Finite Group Databasebeta

← #27161711

Subgroups — #27161711

Order 12608.

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×C22⋊D197C212yesmaximal, complemented
2C2×D4×D197C212yesmaximal, complemented
3#20263322C211yesmaximal, characteristic, complemented
4#20263299C212yesmaximal, complemented
5#20263298C212yesmaximal, complemented
6#20263296C212yesmaximal, complemented
7#20263294C211yesmaximal, characteristic, complemented
8#20263286C212yesmaximal, complemented
9#120×C197C211yesmaximal, characteristic, complemented
10C23×D197C2211yescharacteristic, complemented
11C23×D197C2212yescomplemented
12C2×D788C2212yescomplemented
13C2×D788C2211yescharacteristic, complemented
14C2×C22⋊D197—28no—
15C2×C22⋊D197C2212yes—
16C2×C22⋊D197C2212yescomplemented
17C2×C22⋊D197C2212yescomplemented
18C2×C22⋊D197C2212yescomplemented
19C2×C22⋊D197C2212yescomplemented
20D4×D197—28no—
21C22×Dic197C2212yescomplemented
22C2×C4×D197C2212yescomplemented
23#15691337C2212yescomplemented
24C22⋊Dic197C2211yescharacteristic, complemented
25#15691335C2212yescomplemented
26#15691335C2212yescomplemented
27C4×Dic197C2211yescharacteristic, complemented
28C24×C197C2211yescharacteristic, complemented
29C2×D4×C197C2212yescomplemented
30C2×D4×C197C2211yescharacteristic, complemented
31C22⋊C4×C197C2212yescomplemented
32C22⋊C4×C197C2211yescharacteristic, complemented
33C22×D197—24no—
34C22×D197—24no—
35C22×D197—24no—
36C22×D197—41no—
37C22×D197—42no—
38C22×D197C2312yes—
39C22×D197C2312yes—
40D788—24no—
41D788—41no—
42C22⋊D197—24no—
43C22⋊D197—24no—
44C22⋊D197—24no—
45C22⋊D197—24no—
46C22⋊D197D418yescomplemented
47C2×Dic197—24no—
48C2×Dic197C2312yes—
49C2×Dic197C2312yes—
50C4×D197—24no—
51C23×C197—24no—
52C23×C197—41no—
53C23×C197C2311yescharacteristic
54C23×C197C2311yescharacteristic
55C23×C197C2312yes—
56D4×C197—24no—
57D4×C197—41no—
58C2×C4×C197C2311yescharacteristic
59C2×C4×C197C2312yes—
60C22×C197—22no—
61C22×C197—22no—
62C22×C197—22no—
63C22×C197—22no—
64C22×C197—21no—
65C22×C197—21no—
66C22×C197—44no—
67C22×C197C2×D414yescomplemented
68C22×C197C2411yescharacteristic
69C2×D197—22no—
70C2×D197—22no—
71C2×D197—22no—
72C2×D197—22no—
73C2×D197—22no—
74C2×D197—44no—
75C2×D197—48no—
76C2×D197C2×D414yescomplemented
77C4×C197—22no—
78C4×C197—21no—
79Dic197—22no—
80Dic197C2×D414yescomplemented
81C2×C197—24no—
82C2×C197—41no—
83C2×C197—41no—
84C2×C197C22×D412yes—
85C2×C197H(5,𝔽2)11yescharacteristic
86D197—24no—
87D197—42no—
88C197D4211yescharacteristic, complemented, Sylow
89D42—1971nomaximal, Sylow
90C22×D4—1972no—
91C22×D4—1972no—
92#127—1971no—
93#121—1972no—
94#121—1972no—
95#120—1971no—
96#120—1972no—
97#120D19711yescharacteristic, complemented
98C4×D4—1972no—
99C24—1971no—
100C24—1972no—
101C24C2×D19711yescharacteristic, complemented
102C2×D4—3948no—
103C2×D4—3948no—
104C2×D4—1972no—
105C2×D4—1972no—
106C2×D4—1972no—
107C2×D4—1972no—
108C2×D4—1972no—
109C2×D4—1971no—
110C2×D4—1972no—
111C2×D4C2×D19712yescomplemented
112C2×D4C2×D19711yescharacteristic, complemented
113C22×C4—1972no—
114C22×C4—1972no—
115C4⋊C4—1972no—
116C22⋊C4—1971no—
117C22⋊C4—1972no—
118C22⋊C4—1972no—
119C22⋊C4C2×D19712yescomplemented
120C22⋊C4C2×D19711yescharacteristic, complemented
121C42—1971no—
122C23—24no—
123C23—41no—
124C23—7882no—
125C23—7881no—
126C23—3944no—
127C23—3944no—
128C23—3944no—
129C23—1972no—
130C23—1972no—
131C23C22×D19711yescharacteristic
132C23C22×D19711yescharacteristic
133C23C22×D19712yes—
134D4—24no—
135D4—41no—
136D4—7881no—
137D4—3944no—
138D4—3944no—
139D4—3944no—
140D4—3944no—
141D4—3944no—
142D4—1978no—
143C2×C4—3944no—
144C2×C4—3944no—
145C2×C4—1972no—
146C2×C4—1972no—
147C2×C4C22×D19711yescharacteristic
148C2×C4C22×D19712yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—7888no—
157C22—7884no—
158C22—3942no—
159C22—3942no—
160C22—3942no—
161C22—3942no—
162C22—3942no—
163C22—1974no—
164C22D4×D19714yescomplemented
165C22C23×D19711yescharacteristic
166C4—22no—
167C4—21no—
168C4—3942no—
169C4—1974no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—7882no—
174C2—3944no—
175C2C2×D4×D19712yes—
176C2#2026337111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#120×C197C2
hypercenter#120D197
hypocenterC197D42
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. #20263322C2semidirect
4. #20263299C2semidirect
5. #20263298C2semidirect
6. #20263296C2semidirect
7. #20263294C2semidirect
8. #20263286C2semidirect
9. #120×C197C2semidirect
10. C23×D197C22semidirect
11. C23×D197C22semidirect
12. C2×D788C22semidirect
13. C2×D788C22semidirect
16. C2×C22⋊D197C22semidirect
17. C2×C22⋊D197C22semidirect
18. C2×C22⋊D197C22semidirect
19. C2×C22⋊D197C22semidirect
21. C22×Dic197C22semidirect
22. C2×C4×D197C22semidirect
23. #15691337C22semidirect
24. C22⋊Dic197C22semidirect
25. #15691335C22semidirect
26. #15691335C22semidirect
27. C4×Dic197C22semidirect
28. C24×C197C22semidirect
29. C2×D4×C197C22semidirect
30. C2×D4×C197C22semidirect
31. C22⋊C4×C197C22semidirect
32. C22⋊C4×C197C22semidirect
46. C22⋊D197D4semidirect
67. C22×C197C2×D4semidirect
76. C2×D197C2×D4semidirect
80. Dic197C2×D4semidirect
88. C197D42semidirect
97. #120D197semidirect
101. C24C2×D197semidirect
111. C2×D4C2×D197semidirect
112. C2×D4C2×D197semidirect
119. C22⋊C4C2×D197semidirect
120. C22⋊C4C2×D197semidirect
164. C22D4×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
#202633222
#202632992
#202632982
#202632962
#202632942
#202632862
#120×C1972
D42197