Finite Group Databasebeta

← C2×#27163141 #38763352

Subgroups — C2×#27163141 #38763352

Order 25408.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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#27163141C2116yesmaximal, direct product factor
2C2×#20263731C212yesmaximal, complemented
3C2×#20263730C214yesmaximal, complemented
4C22×C22⋊D397C214yesmaximal, complemented
5C2×D4×D397C214yesmaximal, complemented
6C22×D4×C397C211yesmaximal, characteristic, complemented
7C23×D397C2214yescomplemented
8#20263731C22116yescomplemented
9#20263730C22132yescomplemented
10C2×Dic794C2212yescomplemented
11C2×D1588C2212yescomplemented
12C2×C22⋊D397C2218yescomplemented
13C2×C22⋊D397C2214yescomplemented
14C2×C22⋊D397C22132yescomplemented
15D4×D397C22132yescomplemented
16C22×Dic397C2214yescomplemented
17C2×C4×D397C2214yescomplemented
18C24×C397C2212yescomplemented
19C2×D4×C397C2214yescomplemented
20C2×D4×C397C2218yescomplemented
21C22×C4×C397C2211yescharacteristic, complemented
22C22×D397—48no—
23C22×D397C2314yes—
24C22×D397C23116yescomplemented
25Dic794C2318yescomplemented
26D1588C2318yescomplemented
27C22⋊D397C23132yescomplemented
28C22⋊D397C23116yescomplemented
29C2×Dic397C2314yes—
30C2×Dic397C23116yescomplemented
31C4×D397C23116yescomplemented
32C23×C397—44no—
33C23×C397C2311yescharacteristic
34C23×C397C2314yes—
35C23×C397C2318yescomplemented
36D4×C397C23116yescomplemented
37C2×C4×C397C2312yes—
38C2×C4×C397C2314yescomplemented
39C22×C397—22no—
40C22×C397—28no—
41C22×C397—48no—
42C22×C397C2412yes—
43C22×C397C2418yes—
44C22×C397C2411yescharacteristic
45C2×D397—28no—
46C2×D397—416no—
47C2×D397C2418yes—
48C4×C397C2414yes—
49Dic397C2418yes—
50C2×C397—28no—
51C2×C397—22no—
52C2×C397H(5,𝔽2)12yescomplemented
53C2×C397C2511yescharacteristic
54D397—28no—
55C397C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
56C2×H(5,𝔽2)—3971nomaximal, Sylow
57H(5,𝔽2)—39716no—
58C2×#41—3974no—
59C2×#41—3972no—
60C22×D4—3974no—
61C22×D4—3974no—
62C22×D4D39711yescharacteristic, complemented
63C24—3974no—
64C24C2×D39712yescomplemented
65#41—39732no—
66#41—39716no—
67C2×Q8—3972no—
68C2×D4—3978no—
69C2×D4—3974no—
70C2×D4—39732no—
71C2×D4—39732no—
72C2×D4—3972no—
73C2×D4C2×D39714yescomplemented
74C2×D4C2×D39718yescomplemented
75C22×C4—3974no—
76C22×C4—3974no—
77C22×C4C2×D39711yescharacteristic, complemented
78C23—44no—
79C23—15888no—
80C23—3974no—
81C23—39716no—
82C23C22×D39714yes—
83C23C22×D39711yescharacteristic
84C23C22×D39718yescomplemented
85Q8—3978no—
86D4—39732no—
87D4—39716no—
88D4—3978no—
89D4C22×D397116yescomplemented
90C2×C4—39716no—
91C2×C4—3974no—
92C2×C4—39716no—
93C2×C4C22×D39712yes—
94C2×C4C22×D39714yescomplemented
95C22—22no—
96C22—28no—
97C22—48no—
98C22—7948no—
99C22—158816no—
100C22—3978no—
101C22C23×D39718yes—
102C22C23×D39712yes—
103C22C23×D39711yescharacteristic
104C4—3978no—
105C4C23×D39714yes—
106C2—22no—
107C2—28no—
108C2—7948no—
109C2#2716314112yesdirect product factor
110C2C24×D39711yescharacteristic
subgroupG/H
centerC22C23×D397
commutator, cosocleC2×C397C25
FrattiniC2C24×D397
FittingC22×D4×C397C2
socleC22×C397C24
hypercenterC22×D4D397
hypocenterC397C2×H(5,𝔽2)
LCSG ▷ C2×C397 ▷ C397
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #27163141C2direct (N × H)
2. C2×#20263731C2semidirect
3. C2×#20263730C2semidirect
4. C22×C22⋊D397C2semidirect
5. C2×D4×D397C2semidirect
6. C22×D4×C397C2semidirect
7. C23×D397C22semidirect
8. #20263731C22semidirect
9. #20263730C22semidirect
10. C2×Dic794C22semidirect
11. C2×D1588C22semidirect
12. C2×C22⋊D397C22semidirect
13. C2×C22⋊D397C22semidirect
14. C2×C22⋊D397C22semidirect
15. D4×D397C22semidirect
16. C22×Dic397C22semidirect
17. C2×C4×D397C22semidirect
18. C24×C397C22semidirect
19. C2×D4×C397C22semidirect
20. C2×D4×C397C22semidirect
21. C22×C4×C397C22semidirect
24. C22×D397C23semidirect
25. Dic794C23semidirect
26. D1588C23semidirect
27. C22⋊D397C23semidirect
28. C22⋊D397C23semidirect
30. C2×Dic397C23semidirect
31. C4×D397C23semidirect
35. C23×C397C23semidirect
36. D4×C397C23semidirect
38. C2×C4×C397C23semidirect
52. C2×C397H(5,𝔽2)semidirect
55. C397C2×H(5,𝔽2)semidirect
62. C22×D4D397semidirect
64. C24C2×D397semidirect
73. C2×D4C2×D397semidirect
74. C2×D4C2×D397semidirect
77. C22×C4C2×D397semidirect
84. C23C22×D397semidirect
89. D4C22×D397semidirect
94. C2×C4C22×D397semidirect
109. C2#27163141direct (N × H)

Maximal subgroups

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

subgroupindex
#271631412
C2×#202637312
C2×#202637302
C22×C22⋊D3972
C2×D4×D3972
C22×D4×C3972
C2×H(5,𝔽2)397