Finite Group Databasebeta

← #26012294

Subgroups — #26012294

Order 11072.

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#19149725C211yesmaximal, characteristic, complemented
2#19149705C213yesmaximal, complemented
3#19149704C213yesmaximal, complemented
4#19149699C213yesmaximal, complemented
5C2×#14583811C213yesmaximal, complemented
6#19149686C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C173C211yesmaximal, characteristic, complemented
8#14583827—21no—
9#14583826C2213yescomplemented
10C2×D692C2213yescomplemented
11D4×D173—23no—
12#14583816C2211yescharacteristic, complemented
13#14583814C2213yescomplemented
14#14583813C2213yescomplemented
15#14583811C2213yescomplemented
16#14583811C2216yescomplemented
17#14583810C2213yescomplemented
18C2×#13063436C2213yescomplemented
19#41×C173—21no—
20#41×C173C2211yescharacteristic
21#41×C173C2213yescomplemented
22C2×D4×C173—23no—
23C2×D4×C173C2213yescomplemented
24C22×D173—23no—
25Dic346C2311yescharacteristic
26D692—23no—
27D692C2313yes—
28C22⋊D173—23no—
29C4×D173—23no—
30#13063436C2311yescharacteristic
31#13063436C2313yes—
32C23×C173—23no—
33Q8×C173D411yescharacteristic
34Q8×C173C2311yescharacteristic
35D4×C173—23no—
36D4×C173—23no—
37D4×C173D413yes—
38D4×C173C2313yes—
39C2×C4×C173—23no—
40C2×C4×C173C2313yes—
41C22×C173—23no—
42C22×C173—83no—
43C22×C173C2×D413yes—
44C2×D173—23no—
45C2×D173—83no—
46C4×C173—21no—
47C4×C173C2×D413yes—
48C4×C173C2411yescharacteristic
49Dic173—21no—
50C2×C173—23no—
51C2×C173—43no—
52C2×C173C22×D411yescharacteristic
53D173—43no—
54C173#57611yescharacteristic, complemented, Sylow
55#576—1731nomaximal, Sylow
56H(5,𝔽2)—1731no—
57H(5,𝔽2)D17311yescharacteristic, complemented
58AGL(1,ℤ/8)—1733no—
59AGL(1,ℤ/8)—1733no—
60#135—1733no—
61C2×D8—1733no—
62#131—1731no—
63#41—21no—
64#41—3461no—
65#41—1733no—
66#41C2×D17311yescharacteristic
67#41C2×D17313yescomplemented
68C2×D4—23no—
69C2×D4—3463no—
70C2×D4—1733no—
71C2×D4C2×D17313yescomplemented
72Q16—1731no—
73SD16—1733no—
74SD16—1733no—
75D8—1733no—
76D8—1736no—
77M16—1733no—
78C2×C8—1733no—
79C23—23no—
80C23—3463no—
81Q8—1731no—
82Q8C22⋊D17311yescharacteristic
83Q8C22×D17311yescharacteristic
84D4—23no—
85D4—23no—
86D4—3463no—
87D4—3463no—
88D4—1733no—
89D4C22⋊D17313yes—
90D4C22×D17313yes—
91C2×C4—23no—
92C2×C4—3463no—
93C2×C4C22×D17313yes—
94C8—1731no—
95C8—1733no—
96C22—23no—
97C22—83no—
98C22—3463no—
99C22—13843no—
100C22C2×C22⋊D17313yes—
101C4—21no—
102C4—3461no—
103C4C2×C22⋊D17313yes—
104C4C23×D17311yescharacteristic
105C2—23no—
106C2—43no—
107C2—6923no—
108C2C22×C22⋊D17311yescharacteristic
subgroupG/H
centerC2C22×C22⋊D173
commutator, cosocleC4×C173C24
FrattiniC4C23×D173
FittingH(5,𝔽2)×C173C2
socleC2×C173C22×D4
hypercenterH(5,𝔽2)D173
hypocenterC173#576
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #19149725C2semidirect
2. #19149705C2semidirect
3. #19149704C2semidirect
4. #19149699C2semidirect
5. C2×#14583811C2semidirect
6. #19149686C2semidirect
7. H(5,𝔽2)×C173C2semidirect
9. #14583826C22semidirect
10. C2×D692C22semidirect
12. #14583816C22semidirect
13. #14583814C22semidirect
14. #14583813C22semidirect
15. #14583811C22semidirect
16. #14583811C22semidirect
17. #14583810C22semidirect
18. C2×#13063436C22semidirect
21. #41×C173C22semidirect
23. C2×D4×C173C22semidirect
54. C173#576semidirect
57. H(5,𝔽2)D173semidirect
67. #41C2×D173semidirect
71. C2×D4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#191497252
#191497052
#191497042
#191496992
C2×#145838112
#191496862
H(5,𝔽2)×C1732
#576173