Finite Group Databasebeta

← #27161882

Subgroups — #27161882

Order 12608.

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

normal subgroups only

This group has 268 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#20263372C211yesmaximal, characteristic, complemented
2#20263352C213yesmaximal, complemented
3#20263351C213yesmaximal, complemented
4#20263346C213yesmaximal, complemented
5C2×#15691343C213yesmaximal, complemented
6#20263333C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C197C211yesmaximal, characteristic, complemented
8#15691359—21no—
9#15691358C2213yescomplemented
10C2×D788C2213yescomplemented
11D4×D197—23no—
12#15691348C2211yescharacteristic, complemented
13#15691346C2213yescomplemented
14#15691345C2213yescomplemented
15#15691343C2213yescomplemented
16#15691343C2216yescomplemented
17#15691342C2213yescomplemented
18C2×#13096802C2213yescomplemented
19#41×C197—21no—
20#41×C197C2211yescharacteristic
21#41×C197C2213yescomplemented
22C2×D4×C197—23no—
23C2×D4×C197C2213yescomplemented
24C22×D197—23no—
25Dic394C2311yescharacteristic
26D788—23no—
27D788C2313yes—
28C22⋊D197—23no—
29C4×D197—23no—
30#13096802C2311yescharacteristic
31#13096802C2313yes—
32C23×C197—23no—
33Q8×C197D411yescharacteristic
34Q8×C197C2311yescharacteristic
35D4×C197—23no—
36D4×C197—23no—
37D4×C197D413yes—
38D4×C197C2313yes—
39C2×C4×C197—23no—
40C2×C4×C197C2313yes—
41C22×C197—23no—
42C22×C197—83no—
43C22×C197C2×D413yes—
44C2×D197—23no—
45C2×D197—83no—
46C4×C197—21no—
47C4×C197C2×D413yes—
48C4×C197C2411yescharacteristic
49Dic197—21no—
50C2×C197—23no—
51C2×C197—43no—
52C2×C197C22×D411yescharacteristic
53D197—43no—
54C197#57611yescharacteristic, complemented, Sylow
55#576—1971nomaximal, Sylow
56H(5,𝔽2)—1971no—
57H(5,𝔽2)D19711yescharacteristic, complemented
58AGL(1,ℤ/8)—1973no—
59AGL(1,ℤ/8)—1973no—
60#135—1973no—
61C2×D8—1973no—
62#131—1971no—
63#41—21no—
64#41—3941no—
65#41—1973no—
66#41C2×D19711yescharacteristic
67#41C2×D19713yescomplemented
68C2×D4—23no—
69C2×D4—3943no—
70C2×D4—1973no—
71C2×D4C2×D19713yescomplemented
72Q16—1971no—
73SD16—1973no—
74SD16—1973no—
75D8—1973no—
76D8—1976no—
77M16—1973no—
78C2×C8—1973no—
79C23—23no—
80C23—3943no—
81Q8—1971no—
82Q8C22⋊D19711yescharacteristic
83Q8C22×D19711yescharacteristic
84D4—23no—
85D4—23no—
86D4—3943no—
87D4—3943no—
88D4—1973no—
89D4C22⋊D19713yes—
90D4C22×D19713yes—
91C2×C4—23no—
92C2×C4—3943no—
93C2×C4C22×D19713yes—
94C8—1971no—
95C8—1973no—
96C22—23no—
97C22—83no—
98C22—15763no—
99C22—3943no—
100C22C2×C22⋊D19713yes—
101C4—21no—
102C4—3941no—
103C4C2×C22⋊D19713yes—
104C4C23×D19711yescharacteristic
105C2—23no—
106C2—43no—
107C2—7883no—
108C2C22×C22⋊D19711yescharacteristic
subgroupG/H
centerC2C22×C22⋊D197
commutator, cosocleC4×C197C24
FrattiniC4C23×D197
FittingH(5,𝔽2)×C197C2
socleC2×C197C22×D4
hypercenterH(5,𝔽2)D197
hypocenterC197#576
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20263372C2semidirect
2. #20263352C2semidirect
3. #20263351C2semidirect
4. #20263346C2semidirect
5. C2×#15691343C2semidirect
6. #20263333C2semidirect
7. H(5,𝔽2)×C197C2semidirect
9. #15691358C22semidirect
10. C2×D788C22semidirect
12. #15691348C22semidirect
13. #15691346C22semidirect
14. #15691345C22semidirect
15. #15691343C22semidirect
16. #15691343C22semidirect
17. #15691342C22semidirect
18. C2×#13096802C22semidirect
21. #41×C197C22semidirect
23. C2×D4×C197C22semidirect
54. C197#576semidirect
57. H(5,𝔽2)D197semidirect
67. #41C2×D197semidirect
71. C2×D4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633722
#202633522
#202633512
#202633462
C2×#156913432
#202633332
H(5,𝔽2)×C1972
#576197