Finite Group Databasebeta

← #27161891

Subgroups — #27161891

Order 12608.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#20263373C211yesmaximal, characteristic, complemented
2#20263372C211yesmaximal, characteristic, complemented
3#20263359C213yesmaximal, complemented
4#20263354C213yesmaximal, complemented
5#20263348C213yesmaximal, complemented
6C2×#15691347C213yesmaximal, complemented
7#131×C197C211yesmaximal, characteristic, complemented
8#15691359—21no—
9#15691358C2213yes—
10#15691358C2213yescomplemented
11#15691357—23no—
12C2×Dic394C2213yescomplemented
13Q8×D197—21no—
14C2×D788C2213yes—
15D4×D197—23no—
16Dic788C2213yescomplemented
17#15691347C2213yescomplemented
18#15691347C2216yescomplemented
19#15691347C2211yescharacteristic, complemented
20D1576C2213yescomplemented
21#41×C197C2211yescharacteristic
22M16×C197C2213yescomplemented
23C2×C8×C197C2213yescomplemented
24C22×D197—23no—
25Dic394—23no—
26Dic394C2313yes—
27Dic394C2311yescharacteristic
28D788—23no—
29D788C2311yescharacteristic
30D788C2313yes—
31C22⋊D197—23no—
32C22⋊D197—23no—
33C2×Dic197—23no—
34C4×D197—23no—
35C4×D197—23no—
36Q8×C197D411yescharacteristic
37D4×C197D413yes—
38C2×C4×C197C2313yes—
39C8×C197C2311yescharacteristic
40C8×C197C2313yes—
41C22×C197C2×D413yes—
42C2×D197—21no—
43C2×D197—23no—
44C2×D197—83no—
45C4×C197C2×D413yes—
46C4×C197C2411yescharacteristic
47Dic197—21no—
48Dic197—23no—
49C2×C197—23no—
50C2×C197C22×D411yescharacteristic
51D197—41no—
52D197—43no—
53C197#57711yescharacteristic, complemented, Sylow
54#577—1971nomaximal, Sylow
552−1+4—1971no—
56H(5,𝔽2)—1971no—
57#137—1973no—
58AGL(1,ℤ/8)—1973no—
59#135—1973no—
60C2×SD16—1973no—
61#131D19711yescharacteristic, complemented
62#41—3941no—
63#41—3943no—
64#41—1973no—
65#41—1973no—
66#41C2×D19711yescharacteristic
67C2×Q8—3941no—
68C2×Q8—1973no—
69C2×D4—3943no—
70C2×D4—1973no—
71Q16—1973no—
72SD16—1973no—
73SD16—1976no—
74SD16—1971no—
75D8—1973no—
76M16C2×D19713yescomplemented
77C2×C8C2×D19713yescomplemented
78C23—3943no—
79Q8—3943no—
80Q8—1973no—
81Q8—1971no—
82Q8D78811yescharacteristic
83D4—3943no—
84D4—3943no—
85D4—3943no—
86D4—1971no—
87D4—1973no—
88D4D78813yes—
89C2×C4—3943no—
90C2×C4—3943no—
91C2×C4—3943no—
92C2×C4C22×D19713yes—
93C8C22×D19711yescharacteristic
94C8C22×D19713yes—
95C22—15763no—
96C22—3941no—
97C22—3943no—
98C22C2×D78813yes—
99C4—3941no—
100C4—3943no—
101C4C2×D78813yes—
102C4C23×D19711yescharacteristic
103C2—23no—
104C2—7883no—
105C2—7881no—
106C2C22×D78811yescharacteristic
subgroupG/H
centerC2C22×D788
commutator, cosocleC4×C197C24
FrattiniC4C23×D197
Fitting#131×C197C2
socleC2×C197C22×D4
hypercenter#131D197
hypocenterC197#577
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #20263373C2semidirect
2. #20263372C2semidirect
3. #20263359C2semidirect
4. #20263354C2semidirect
5. #20263348C2semidirect
6. C2×#15691347C2semidirect
7. #131×C197C2semidirect
10. #15691358C22semidirect
12. C2×Dic394C22semidirect
16. Dic788C22semidirect
17. #15691347C22semidirect
18. #15691347C22semidirect
19. #15691347C22semidirect
20. D1576C22semidirect
22. M16×C197C22semidirect
23. C2×C8×C197C22semidirect
53. C197#577semidirect
61. #131D197semidirect
76. M16C2×D197semidirect
77. C2×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633732
#202633722
#202633592
#202633542
#202633482
C2×#156913472
#131×C1972
#577197