Finite Group Databasebeta

← #35204034

Subgroups — #35204034

Order 20288.

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

normal subgroups only

This group has 258 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#23778827C211yesmaximal, characteristic, complemented
2#23778810C213yesmaximal, complemented
3#23778806C213yesmaximal, complemented
4#23778800C213yesmaximal, complemented
5C2×#19099562C213yesmaximal, complemented
6#23778787C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C317C211yesmaximal, characteristic, complemented
8#19099574C2213yescomplemented
9#19099573—23no—
10C2×Dic634C2213yescomplemented
11Q8×D317—21no—
12#19099564C2213yescomplemented
13#19099562C2213yescomplemented
14#19099562C2216yescomplemented
15#19099561C2211yescharacteristic, complemented
16#19099559C2213yescomplemented
17#19099558C2213yescomplemented
18C2×#14579261C2213yescomplemented
19#41×C317—21no—
20#41×C317C2213yes—
21#41×C317C2211yescharacteristic
22C2×D4×C317—23no—
23C2×D4×C317C2213yes—
24Dic634—23no—
25Dic634C2313yes—
26D1268C2311yescharacteristic
27C22⋊D317—23no—
28C2×Dic317—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—21no—
45C4×C317—21no—
46C4×C317C2×D413yes—
47C4×C317C2411yescharacteristic
48Dic317—23no—
49C2×C317—23no—
50C2×C317—43no—
51C2×C317C22×D411yescharacteristic
52D317—41no—
53C317#57711yescharacteristic, complemented, Sylow
54#577—3171nomaximal, Sylow
552−1+4—3171no—
56H(5,𝔽2)D31711yescharacteristic, complemented
57#137—3173no—
58AGL(1,ℤ/8)—3173no—
59#135—3173no—
60C2×SD16—3173no—
61#131—3171no—
62#41—21no—
63#41—3173no—
64#41—6343no—
65#41C2×D31713yes—
66#41C2×D31711yescharacteristic
67C2×Q8—3173no—
68C2×Q8—6341no—
69C2×D4—23no—
70C2×D4C2×D31713yes—
71Q16—3173no—
72SD16—3171no—
73SD16—3173no—
74SD16—3176no—
75D8—3173no—
76M16—3173no—
77C2×C8—3173no—
78C23—23no—
79Q8—3173no—
80Q8—6343no—
81Q8C22⋊D31711yescharacteristic
82Q8C22×D31711yescharacteristic
83D4—23no—
84D4—23no—
85D4—3171no—
86D4—6343no—
87D4C22⋊D31713yes—
88D4C22×D31713yes—
89C2×C4—23no—
90C2×C4—6343no—
91C2×C4—6343no—
92C2×C4C22×D31713yes—
93C8—3171no—
94C8—3173no—
95C22—23no—
96C22—83no—
97C22—6341no—
98C22C2×C22⋊D31713yes—
99C4—21no—
100C4—6343no—
101C4C2×C22⋊D31713yes—
102C4C23×D31711yescharacteristic
103C2—23no—
104C2—43no—
105C2—12681no—
106C2C22×C22⋊D31711yescharacteristic
subgroupG/H
centerC2C22×C22⋊D317
commutator, cosocleC4×C317C24
FrattiniC4C23×D317
FittingH(5,𝔽2)×C317C2
socleC2×C317C22×D4
hypercenterH(5,𝔽2)D317
hypocenterC317#577
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. #23778827C2semidirect
2. #23778810C2semidirect
3. #23778806C2semidirect
4. #23778800C2semidirect
5. C2×#19099562C2semidirect
6. #23778787C2semidirect
7. H(5,𝔽2)×C317C2semidirect
8. #19099574C22semidirect
10. C2×Dic634C22semidirect
12. #19099564C22semidirect
13. #19099562C22semidirect
14. #19099562C22semidirect
15. #19099561C22semidirect
16. #19099559C22semidirect
17. #19099558C22semidirect
18. C2×#14579261C22semidirect
53. C317#577semidirect
56. H(5,𝔽2)D317semidirect

Maximal subgroups

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

subgroupindex
#237788272
#237788102
#237788062
#237788002
C2×#190995622
#237787872
H(5,𝔽2)×C3172
#577317