Finite Group Databasebeta

← #27136062

Subgroups — #27136062

Order 12352.

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#20259945C211yesmaximal, characteristic, complemented
2#20259925C213yesmaximal, complemented
3#20259924C213yesmaximal, complemented
4#20259919C213yesmaximal, complemented
5C2×#15690540C213yesmaximal, complemented
6#20259906C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C193C211yesmaximal, characteristic, complemented
8#15690556—21no—
9#15690555C2213yescomplemented
10C2×D772C2213yescomplemented
11D4×D193—23no—
12#15690545C2211yescharacteristic, complemented
13#15690543C2213yescomplemented
14#15690542C2213yescomplemented
15#15690540C2213yescomplemented
16#15690540C2216yescomplemented
17#15690539C2213yescomplemented
18C2×#13095393C2213yescomplemented
19#41×C193—21no—
20#41×C193C2211yescharacteristic
21#41×C193C2213yescomplemented
22C2×D4×C193—23no—
23C2×D4×C193C2213yescomplemented
24C22×D193—23no—
25Dic386C2311yescharacteristic
26D772—23no—
27D772C2313yes—
28C22⋊D193—23no—
29C4×D193—23no—
30#13095393C2311yescharacteristic
31#13095393C2313yes—
32C23×C193—23no—
33Q8×C193D411yescharacteristic
34Q8×C193C2311yescharacteristic
35D4×C193—23no—
36D4×C193—23no—
37D4×C193D413yes—
38D4×C193C2313yes—
39C2×C4×C193—23no—
40C2×C4×C193C2313yes—
41C22×C193—23no—
42C22×C193—83no—
43C22×C193C2×D413yes—
44C2×D193—23no—
45C2×D193—83no—
46C4×C193—21no—
47C4×C193C2×D413yes—
48C4×C193C2411yescharacteristic
49Dic193—21no—
50C2×C193—23no—
51C2×C193—43no—
52C2×C193C22×D411yescharacteristic
53D193—43no—
54C193#57611yescharacteristic, complemented, Sylow
55#576—1931nomaximal, Sylow
56H(5,𝔽2)—1931no—
57H(5,𝔽2)D19311yescharacteristic, complemented
58AGL(1,ℤ/8)—1933no—
59AGL(1,ℤ/8)—1933no—
60#135—1933no—
61C2×D8—1933no—
62#131—1931no—
63#41—21no—
64#41—3861no—
65#41—1933no—
66#41C2×D19311yescharacteristic
67#41C2×D19313yescomplemented
68C2×D4—23no—
69C2×D4—3863no—
70C2×D4—1933no—
71C2×D4C2×D19313yescomplemented
72Q16—1931no—
73SD16—1933no—
74SD16—1933no—
75D8—1933no—
76D8—1936no—
77M16—1933no—
78C2×C8—1933no—
79C23—23no—
80C23—3863no—
81Q8—1931no—
82Q8C22⋊D19311yescharacteristic
83Q8C22×D19311yescharacteristic
84D4—23no—
85D4—23no—
86D4—3863no—
87D4—3863no—
88D4—1933no—
89D4C22⋊D19313yes—
90D4C22×D19313yes—
91C2×C4—23no—
92C2×C4—3863no—
93C2×C4C22×D19313yes—
94C8—1931no—
95C8—1933no—
96C22—23no—
97C22—83no—
98C22—15443no—
99C22—3863no—
100C22C2×C22⋊D19313yes—
101C4—21no—
102C4—3861no—
103C4C2×C22⋊D19313yes—
104C4C23×D19311yescharacteristic
105C2—23no—
106C2—43no—
107C2—7723no—
108C2C22×C22⋊D19311yescharacteristic
subgroupG/H
centerC2C22×C22⋊D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
FittingH(5,𝔽2)×C193C2
socleC2×C193C22×D4
hypercenterH(5,𝔽2)D193
hypocenterC193#576
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20259945C2semidirect
2. #20259925C2semidirect
3. #20259924C2semidirect
4. #20259919C2semidirect
5. C2×#15690540C2semidirect
6. #20259906C2semidirect
7. H(5,𝔽2)×C193C2semidirect
9. #15690555C22semidirect
10. C2×D772C22semidirect
12. #15690545C22semidirect
13. #15690543C22semidirect
14. #15690542C22semidirect
15. #15690540C22semidirect
16. #15690540C22semidirect
17. #15690539C22semidirect
18. C2×#13095393C22semidirect
21. #41×C193C22semidirect
23. C2×D4×C193C22semidirect
54. C193#576semidirect
57. H(5,𝔽2)D193semidirect
67. #41C2×D193semidirect
71. C2×D4C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599252
#202599242
#202599192
C2×#156905402
#202599062
H(5,𝔽2)×C1932
#576193