Finite Group Databasebeta

← C2×#27211763 #41001191

Subgroups — C2×#27211763 #41001191

Order 26816.

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#27211763C2116yesmaximal, direct product factor
2C2×#20269411C212yesmaximal, complemented
3C2×#20269410C214yesmaximal, complemented
4C22×C22⋊D419C214yesmaximal, complemented
5C2×D4×D419C214yesmaximal, complemented
6C22×D4×C419C211yesmaximal, characteristic, complemented
7C23×D419C2214yescomplemented
8#20269411C22116yescomplemented
9#20269410C22132yescomplemented
10C2×Dic838C2212yescomplemented
11C2×D1676C2212yescomplemented
12C2×C22⋊D419C2218yescomplemented
13C2×C22⋊D419C2214yescomplemented
14C2×C22⋊D419C22132yescomplemented
15D4×D419C22132yescomplemented
16C22×Dic419C2214yescomplemented
17C2×C4×D419C2214yescomplemented
18C24×C419C2212yescomplemented
19C2×D4×C419C2214yescomplemented
20C2×D4×C419C2218yescomplemented
21C22×C4×C419C2211yescharacteristic, complemented
22C22×D419—48no—
23C22×D419C2314yes—
24C22×D419C23116yescomplemented
25Dic838C2318yescomplemented
26D1676C2318yescomplemented
27C22⋊D419C23132yescomplemented
28C22⋊D419C23116yescomplemented
29C2×Dic419C2314yes—
30C2×Dic419C23116yescomplemented
31C4×D419C23116yescomplemented
32C23×C419—44no—
33C23×C419C2311yescharacteristic
34C23×C419C2314yes—
35C23×C419C2318yescomplemented
36D4×C419C23116yescomplemented
37C2×C4×C419C2312yes—
38C2×C4×C419C2314yescomplemented
39C22×C419—22no—
40C22×C419—28no—
41C22×C419—48no—
42C22×C419C2412yes—
43C22×C419C2418yes—
44C22×C419C2411yescharacteristic
45C2×D419—28no—
46C2×D419—416no—
47C2×D419C2418yes—
48C4×C419C2414yes—
49Dic419C2418yes—
50C2×C419—28no—
51C2×C419—22no—
52C2×C419H(5,𝔽2)12yescomplemented
53C2×C419C2511yescharacteristic
54D419—28no—
55C419C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
56C2×H(5,𝔽2)—4191nomaximal, Sylow
57H(5,𝔽2)—41916no—
58C2×#41—4194no—
59C2×#41—4192no—
60C22×D4—4194no—
61C22×D4—4194no—
62C22×D4D41911yescharacteristic, complemented
63C24—4194no—
64C24C2×D41912yescomplemented
65#41—41932no—
66#41—41916no—
67C2×Q8—4192no—
68C2×D4—4198no—
69C2×D4—4194no—
70C2×D4—41932no—
71C2×D4—41932no—
72C2×D4—4192no—
73C2×D4C2×D41914yescomplemented
74C2×D4C2×D41918yescomplemented
75C22×C4—4194no—
76C22×C4—4194no—
77C22×C4C2×D41911yescharacteristic, complemented
78C23—44no—
79C23—16768no—
80C23—4194no—
81C23—41916no—
82C23C22×D41914yes—
83C23C22×D41911yescharacteristic
84C23C22×D41918yescomplemented
85Q8—4198no—
86D4—41932no—
87D4—41916no—
88D4—4198no—
89D4C22×D419116yescomplemented
90C2×C4—41916no—
91C2×C4—4194no—
92C2×C4—41916no—
93C2×C4C22×D41912yes—
94C2×C4C22×D41914yescomplemented
95C22—22no—
96C22—28no—
97C22—48no—
98C22—8388no—
99C22—167616no—
100C22—4198no—
101C22C23×D41918yes—
102C22C23×D41912yes—
103C22C23×D41911yescharacteristic
104C4—4198no—
105C4C23×D41914yes—
106C2—22no—
107C2—28no—
108C2—8388no—
109C2#2721176312yesdirect product factor
110C2C24×D41911yescharacteristic
subgroupG/H
centerC22C23×D419
commutator, cosocleC2×C419C25
FrattiniC2C24×D419
FittingC22×D4×C419C2
socleC22×C419C24
hypercenterC22×D4D419
hypocenterC419C2×H(5,𝔽2)
LCSG ▷ C2×C419 ▷ C419
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #27211763C2direct (N × H)
2. C2×#20269411C2semidirect
3. C2×#20269410C2semidirect
4. C22×C22⋊D419C2semidirect
5. C2×D4×D419C2semidirect
6. C22×D4×C419C2semidirect
7. C23×D419C22semidirect
8. #20269411C22semidirect
9. #20269410C22semidirect
10. C2×Dic838C22semidirect
11. C2×D1676C22semidirect
12. C2×C22⋊D419C22semidirect
13. C2×C22⋊D419C22semidirect
14. C2×C22⋊D419C22semidirect
15. D4×D419C22semidirect
16. C22×Dic419C22semidirect
17. C2×C4×D419C22semidirect
18. C24×C419C22semidirect
19. C2×D4×C419C22semidirect
20. C2×D4×C419C22semidirect
21. C22×C4×C419C22semidirect
24. C22×D419C23semidirect
25. Dic838C23semidirect
26. D1676C23semidirect
27. C22⋊D419C23semidirect
28. C22⋊D419C23semidirect
30. C2×Dic419C23semidirect
31. C4×D419C23semidirect
35. C23×C419C23semidirect
36. D4×C419C23semidirect
38. C2×C4×C419C23semidirect
52. C2×C419H(5,𝔽2)semidirect
55. C419C2×H(5,𝔽2)semidirect
62. C22×D4D419semidirect
64. C24C2×D419semidirect
73. C2×D4C2×D419semidirect
74. C2×D4C2×D419semidirect
77. C22×C4C2×D419semidirect
84. C23C22×D419semidirect
89. D4C22×D419semidirect
94. C2×C4C22×D419semidirect
109. C2#27211763direct (N × H)

Maximal subgroups

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

subgroupindex
#272117632
C2×#202694112
C2×#202694102
C22×C22⋊D4192
C2×D4×D4192
C22×D4×C4192
C2×H(5,𝔽2)419