Finite Group Databasebeta

← #41003643

Subgroups — #41003643

Order 26944.

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#27212412C211yesmaximal, characteristic, complemented
2#27212392C213yesmaximal, complemented
3#27212391C213yesmaximal, complemented
4#27212386C213yesmaximal, complemented
5C2×#20269756C213yesmaximal, complemented
6#27212373C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C421C211yesmaximal, characteristic, complemented
8#20269772—21no—
9#20269771C2213yescomplemented
10C2×D1684C2213yescomplemented
11D4×D421—23no—
12#20269761C2211yescharacteristic, complemented
13#20269759C2213yescomplemented
14#20269758C2213yescomplemented
15#20269756C2213yescomplemented
16#20269756C2216yescomplemented
17#20269755C2213yescomplemented
18C2×#16809890C2213yescomplemented
19#41×C421—21no—
20#41×C421C2211yescharacteristic
21#41×C421C2213yescomplemented
22C2×D4×C421—23no—
23C2×D4×C421C2213yescomplemented
24C22×D421—23no—
25Dic842C2311yescharacteristic
26D1684—23no—
27D1684C2313yes—
28C22⋊D421—23no—
29C4×D421—23no—
30#16809890C2311yescharacteristic
31#16809890C2313yes—
32C23×C421—23no—
33Q8×C421D411yescharacteristic
34Q8×C421C2311yescharacteristic
35D4×C421—23no—
36D4×C421—23no—
37D4×C421D413yes—
38D4×C421C2313yes—
39C2×C4×C421—23no—
40C2×C4×C421C2313yes—
41C22×C421—23no—
42C22×C421—83no—
43C22×C421C2×D413yes—
44C2×D421—23no—
45C2×D421—83no—
46C4×C421—21no—
47C4×C421C2×D413yes—
48C4×C421C2411yescharacteristic
49Dic421—21no—
50C2×C421—23no—
51C2×C421—43no—
52C2×C421C22×D411yescharacteristic
53D421—43no—
54C421#57611yescharacteristic, complemented, Sylow
55#576—4211nomaximal, Sylow
56H(5,𝔽2)—4211no—
57H(5,𝔽2)D42111yescharacteristic, complemented
58AGL(1,ℤ/8)—4213no—
59AGL(1,ℤ/8)—4213no—
60#135—4213no—
61C2×D8—4213no—
62#131—4211no—
63#41—21no—
64#41—8421no—
65#41—4213no—
66#41C2×D42111yescharacteristic
67#41C2×D42113yescomplemented
68C2×D4—23no—
69C2×D4—8423no—
70C2×D4—4213no—
71C2×D4C2×D42113yescomplemented
72Q16—4211no—
73SD16—4213no—
74SD16—4213no—
75D8—4213no—
76D8—4216no—
77M16—4213no—
78C2×C8—4213no—
79C23—23no—
80C23—8423no—
81Q8—4211no—
82Q8C22⋊D42111yescharacteristic
83Q8C22×D42111yescharacteristic
84D4—23no—
85D4—23no—
86D4—8423no—
87D4—8423no—
88D4—4213no—
89D4C22⋊D42113yes—
90D4C22×D42113yes—
91C2×C4—23no—
92C2×C4—8423no—
93C2×C4C22×D42113yes—
94C8—4211no—
95C8—4213no—
96C22—23no—
97C22—83no—
98C22—33683no—
99C22—8423no—
100C22C2×C22⋊D42113yes—
101C4—21no—
102C4—8421no—
103C4C2×C22⋊D42113yes—
104C4C23×D42111yescharacteristic
105C2—23no—
106C2—43no—
107C2—16843no—
108C2C22×C22⋊D42111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
FittingH(5,𝔽2)×C421C2
socleC2×C421C22×D4
hypercenterH(5,𝔽2)D421
hypocenterC421#576
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #27212412C2semidirect
2. #27212392C2semidirect
3. #27212391C2semidirect
4. #27212386C2semidirect
5. C2×#20269756C2semidirect
6. #27212373C2semidirect
7. H(5,𝔽2)×C421C2semidirect
9. #20269771C22semidirect
10. C2×D1684C22semidirect
12. #20269761C22semidirect
13. #20269759C22semidirect
14. #20269758C22semidirect
15. #20269756C22semidirect
16. #20269756C22semidirect
17. #20269755C22semidirect
18. C2×#16809890C22semidirect
21. #41×C421C22semidirect
23. C2×D4×C421C22semidirect
54. C421#576semidirect
57. H(5,𝔽2)D421semidirect
67. #41C2×D421semidirect
71. C2×D4C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272124122
#272123922
#272123912
#272123862
C2×#202697562
#272123732
H(5,𝔽2)×C4212
#576421