Finite Group Databasebeta

← #41003651

Subgroups — #41003651

Order 26944.

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#27212413C211yesmaximal, characteristic, complemented
2#27212396C213yesmaximal, complemented
3#27212392C213yesmaximal, complemented
4#27212386C213yesmaximal, complemented
5C2×#20269759C213yesmaximal, complemented
6#27212373C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C421C211yesmaximal, characteristic, complemented
8#20269771C2213yescomplemented
9#20269770—23no—
10C2×Dic842C2213yescomplemented
11Q8×D421—21no—
12#20269761C2213yescomplemented
13#20269759C2213yescomplemented
14#20269759C2216yescomplemented
15#20269758C2211yescharacteristic, complemented
16#20269756C2213yescomplemented
17#20269755C2213yescomplemented
18C2×#16809890C2213yescomplemented
19#41×C421—21no—
20#41×C421C2213yes—
21#41×C421C2211yescharacteristic
22C2×D4×C421—23no—
23C2×D4×C421C2213yes—
24Dic842—23no—
25Dic842C2313yes—
26D1684C2311yescharacteristic
27C22⋊D421—23no—
28C2×Dic421—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—21no—
45C4×C421—21no—
46C4×C421C2×D413yes—
47C4×C421C2411yescharacteristic
48Dic421—23no—
49C2×C421—23no—
50C2×C421—43no—
51C2×C421C22×D411yescharacteristic
52D421—41no—
53C421#57711yescharacteristic, complemented, Sylow
54#577—4211nomaximal, Sylow
552−1+4—4211no—
56H(5,𝔽2)D42111yescharacteristic, complemented
57#137—4213no—
58AGL(1,ℤ/8)—4213no—
59#135—4213no—
60C2×SD16—4213no—
61#131—4211no—
62#41—21no—
63#41—8423no—
64#41—4213no—
65#41C2×D42113yes—
66#41C2×D42111yescharacteristic
67C2×Q8—8421no—
68C2×Q8—4213no—
69C2×D4—23no—
70C2×D4C2×D42113yes—
71Q16—4213no—
72SD16—4211no—
73SD16—4213no—
74SD16—4216no—
75D8—4213no—
76M16—4213no—
77C2×C8—4213no—
78C23—23no—
79Q8—8423no—
80Q8—4213no—
81Q8C22⋊D42111yescharacteristic
82Q8C22×D42111yescharacteristic
83D4—23no—
84D4—23no—
85D4—8423no—
86D4—4211no—
87D4C22⋊D42113yes—
88D4C22×D42113yes—
89C2×C4—23no—
90C2×C4—8423no—
91C2×C4—8423no—
92C2×C4C22×D42113yes—
93C8—4211no—
94C8—4213no—
95C22—23no—
96C22—83no—
97C22—8421no—
98C22C2×C22⋊D42113yes—
99C4—21no—
100C4—8423no—
101C4C2×C22⋊D42113yes—
102C4C23×D42111yescharacteristic
103C2—23no—
104C2—43no—
105C2—16841no—
106C2C22×C22⋊D42111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
FittingH(5,𝔽2)×C421C2
socleC2×C421C22×D4
hypercenterH(5,𝔽2)D421
hypocenterC421#577
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. #27212413C2semidirect
2. #27212396C2semidirect
3. #27212392C2semidirect
4. #27212386C2semidirect
5. C2×#20269759C2semidirect
6. #27212373C2semidirect
7. H(5,𝔽2)×C421C2semidirect
8. #20269771C22semidirect
10. C2×Dic842C22semidirect
12. #20269761C22semidirect
13. #20269759C22semidirect
14. #20269759C22semidirect
15. #20269758C22semidirect
16. #20269756C22semidirect
17. #20269755C22semidirect
18. C2×#16809890C22semidirect
53. C421#577semidirect
56. H(5,𝔽2)D421semidirect

Maximal subgroups

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

subgroupindex
#272124132
#272123962
#272123922
#272123862
C2×#202697592
#272123732
H(5,𝔽2)×C4212
#577421