Finite Group Databasebeta

← #41001164

Subgroups — #41001164

Order 26816.

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#27211764C211yesmaximal, characteristic, complemented
2#27211744C213yesmaximal, complemented
3#27211743C213yesmaximal, complemented
4#27211738C213yesmaximal, complemented
5C2×#20269396C213yesmaximal, complemented
6#27211725C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C419C211yesmaximal, characteristic, complemented
8#20269412—21no—
9#20269411C2213yescomplemented
10C2×D1676C2213yescomplemented
11D4×D419—23no—
12#20269401C2211yescharacteristic, complemented
13#20269399C2213yescomplemented
14#20269398C2213yescomplemented
15#20269396C2213yescomplemented
16#20269396C2216yescomplemented
17#20269395C2213yescomplemented
18C2×C419⋊C8C2213yescomplemented
19#41×C419—21no—
20#41×C419C2211yescharacteristic
21#41×C419C2213yescomplemented
22C2×D4×C419—23no—
23C2×D4×C419C2213yescomplemented
24C22×D419—23no—
25Dic838C2311yescharacteristic
26D1676—23no—
27D1676C2313yes—
28C22⋊D419—23no—
29C4×D419—23no—
30C419⋊C8C2311yescharacteristic
31C419⋊C8C2313yes—
32C23×C419—23no—
33Q8×C419D411yescharacteristic
34Q8×C419C2311yescharacteristic
35D4×C419—23no—
36D4×C419—23no—
37D4×C419D413yes—
38D4×C419C2313yes—
39C2×C4×C419—23no—
40C2×C4×C419C2313yes—
41C22×C419—23no—
42C22×C419—83no—
43C22×C419C2×D413yes—
44C2×D419—23no—
45C2×D419—83no—
46C4×C419—21no—
47C4×C419C2×D413yes—
48C4×C419C2411yescharacteristic
49Dic419—21no—
50C2×C419—23no—
51C2×C419—43no—
52C2×C419C22×D411yescharacteristic
53D419—43no—
54C419#57611yescharacteristic, complemented, Sylow
55#576—4191nomaximal, Sylow
56H(5,𝔽2)—4191no—
57H(5,𝔽2)D41911yescharacteristic, complemented
58AGL(1,ℤ/8)—4193no—
59AGL(1,ℤ/8)—4193no—
60#135—4193no—
61C2×D8—4193no—
62#131—4191no—
63#41—21no—
64#41—8381no—
65#41—4193no—
66#41C2×D41911yescharacteristic
67#41C2×D41913yescomplemented
68C2×D4—23no—
69C2×D4—8383no—
70C2×D4—4193no—
71C2×D4C2×D41913yescomplemented
72Q16—4191no—
73SD16—4193no—
74SD16—4193no—
75D8—4193no—
76D8—4196no—
77M16—4193no—
78C2×C8—4193no—
79C23—23no—
80C23—8383no—
81Q8—4191no—
82Q8C22⋊D41911yescharacteristic
83Q8C22×D41911yescharacteristic
84D4—23no—
85D4—23no—
86D4—8383no—
87D4—8383no—
88D4—4193no—
89D4C22⋊D41913yes—
90D4C22×D41913yes—
91C2×C4—23no—
92C2×C4—8383no—
93C2×C4C22×D41913yes—
94C8—4191no—
95C8—4193no—
96C22—23no—
97C22—83no—
98C22—33523no—
99C22—8383no—
100C22C2×C22⋊D41913yes—
101C4—21no—
102C4—8381no—
103C4C2×C22⋊D41913yes—
104C4C23×D41911yescharacteristic
105C2—23no—
106C2—43no—
107C2—16763no—
108C2C22×C22⋊D41911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D419
commutator, cosocleC4×C419C24
FrattiniC4C23×D419
FittingH(5,𝔽2)×C419C2
socleC2×C419C22×D4
hypercenterH(5,𝔽2)D419
hypocenterC419#576
LCSG ▷ C4×C419 ▷ C2×C419 ▷ C419
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #27211764C2semidirect
2. #27211744C2semidirect
3. #27211743C2semidirect
4. #27211738C2semidirect
5. C2×#20269396C2semidirect
6. #27211725C2semidirect
7. H(5,𝔽2)×C419C2semidirect
9. #20269411C22semidirect
10. C2×D1676C22semidirect
12. #20269401C22semidirect
13. #20269399C22semidirect
14. #20269398C22semidirect
15. #20269396C22semidirect
16. #20269396C22semidirect
17. #20269395C22semidirect
18. C2×C419⋊C8C22semidirect
21. #41×C419C22semidirect
23. C2×D4×C419C22semidirect
54. C419#576semidirect
57. H(5,𝔽2)D419semidirect
67. #41C2×D419semidirect
71. C2×D4C2×D419semidirect

Maximal subgroups

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

subgroupindex
#272117642
#272117442
#272117432
#272117382
C2×#202693962
#272117252
H(5,𝔽2)×C4192
#576419