Finite Group Databasebeta

← #31774700

Subgroups — #31774700

Order 17344.

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#22576206C211yesmaximal, characteristic, complemented
2#22576186C213yesmaximal, complemented
3#22576185C213yesmaximal, complemented
4#22576180C213yesmaximal, complemented
5C2×#16864639C213yesmaximal, complemented
6#22576167C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C271C211yesmaximal, characteristic, complemented
8#16864655—21no—
9#16864654C2213yescomplemented
10C2×D1084C2213yescomplemented
11D4×D271—23no—
12#16864644C2211yescharacteristic, complemented
13#16864642C2213yescomplemented
14#16864641C2213yescomplemented
15#16864639C2213yescomplemented
16#16864639C2216yescomplemented
17#16864638C2213yescomplemented
18C2×C271⋊C8C2213yescomplemented
19#41×C271—21no—
20#41×C271C2211yescharacteristic
21#41×C271C2213yescomplemented
22C2×D4×C271—23no—
23C2×D4×C271C2213yescomplemented
24C22×D271—23no—
25Dic542C2311yescharacteristic
26D1084—23no—
27D1084C2313yes—
28C22⋊D271—23no—
29C4×D271—23no—
30C271⋊C8C2311yescharacteristic
31C271⋊C8C2313yes—
32C23×C271—23no—
33Q8×C271D411yescharacteristic
34Q8×C271C2311yescharacteristic
35D4×C271—23no—
36D4×C271—23no—
37D4×C271D413yes—
38D4×C271C2313yes—
39C2×C4×C271—23no—
40C2×C4×C271C2313yes—
41C22×C271—23no—
42C22×C271—83no—
43C22×C271C2×D413yes—
44C2×D271—23no—
45C2×D271—83no—
46C4×C271—21no—
47C4×C271C2×D413yes—
48C4×C271C2411yescharacteristic
49Dic271—21no—
50C2×C271—23no—
51C2×C271—43no—
52C2×C271C22×D411yescharacteristic
53D271—43no—
54C271#57611yescharacteristic, complemented, Sylow
55#576—2711nomaximal, Sylow
56H(5,𝔽2)—2711no—
57H(5,𝔽2)D27111yescharacteristic, complemented
58AGL(1,ℤ/8)—2713no—
59AGL(1,ℤ/8)—2713no—
60#135—2713no—
61C2×D8—2713no—
62#131—2711no—
63#41—21no—
64#41—2713no—
65#41—5421no—
66#41C2×D27111yescharacteristic
67#41C2×D27113yescomplemented
68C2×D4—23no—
69C2×D4—2713no—
70C2×D4—5423no—
71C2×D4C2×D27113yescomplemented
72Q16—2711no—
73SD16—2713no—
74SD16—2713no—
75D8—2713no—
76D8—2716no—
77M16—2713no—
78C2×C8—2713no—
79C23—23no—
80C23—5423no—
81Q8—2711no—
82Q8C22⋊D27111yescharacteristic
83Q8C22×D27111yescharacteristic
84D4—23no—
85D4—23no—
86D4—2713no—
87D4—5423no—
88D4—5423no—
89D4C22⋊D27113yes—
90D4C22×D27113yes—
91C2×C4—23no—
92C2×C4—5423no—
93C2×C4C22×D27113yes—
94C8—2711no—
95C8—2713no—
96C22—23no—
97C22—83no—
98C22—5423no—
99C22—21683no—
100C22C2×C22⋊D27113yes—
101C4—21no—
102C4—5421no—
103C4C2×C22⋊D27113yes—
104C4C23×D27111yescharacteristic
105C2—23no—
106C2—43no—
107C2—10843no—
108C2C22×C22⋊D27111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D271
commutator, cosocleC4×C271C24
FrattiniC4C23×D271
FittingH(5,𝔽2)×C271C2
socleC2×C271C22×D4
hypercenterH(5,𝔽2)D271
hypocenterC271#576
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22576206C2semidirect
2. #22576186C2semidirect
3. #22576185C2semidirect
4. #22576180C2semidirect
5. C2×#16864639C2semidirect
6. #22576167C2semidirect
7. H(5,𝔽2)×C271C2semidirect
9. #16864654C22semidirect
10. C2×D1084C22semidirect
12. #16864644C22semidirect
13. #16864642C22semidirect
14. #16864641C22semidirect
15. #16864639C22semidirect
16. #16864639C22semidirect
17. #16864638C22semidirect
18. C2×C271⋊C8C22semidirect
21. #41×C271C22semidirect
23. C2×D4×C271C22semidirect
54. C271#576semidirect
57. H(5,𝔽2)D271semidirect
67. #41C2×D271semidirect
71. C2×D4C2×D271semidirect

Maximal subgroups

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

subgroupindex
#225762062
#225761862
#225761852
#225761802
C2×#168646392
#225761672
H(5,𝔽2)×C2712
#576271