Finite Group Databasebeta

← #45656041

Subgroups — #45656041

Order 32576.

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#30654261C211yesmaximal, characteristic, complemented
2#30654241C213yesmaximal, complemented
3#30654240C213yesmaximal, complemented
4#30654235C213yesmaximal, complemented
5C2×#22550846C213yesmaximal, complemented
6#30654222C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C509C211yesmaximal, characteristic, complemented
8#22550862—21no—
9#22550861C2213yescomplemented
10C2×D2036C2213yescomplemented
11D4×D509—23no—
12#22550851C2211yescharacteristic, complemented
13#22550849C2213yescomplemented
14#22550848C2213yescomplemented
15#22550846C2213yescomplemented
16#22550846C2216yescomplemented
17#22550845C2213yescomplemented
18C2×#16861325C2213yescomplemented
19#41×C509—21no—
20#41×C509C2211yescharacteristic
21#41×C509C2213yescomplemented
22C2×D4×C509—23no—
23C2×D4×C509C2213yescomplemented
24C22×D509—23no—
25Dic1018C2311yescharacteristic
26D2036—23no—
27D2036C2313yes—
28C22⋊D509—23no—
29C4×D509—23no—
30#16861325C2311yescharacteristic
31#16861325C2313yes—
32C23×C509—23no—
33Q8×C509D411yescharacteristic
34Q8×C509C2311yescharacteristic
35D4×C509—23no—
36D4×C509—23no—
37D4×C509D413yes—
38D4×C509C2313yes—
39C2×C4×C509—23no—
40C2×C4×C509C2313yes—
41C22×C509—23no—
42C22×C509—83no—
43C22×C509C2×D413yes—
44C2×D509—23no—
45C2×D509—83no—
46C4×C509—21no—
47C4×C509C2×D413yes—
48C4×C509C2411yescharacteristic
49Dic509—21no—
50C2×C509—23no—
51C2×C509—43no—
52C2×C509C22×D411yescharacteristic
53D509—43no—
54C509#57611yescharacteristic, complemented, Sylow
55#576—5091nomaximal, Sylow
56H(5,𝔽2)—5091no—
57H(5,𝔽2)D50911yescharacteristic, complemented
58AGL(1,ℤ/8)—5093no—
59AGL(1,ℤ/8)—5093no—
60#135—5093no—
61C2×D8—5093no—
62#131—5091no—
63#41—21no—
64#41—10181no—
65#41—5093no—
66#41C2×D50911yescharacteristic
67#41C2×D50913yescomplemented
68C2×D4—23no—
69C2×D4—10183no—
70C2×D4—5093no—
71C2×D4C2×D50913yescomplemented
72Q16—5091no—
73SD16—5093no—
74SD16—5093no—
75D8—5093no—
76D8—5096no—
77M16—5093no—
78C2×C8—5093no—
79C23—23no—
80C23—10183no—
81Q8—5091no—
82Q8C22⋊D50911yescharacteristic
83Q8C22×D50911yescharacteristic
84D4—23no—
85D4—23no—
86D4—10183no—
87D4—10183no—
88D4—5093no—
89D4C22⋊D50913yes—
90D4C22×D50913yes—
91C2×C4—23no—
92C2×C4—10183no—
93C2×C4C22×D50913yes—
94C8—5091no—
95C8—5093no—
96C22—23no—
97C22—83no—
98C22—40723no—
99C22—10183no—
100C22C2×C22⋊D50913yes—
101C4—21no—
102C4—10181no—
103C4C2×C22⋊D50913yes—
104C4C23×D50911yescharacteristic
105C2—23no—
106C2—43no—
107C2—20363no—
108C2C22×C22⋊D50911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D509
commutator, cosocleC4×C509C24
FrattiniC4C23×D509
FittingH(5,𝔽2)×C509C2
socleC2×C509C22×D4
hypercenterH(5,𝔽2)D509
hypocenterC509#576
LCSG ▷ C4×C509 ▷ C2×C509 ▷ C509
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #30654261C2semidirect
2. #30654241C2semidirect
3. #30654240C2semidirect
4. #30654235C2semidirect
5. C2×#22550846C2semidirect
6. #30654222C2semidirect
7. H(5,𝔽2)×C509C2semidirect
9. #22550861C22semidirect
10. C2×D2036C22semidirect
12. #22550851C22semidirect
13. #22550849C22semidirect
14. #22550848C22semidirect
15. #22550846C22semidirect
16. #22550846C22semidirect
17. #22550845C22semidirect
18. C2×#16861325C22semidirect
21. #41×C509C22semidirect
23. C2×D4×C509C22semidirect
54. C509#576semidirect
57. H(5,𝔽2)D509semidirect
67. #41C2×D509semidirect
71. C2×D4C2×D509semidirect

Maximal subgroups

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

subgroupindex
#306542612
#306542412
#306542402
#306542352
C2×#225508462
#306542222
H(5,𝔽2)×C5092
#576509