Finite Group Databasebeta

← D4×D788 #27161715

Subgroups — D4×D788 #27161715

Order 12608.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×D788C212yesmaximal, complemented
2C2×D4×D197C212yesmaximal, complemented
3#20263323C211yesmaximal, characteristic, complemented
4#20263301C212yesmaximal, complemented
5#20263300C212yesmaximal, complemented
6#20263296C214yesmaximal, complemented
7C4×D788C211yesmaximal, characteristic, complemented
8C4×D4×C197C211yesmaximal, characteristic, complemented
9C23×D197C2214yescomplemented
10C2×D788—28no—
11C2×D788C2211yescharacteristic
12C2×D788C2214yescomplemented
13C2×D788C2212yescomplemented
14C2×D788C2212yescomplemented
15C2×D788C2212yescomplemented
16C2×C22⋊D197C2214yescomplemented
17D4×D197—28no—
18C2×C4×D197C2212yescomplemented
19C4⋊Dic197C2211yescharacteristic, complemented
20#15691335C2212yescomplemented
21#15691335C2214yescomplemented
22C2×D4×C197C2211yescharacteristic
23C22×C4×C197C2212yescomplemented
24C4⋊C4×C197C2211yescharacteristic, complemented
25C22⋊C4×C197C2212yescomplemented
26C42×C197C2211yescharacteristic, complemented
27C22×D197—28no—
28C22×D197—28no—
29C22×D197—44no—
30C22×D197C2314yes—
31C22×D197C2312yes—
32D788—28no—
33D788—24no—
34D788—24no—
35D788—42no—
36D788D414yesdirect product factor
37C22⋊D197—28no—
38C2×Dic197C2312yes—
39C4×D197—24no—
40C23×C197C2312yes—
41D4×C197D414yescomplemented
42C2×C4×C197—24no—
43C2×C4×C197C2311yescharacteristic
44C2×C4×C197C2312yes—
45C2×C4×C197C2311yescharacteristic
46C2×C4×C197C2311yescharacteristic
47C22×C197—22no—
48C22×C197—22no—
49C22×C197C2×D414yescomplemented
50C22×C197C2411yescharacteristic
51C2×D197—22no—
52C2×D197—22no—
53C2×D197—24no—
54C2×D197—24no—
55C2×D197—24no—
56C2×D197—416no—
57C2×D197C2×D414yescomplemented
58C4×C197—22no—
59C4×C197—21no—
60C4×C197C2×D412yescomplemented
61C4×C197C2×D412yescomplemented
62Dic197—22no—
63C2×C197—24no—
64C2×C197C22×D411yescharacteristic
65C2×C197C22×D411yescharacteristic
66C2×C197H(5,𝔽2)11yescharacteristic
67D197—24no—
68D197—44no—
69C197D4211yescharacteristic, complemented, Sylow
70D42—1971nomaximal, Sylow
71C22×D4—1972no—
72C22×D4—1972no—
73#127—1971no—
74#121—1972no—
75#121—1972no—
76#120—1974no—
77C4×D4—1971no—
78C4×D4D19711yescharacteristic, complemented
79C24—1974no—
80C2×D4—3948no—
81C2×D4—3948no—
82C2×D4—1974no—
83C2×D4—1972no—
84C2×D4—1974no—
85C2×D4—1972no—
86C2×D4—1972no—
87C2×D4—1971no—
88C2×D4C2×D19711yescharacteristic
89C22×C4—1972no—
90C22×C4C2×D19712yescomplemented
91C4⋊C4—1971no—
92C4⋊C4C2×D19711yescharacteristic, complemented
93C22⋊C4—1972no—
94C22⋊C4—1974no—
95C22⋊C4C2×D19712yescomplemented
96C42C2×D19711yescharacteristic, complemented
97C23—7884no—
98C23—3948no—
99C23—3948no—
100C23—1974no—
101C23—1972no—
102C23C22×D19712yes—
103D4—7882no—
104D4—3948no—
105D4—3948no—
106D4—3944no—
107D4—3944no—
108D4—1974no—
109D4D78814yesdirect product factor
110C2×C4—24no—
111C2×C4—3944no—
112C2×C4—1972no—
113C2×C4C22×D19711yescharacteristic
114C2×C4C22×D19712yes—
115C2×C4C22×D19711yescharacteristic
116C2×C4C22×D19711yescharacteristic
117C22—22no—
118C22—22no—
119C22—78816no—
120C22—3944no—
121C22—3944no—
122C22—3944no—
123C22—3942no—
124C22—3942no—
125C22—1974no—
126C22C2×D78814yescomplemented
127C22C23×D19711yescharacteristic
128C4—22no—
129C4—21no—
130C4—3942no—
131C4D4×D19712yescomplemented
132C4C2×D78812yescomplemented
133C2—24no—
134C2—7884no—
135C2—3944no—
136C2C2×D4×D19711yescharacteristic
137C2C22×D78811yescharacteristic
138C2#2026337211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
FittingC4×D4×C197C2
hypercenterC4×D4D197
hypocenterC197D42
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×D788C2semidirect
2. C2×D4×D197C2semidirect
3. #20263323C2semidirect
4. #20263301C2semidirect
5. #20263300C2semidirect
6. #20263296C2semidirect
7. C4×D788C2semidirect
8. C4×D4×C197C2semidirect
9. C23×D197C22semidirect
12. C2×D788C22semidirect
13. C2×D788C22semidirect
14. C2×D788C22semidirect
15. C2×D788C22semidirect
16. C2×C22⋊D197C22semidirect
18. C2×C4×D197C22semidirect
19. C4⋊Dic197C22semidirect
20. #15691335C22semidirect
21. #15691335C22semidirect
23. C22×C4×C197C22semidirect
24. C4⋊C4×C197C22semidirect
25. C22⋊C4×C197C22semidirect
26. C42×C197C22semidirect
36. D788D4direct (N × H)
41. D4×C197D4semidirect
49. C22×C197C2×D4semidirect
57. C2×D197C2×D4semidirect
60. C4×C197C2×D4semidirect
61. C4×C197C2×D4semidirect
69. C197D42semidirect
78. C4×D4D197semidirect
90. C22×C4C2×D197semidirect
92. C4⋊C4C2×D197semidirect
95. C22⋊C4C2×D197semidirect
96. C42C2×D197semidirect
109. D4D788direct (N × H)
126. C22C2×D788semidirect
131. C4D4×D197semidirect
132. C4C2×D788semidirect

Maximal subgroups

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

subgroupindex
C22×D7882
C2×D4×D1972
#202633232
#202633012
#202633002
#202632962
C4×D7882
C4×D4×C1972
D42197