Finite Group Databasebeta

← #39910050

Subgroups — #39910050

Order 26176.

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#27189763C211yesmaximal, characteristic, complemented
2#27189743C213yesmaximal, complemented
3#27189742C213yesmaximal, complemented
4#27189737C213yesmaximal, complemented
5C2×#20266973C213yesmaximal, complemented
6#27189724C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C409C211yesmaximal, characteristic, complemented
8#20266989—21no—
9#20266988C2213yescomplemented
10C2×D1636C2213yescomplemented
11D4×D409—23no—
12#20266978C2211yescharacteristic, complemented
13#20266976C2213yescomplemented
14#20266975C2213yescomplemented
15#20266973C2213yescomplemented
16#20266973C2216yescomplemented
17#20266972C2213yescomplemented
18C2×#15692543C2213yescomplemented
19#41×C409—21no—
20#41×C409C2211yescharacteristic
21#41×C409C2213yescomplemented
22C2×D4×C409—23no—
23C2×D4×C409C2213yescomplemented
24C22×D409—23no—
25Dic818C2311yescharacteristic
26D1636—23no—
27D1636C2313yes—
28C22⋊D409—23no—
29C4×D409—23no—
30#15692543C2311yescharacteristic
31#15692543C2313yes—
32C23×C409—23no—
33Q8×C409D411yescharacteristic
34Q8×C409C2311yescharacteristic
35D4×C409—23no—
36D4×C409—23no—
37D4×C409D413yes—
38D4×C409C2313yes—
39C2×C4×C409—23no—
40C2×C4×C409C2313yes—
41C22×C409—23no—
42C22×C409—83no—
43C22×C409C2×D413yes—
44C2×D409—23no—
45C2×D409—83no—
46C4×C409—21no—
47C4×C409C2×D413yes—
48C4×C409C2411yescharacteristic
49Dic409—21no—
50C2×C409—23no—
51C2×C409—43no—
52C2×C409C22×D411yescharacteristic
53D409—43no—
54C409#57611yescharacteristic, complemented, Sylow
55#576—4091nomaximal, Sylow
56H(5,𝔽2)—4091no—
57H(5,𝔽2)D40911yescharacteristic, complemented
58AGL(1,ℤ/8)—4093no—
59AGL(1,ℤ/8)—4093no—
60#135—4093no—
61C2×D8—4093no—
62#131—4091no—
63#41—21no—
64#41—8181no—
65#41—4093no—
66#41C2×D40911yescharacteristic
67#41C2×D40913yescomplemented
68C2×D4—23no—
69C2×D4—8183no—
70C2×D4—4093no—
71C2×D4C2×D40913yescomplemented
72Q16—4091no—
73SD16—4093no—
74SD16—4093no—
75D8—4093no—
76D8—4096no—
77M16—4093no—
78C2×C8—4093no—
79C23—23no—
80C23—8183no—
81Q8—4091no—
82Q8C22⋊D40911yescharacteristic
83Q8C22×D40911yescharacteristic
84D4—23no—
85D4—23no—
86D4—8183no—
87D4—8183no—
88D4—4093no—
89D4C22⋊D40913yes—
90D4C22×D40913yes—
91C2×C4—23no—
92C2×C4—8183no—
93C2×C4C22×D40913yes—
94C8—4091no—
95C8—4093no—
96C22—23no—
97C22—83no—
98C22—8183no—
99C22—32723no—
100C22C2×C22⋊D40913yes—
101C4—21no—
102C4—8181no—
103C4C2×C22⋊D40913yes—
104C4C23×D40911yescharacteristic
105C2—23no—
106C2—43no—
107C2—16363no—
108C2C22×C22⋊D40911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
FittingH(5,𝔽2)×C409C2
socleC2×C409C22×D4
hypercenterH(5,𝔽2)D409
hypocenterC409#576
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #27189763C2semidirect
2. #27189743C2semidirect
3. #27189742C2semidirect
4. #27189737C2semidirect
5. C2×#20266973C2semidirect
6. #27189724C2semidirect
7. H(5,𝔽2)×C409C2semidirect
9. #20266988C22semidirect
10. C2×D1636C22semidirect
12. #20266978C22semidirect
13. #20266976C22semidirect
14. #20266975C22semidirect
15. #20266973C22semidirect
16. #20266973C22semidirect
17. #20266972C22semidirect
18. C2×#15692543C22semidirect
21. #41×C409C22semidirect
23. C2×D4×C409C22semidirect
54. C409#576semidirect
57. H(5,𝔽2)D409semidirect
67. #41C2×D409semidirect
71. C2×D4C2×D409semidirect

Maximal subgroups

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

subgroupindex
#271897632
#271897432
#271897422
#271897372
C2×#202669732
#271897242
H(5,𝔽2)×C4092
#576409