Finite Group Databasebeta

← #37554989

Subgroups — #37554989

Order 23488.

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#26044307C211yesmaximal, characteristic, complemented
2#26044287C213yesmaximal, complemented
3#26044286C213yesmaximal, complemented
4#26044281C213yesmaximal, complemented
5C2×#19154562C213yesmaximal, complemented
6#26044268C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C367C211yesmaximal, characteristic, complemented
8#19154578—21no—
9#19154577C2213yescomplemented
10C2×D1468C2213yescomplemented
11D4×D367—23no—
12#19154567C2211yescharacteristic, complemented
13#19154565C2213yescomplemented
14#19154564C2213yescomplemented
15#19154562C2213yescomplemented
16#19154562C2216yescomplemented
17#19154561C2213yescomplemented
18C2×C367⋊C8C2213yescomplemented
19#41×C367—21no—
20#41×C367C2211yescharacteristic
21#41×C367C2213yescomplemented
22C2×D4×C367—23no—
23C2×D4×C367C2213yescomplemented
24C22×D367—23no—
25Dic734C2311yescharacteristic
26D1468—23no—
27D1468C2313yes—
28C22⋊D367—23no—
29C4×D367—23no—
30C367⋊C8C2311yescharacteristic
31C367⋊C8C2313yes—
32C23×C367—23no—
33Q8×C367D411yescharacteristic
34Q8×C367C2311yescharacteristic
35D4×C367—23no—
36D4×C367—23no—
37D4×C367D413yes—
38D4×C367C2313yes—
39C2×C4×C367—23no—
40C2×C4×C367C2313yes—
41C22×C367—23no—
42C22×C367—83no—
43C22×C367C2×D413yes—
44C2×D367—23no—
45C2×D367—83no—
46C4×C367—21no—
47C4×C367C2×D413yes—
48C4×C367C2411yescharacteristic
49Dic367—21no—
50C2×C367—23no—
51C2×C367—43no—
52C2×C367C22×D411yescharacteristic
53D367—43no—
54C367#57611yescharacteristic, complemented, Sylow
55#576—3671nomaximal, Sylow
56H(5,𝔽2)—3671no—
57H(5,𝔽2)D36711yescharacteristic, complemented
58AGL(1,ℤ/8)—3673no—
59AGL(1,ℤ/8)—3673no—
60#135—3673no—
61C2×D8—3673no—
62#131—3671no—
63#41—21no—
64#41—3673no—
65#41—7341no—
66#41C2×D36711yescharacteristic
67#41C2×D36713yescomplemented
68C2×D4—23no—
69C2×D4—3673no—
70C2×D4—7343no—
71C2×D4C2×D36713yescomplemented
72Q16—3671no—
73SD16—3673no—
74SD16—3673no—
75D8—3673no—
76D8—3676no—
77M16—3673no—
78C2×C8—3673no—
79C23—23no—
80C23—7343no—
81Q8—3671no—
82Q8C22⋊D36711yescharacteristic
83Q8C22×D36711yescharacteristic
84D4—23no—
85D4—23no—
86D4—3673no—
87D4—7343no—
88D4—7343no—
89D4C22⋊D36713yes—
90D4C22×D36713yes—
91C2×C4—23no—
92C2×C4—7343no—
93C2×C4C22×D36713yes—
94C8—3671no—
95C8—3673no—
96C22—23no—
97C22—83no—
98C22—29363no—
99C22—7343no—
100C22C2×C22⋊D36713yes—
101C4—21no—
102C4—7341no—
103C4C2×C22⋊D36713yes—
104C4C23×D36711yescharacteristic
105C2—23no—
106C2—43no—
107C2—14683no—
108C2C22×C22⋊D36711yescharacteristic
subgroupG/H
centerC2C22×C22⋊D367
commutator, cosocleC4×C367C24
FrattiniC4C23×D367
FittingH(5,𝔽2)×C367C2
socleC2×C367C22×D4
hypercenterH(5,𝔽2)D367
hypocenterC367#576
LCSG ▷ C4×C367 ▷ C2×C367 ▷ C367
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26044307C2semidirect
2. #26044287C2semidirect
3. #26044286C2semidirect
4. #26044281C2semidirect
5. C2×#19154562C2semidirect
6. #26044268C2semidirect
7. H(5,𝔽2)×C367C2semidirect
9. #19154577C22semidirect
10. C2×D1468C22semidirect
12. #19154567C22semidirect
13. #19154565C22semidirect
14. #19154564C22semidirect
15. #19154562C22semidirect
16. #19154562C22semidirect
17. #19154561C22semidirect
18. C2×C367⋊C8C22semidirect
21. #41×C367C22semidirect
23. C2×D4×C367C22semidirect
54. C367#576semidirect
57. H(5,𝔽2)D367semidirect
67. #41C2×D367semidirect
71. C2×D4C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443072
#260442872
#260442862
#260442812
C2×#191545622
#260442682
H(5,𝔽2)×C3672
#576367