Finite Group Databasebeta

← #27164821

Subgroups — #27164821

Order 12736.

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#20264001C211yesmaximal, characteristic, complemented
2#20263981C213yesmaximal, complemented
3#20263980C213yesmaximal, complemented
4#20263975C213yesmaximal, complemented
5C2×#15691871C213yesmaximal, complemented
6#20263962C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C199C211yesmaximal, characteristic, complemented
8#15691887—21no—
9#15691886C2213yescomplemented
10C2×D796C2213yescomplemented
11D4×D199—23no—
12#15691876C2211yescharacteristic, complemented
13#15691874C2213yescomplemented
14#15691873C2213yescomplemented
15#15691871C2213yescomplemented
16#15691871C2216yescomplemented
17#15691870C2213yescomplemented
18C2×C199⋊C8C2213yescomplemented
19#41×C199—21no—
20#41×C199C2211yescharacteristic
21#41×C199C2213yescomplemented
22C2×D4×C199—23no—
23C2×D4×C199C2213yescomplemented
24C22×D199—23no—
25Dic398C2311yescharacteristic
26D796—23no—
27D796C2313yes—
28C22⋊D199—23no—
29C4×D199—23no—
30C199⋊C8C2311yescharacteristic
31C199⋊C8C2313yes—
32C23×C199—23no—
33Q8×C199D411yescharacteristic
34Q8×C199C2311yescharacteristic
35D4×C199—23no—
36D4×C199—23no—
37D4×C199D413yes—
38D4×C199C2313yes—
39C2×C4×C199—23no—
40C2×C4×C199C2313yes—
41C22×C199—23no—
42C22×C199—83no—
43C22×C199C2×D413yes—
44C2×D199—23no—
45C2×D199—83no—
46C4×C199—21no—
47C4×C199C2×D413yes—
48C4×C199C2411yescharacteristic
49Dic199—21no—
50C2×C199—23no—
51C2×C199—43no—
52C2×C199C22×D411yescharacteristic
53D199—43no—
54C199#57611yescharacteristic, complemented, Sylow
55#576—1991nomaximal, Sylow
56H(5,𝔽2)—1991no—
57H(5,𝔽2)D19911yescharacteristic, complemented
58AGL(1,ℤ/8)—1993no—
59AGL(1,ℤ/8)—1993no—
60#135—1993no—
61C2×D8—1993no—
62#131—1991no—
63#41—21no—
64#41—3981no—
65#41—1993no—
66#41C2×D19911yescharacteristic
67#41C2×D19913yescomplemented
68C2×D4—23no—
69C2×D4—3983no—
70C2×D4—1993no—
71C2×D4C2×D19913yescomplemented
72Q16—1991no—
73SD16—1993no—
74SD16—1993no—
75D8—1993no—
76D8—1996no—
77M16—1993no—
78C2×C8—1993no—
79C23—23no—
80C23—3983no—
81Q8—1991no—
82Q8C22⋊D19911yescharacteristic
83Q8C22×D19911yescharacteristic
84D4—23no—
85D4—23no—
86D4—3983no—
87D4—3983no—
88D4—1993no—
89D4C22⋊D19913yes—
90D4C22×D19913yes—
91C2×C4—23no—
92C2×C4—3983no—
93C2×C4C22×D19913yes—
94C8—1991no—
95C8—1993no—
96C22—23no—
97C22—83no—
98C22—15923no—
99C22—3983no—
100C22C2×C22⋊D19913yes—
101C4—21no—
102C4—3981no—
103C4C2×C22⋊D19913yes—
104C4C23×D19911yescharacteristic
105C2—23no—
106C2—43no—
107C2—7963no—
108C2C22×C22⋊D19911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
FittingH(5,𝔽2)×C199C2
socleC2×C199C22×D4
hypercenterH(5,𝔽2)D199
hypocenterC199#576
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20264001C2semidirect
2. #20263981C2semidirect
3. #20263980C2semidirect
4. #20263975C2semidirect
5. C2×#15691871C2semidirect
6. #20263962C2semidirect
7. H(5,𝔽2)×C199C2semidirect
9. #15691886C22semidirect
10. C2×D796C22semidirect
12. #15691876C22semidirect
13. #15691874C22semidirect
14. #15691873C22semidirect
15. #15691871C22semidirect
16. #15691871C22semidirect
17. #15691870C22semidirect
18. C2×C199⋊C8C22semidirect
21. #41×C199C22semidirect
23. C2×D4×C199C22semidirect
54. C199#576semidirect
57. H(5,𝔽2)D199semidirect
67. #41C2×D199semidirect
71. C2×D4C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640012
#202639812
#202639802
#202639752
C2×#156918712
#202639622
H(5,𝔽2)×C1992
#576199