Finite Group Databasebeta

← #35204026

Subgroups — #35204026

Order 20288.

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#23778826C211yesmaximal, characteristic, complemented
2#23778806C213yesmaximal, complemented
3#23778805C213yesmaximal, complemented
4#23778800C213yesmaximal, complemented
5C2×#19099559C213yesmaximal, complemented
6#23778787C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C317C211yesmaximal, characteristic, complemented
8#19099575—21no—
9#19099574C2213yescomplemented
10C2×D1268C2213yescomplemented
11D4×D317—23no—
12#19099564C2211yescharacteristic, complemented
13#19099562C2213yescomplemented
14#19099561C2213yescomplemented
15#19099559C2213yescomplemented
16#19099559C2216yescomplemented
17#19099558C2213yescomplemented
18C2×#14579261C2213yescomplemented
19#41×C317—21no—
20#41×C317C2211yescharacteristic
21#41×C317C2213yescomplemented
22C2×D4×C317—23no—
23C2×D4×C317C2213yescomplemented
24C22×D317—23no—
25Dic634C2311yescharacteristic
26D1268—23no—
27D1268C2313yes—
28C22⋊D317—23no—
29C4×D317—23no—
30#14579261C2311yescharacteristic
31#14579261C2313yes—
32C23×C317—23no—
33Q8×C317D411yescharacteristic
34Q8×C317C2311yescharacteristic
35D4×C317—23no—
36D4×C317—23no—
37D4×C317D413yes—
38D4×C317C2313yes—
39C2×C4×C317—23no—
40C2×C4×C317C2313yes—
41C22×C317—23no—
42C22×C317—83no—
43C22×C317C2×D413yes—
44C2×D317—23no—
45C2×D317—83no—
46C4×C317—21no—
47C4×C317C2×D413yes—
48C4×C317C2411yescharacteristic
49Dic317—21no—
50C2×C317—23no—
51C2×C317—43no—
52C2×C317C22×D411yescharacteristic
53D317—43no—
54C317#57611yescharacteristic, complemented, Sylow
55#576—3171nomaximal, Sylow
56H(5,𝔽2)—3171no—
57H(5,𝔽2)D31711yescharacteristic, complemented
58AGL(1,ℤ/8)—3173no—
59AGL(1,ℤ/8)—3173no—
60#135—3173no—
61C2×D8—3173no—
62#131—3171no—
63#41—21no—
64#41—3173no—
65#41—6341no—
66#41C2×D31711yescharacteristic
67#41C2×D31713yescomplemented
68C2×D4—23no—
69C2×D4—3173no—
70C2×D4—6343no—
71C2×D4C2×D31713yescomplemented
72Q16—3171no—
73SD16—3173no—
74SD16—3173no—
75D8—3173no—
76D8—3176no—
77M16—3173no—
78C2×C8—3173no—
79C23—23no—
80C23—6343no—
81Q8—3171no—
82Q8C22⋊D31711yescharacteristic
83Q8C22×D31711yescharacteristic
84D4—23no—
85D4—23no—
86D4—3173no—
87D4—6343no—
88D4—6343no—
89D4C22⋊D31713yes—
90D4C22×D31713yes—
91C2×C4—23no—
92C2×C4—6343no—
93C2×C4C22×D31713yes—
94C8—3171no—
95C8—3173no—
96C22—23no—
97C22—83no—
98C22—6343no—
99C22—25363no—
100C22C2×C22⋊D31713yes—
101C4—21no—
102C4—6341no—
103C4C2×C22⋊D31713yes—
104C4C23×D31711yescharacteristic
105C2—23no—
106C2—43no—
107C2—12683no—
108C2C22×C22⋊D31711yescharacteristic
subgroupG/H
centerC2C22×C22⋊D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
FittingH(5,𝔽2)×C317C2
socleC2×C317C22×D4
hypercenterH(5,𝔽2)D317
hypocenterC317#576
LCSG ▷ C4×C317 ▷ C2×C317 ▷ C317
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23778826C2semidirect
2. #23778806C2semidirect
3. #23778805C2semidirect
4. #23778800C2semidirect
5. C2×#19099559C2semidirect
6. #23778787C2semidirect
7. H(5,𝔽2)×C317C2semidirect
9. #19099574C22semidirect
10. C2×D1268C22semidirect
12. #19099564C22semidirect
13. #19099562C22semidirect
14. #19099561C22semidirect
15. #19099559C22semidirect
16. #19099559C22semidirect
17. #19099558C22semidirect
18. C2×#14579261C22semidirect
21. #41×C317C22semidirect
23. C2×D4×C317C22semidirect
54. C317#576semidirect
57. H(5,𝔽2)D317semidirect
67. #41C2×D317semidirect
71. C2×D4C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237788262
#237788062
#237788052
#237788002
C2×#190995592
#237787872
H(5,𝔽2)×C3172
#576317