Finite Group Databasebeta

← #36378598

Subgroups — #36378598

Order 22592.

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#26015039C211yesmaximal, characteristic, complemented
2#26015022C213yesmaximal, complemented
3#26015018C213yesmaximal, complemented
4#26015012C213yesmaximal, complemented
5C2×#19150725C213yesmaximal, complemented
6#26014999C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C353C211yesmaximal, characteristic, complemented
8#19150737C2213yescomplemented
9#19150736—23no—
10C2×Dic706C2213yescomplemented
11Q8×D353—21no—
12#19150727C2213yescomplemented
13#19150725C2213yescomplemented
14#19150725C2216yescomplemented
15#19150724C2211yescharacteristic, complemented
16#19150722C2213yescomplemented
17#19150721C2213yescomplemented
18C2×#15667725C2213yescomplemented
19#41×C353—21no—
20#41×C353C2213yes—
21#41×C353C2211yescharacteristic
22C2×D4×C353—23no—
23C2×D4×C353C2213yes—
24Dic706—23no—
25Dic706C2313yes—
26D1412C2311yescharacteristic
27C22⋊D353—23no—
28C2×Dic353—23no—
29C4×D353—23no—
30#15667725C2311yescharacteristic
31#15667725C2313yes—
32C23×C353—23no—
33Q8×C353D411yescharacteristic
34Q8×C353C2311yescharacteristic
35D4×C353—23no—
36D4×C353—23no—
37D4×C353D413yes—
38D4×C353C2313yes—
39C2×C4×C353—23no—
40C2×C4×C353C2313yes—
41C22×C353—23no—
42C22×C353—83no—
43C22×C353C2×D413yes—
44C2×D353—21no—
45C4×C353—21no—
46C4×C353C2×D413yes—
47C4×C353C2411yescharacteristic
48Dic353—23no—
49C2×C353—23no—
50C2×C353—43no—
51C2×C353C22×D411yescharacteristic
52D353—41no—
53C353#57711yescharacteristic, complemented, Sylow
54#577—3531nomaximal, Sylow
552−1+4—3531no—
56H(5,𝔽2)D35311yescharacteristic, complemented
57#137—3533no—
58AGL(1,ℤ/8)—3533no—
59#135—3533no—
60C2×SD16—3533no—
61#131—3531no—
62#41—21no—
63#41—3533no—
64#41—7063no—
65#41C2×D35313yes—
66#41C2×D35311yescharacteristic
67C2×Q8—3533no—
68C2×Q8—7061no—
69C2×D4—23no—
70C2×D4C2×D35313yes—
71Q16—3533no—
72SD16—3531no—
73SD16—3533no—
74SD16—3536no—
75D8—3533no—
76M16—3533no—
77C2×C8—3533no—
78C23—23no—
79Q8—3533no—
80Q8—7063no—
81Q8C22⋊D35311yescharacteristic
82Q8C22×D35311yescharacteristic
83D4—23no—
84D4—23no—
85D4—3531no—
86D4—7063no—
87D4C22⋊D35313yes—
88D4C22×D35313yes—
89C2×C4—23no—
90C2×C4—7063no—
91C2×C4—7063no—
92C2×C4C22×D35313yes—
93C8—3531no—
94C8—3533no—
95C22—23no—
96C22—83no—
97C22—7061no—
98C22C2×C22⋊D35313yes—
99C4—21no—
100C4—7063no—
101C4C2×C22⋊D35313yes—
102C4C23×D35311yescharacteristic
103C2—23no—
104C2—43no—
105C2—14121no—
106C2C22×C22⋊D35311yescharacteristic
subgroupG/H
centerC2C22×C22⋊D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
FittingH(5,𝔽2)×C353C2
socleC2×C353C22×D4
hypercenterH(5,𝔽2)D353
hypocenterC353#577
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26015039C2semidirect
2. #26015022C2semidirect
3. #26015018C2semidirect
4. #26015012C2semidirect
5. C2×#19150725C2semidirect
6. #26014999C2semidirect
7. H(5,𝔽2)×C353C2semidirect
8. #19150737C22semidirect
10. C2×Dic706C22semidirect
12. #19150727C22semidirect
13. #19150725C22semidirect
14. #19150725C22semidirect
15. #19150724C22semidirect
16. #19150722C22semidirect
17. #19150721C22semidirect
18. C2×#15667725C22semidirect
53. C353#577semidirect
56. H(5,𝔽2)D353semidirect

Maximal subgroups

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

subgroupindex
#260150392
#260150222
#260150182
#260150122
C2×#191507252
#260149992
H(5,𝔽2)×C3532
#577353