026337112yes—
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#120×C197C2
hypercenter#120D197
hypocenterC197#534
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D197C2semidirect
2. C2×D4×D197C2semidirect
3. #20263310C2semidirect
4. #20263308C2semidirect
5. #20263307C2semidirect
6. #20263299C2semidirect
7. #20263297C2semidirect
8. #20263295C2semidirect
9. #20263294C2semidirect
10. C22⋊C4×D197C2semidirect
11. #120×C197C2semidirect
13. C23×D197C22semidirect
14. C2×D788C22semidirect
18. C2×C22⋊D197C22semidirect
19. C2×C22⋊D197C22semidirect
20. C2×C22⋊D197C22semidirect
22. C22×Dic197C22semidirect
23. C2×C4×D197C22semidirect
24. C2×C4×D197C22semidirect
25. C4⋊Dic197C22semidirect
26. #15691337C22semidirect
27. C22⋊Dic197C22semidirect
28. C22⋊Dic197C22semidirect
29. C22⋊Dic197C22semidirect
30. #15691335C22semidirect
31. #15691335C22semidirect
32. C24×C197C22semidirect
34. C2×D4×C197C22semidirect
35. C22⋊C4×C197C22semidirect
36. C22⋊C4×C197C22semidirect
40. C22×D197D4semidirect
95. C197#534semidirect
106. #120D197semidirect
110. C24C2×D197semidirect
120. C2×D4C2×D197semidirect
131. C22⋊C4C2×D197semidirect
132. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D1972
C2×D4×D1972
#202633102
#202633082
#202633072
#202632992
#202632972
#202632952
#202632942
C22⋊C4×D1972
#120×C1972
#534197