Finite Group Databasebeta

← #31774708

Subgroups — #31774708

Order 17344.

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#22576207C211yesmaximal, characteristic, complemented
2#22576190C213yesmaximal, complemented
3#22576186C213yesmaximal, complemented
4#22576180C213yesmaximal, complemented
5C2×#16864642C213yesmaximal, complemented
6#22576167C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C271C211yesmaximal, characteristic, complemented
8#16864654C2213yescomplemented
9#16864653—23no—
10C2×Dic542C2213yescomplemented
11Q8×D271—21no—
12#16864644C2213yescomplemented
13#16864642C2213yescomplemented
14#16864642C2216yescomplemented
15#16864641C2211yescharacteristic, complemented
16#16864639C2213yescomplemented
17#16864638C2213yescomplemented
18C2×C271⋊C8C2213yescomplemented
19#41×C271—21no—
20#41×C271C2213yes—
21#41×C271C2211yescharacteristic
22C2×D4×C271—23no—
23C2×D4×C271C2213yes—
24Dic542—23no—
25Dic542C2313yes—
26D1084C2311yescharacteristic
27C22⋊D271—23no—
28C2×Dic271—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—21no—
45C4×C271—21no—
46C4×C271C2×D413yes—
47C4×C271C2411yescharacteristic
48Dic271—23no—
49C2×C271—23no—
50C2×C271—43no—
51C2×C271C22×D411yescharacteristic
52D271—41no—
53C271#57711yescharacteristic, complemented, Sylow
54#577—2711nomaximal, Sylow
552−1+4—2711no—
56H(5,𝔽2)D27111yescharacteristic, complemented
57#137—2713no—
58AGL(1,ℤ/8)—2713no—
59#135—2713no—
60C2×SD16—2713no—
61#131—2711no—
62#41—21no—
63#41—2713no—
64#41—5423no—
65#41C2×D27113yes—
66#41C2×D27111yescharacteristic
67C2×Q8—2713no—
68C2×Q8—5421no—
69C2×D4—23no—
70C2×D4C2×D27113yes—
71Q16—2713no—
72SD16—2711no—
73SD16—2713no—
74SD16—2716no—
75D8—2713no—
76M16—2713no—
77C2×C8—2713no—
78C23—23no—
79Q8—2713no—
80Q8—5423no—
81Q8C22⋊D27111yescharacteristic
82Q8C22×D27111yescharacteristic
83D4—23no—
84D4—23no—
85D4—2711no—
86D4—5423no—
87D4C22⋊D27113yes—
88D4C22×D27113yes—
89C2×C4—23no—
90C2×C4—5423no—
91C2×C4—5423no—
92C2×C4C22×D27113yes—
93C8—2711no—
94C8—2713no—
95C22—23no—
96C22—83no—
97C22—5421no—
98C22C2×C22⋊D27113yes—
99C4—21no—
100C4—5423no—
101C4C2×C22⋊D27113yes—
102C4C23×D27111yescharacteristic
103C2—23no—
104C2—43no—
105C2—10841no—
106C2C22×C22⋊D27111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D271
commutator, cosocleC4×C271C24
FrattiniC4C23×D271
FittingH(5,𝔽2)×C271C2
socleC2×C271C22×D4
hypercenterH(5,𝔽2)D271
hypocenterC271#577
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. #22576207C2semidirect
2. #22576190C2semidirect
3. #22576186C2semidirect
4. #22576180C2semidirect
5. C2×#16864642C2semidirect
6. #22576167C2semidirect
7. H(5,𝔽2)×C271C2semidirect
8. #16864654C22semidirect
10. C2×Dic542C22semidirect
12. #16864644C22semidirect
13. #16864642C22semidirect
14. #16864642C22semidirect
15. #16864641C22semidirect
16. #16864639C22semidirect
17. #16864638C22semidirect
18. C2×C271⋊C8C22semidirect
53. C271#577semidirect
56. H(5,𝔽2)D271semidirect

Maximal subgroups

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

subgroupindex
#225762072
#225761902
#225761862
#225761802
C2×#168646422
#225761672
H(5,𝔽2)×C2712
#577271