Finite Group Databasebeta

← C2×#20297349 #28384410

Subgroups — C2×#20297349 #28384410

Order 14272.

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#20297349C2116yesmaximal, direct product factor
2C2×#16813170C212yesmaximal, complemented
3C2×#16813169C214yesmaximal, complemented
4C22×C22⋊D223C214yesmaximal, complemented
5C2×D4×D223C214yesmaximal, complemented
6C22×D4×C223C211yesmaximal, characteristic, complemented
7C23×D223C2214yescomplemented
8#16813170C22116yescomplemented
9#16813169C22132yescomplemented
10C2×Dic446C2212yescomplemented
11C2×D892C2212yescomplemented
12C2×C22⋊D223C2218yescomplemented
13C2×C22⋊D223C2214yescomplemented
14C2×C22⋊D223C22132yescomplemented
15D4×D223C22132yescomplemented
16C22×Dic223C2214yescomplemented
17C2×C4×D223C2214yescomplemented
18C24×C223C2212yescomplemented
19C2×D4×C223C2214yescomplemented
20C2×D4×C223C2218yescomplemented
21C22×C4×C223C2211yescharacteristic, complemented
22C22×D223—48no—
23C22×D223C2314yes—
24C22×D223C23116yescomplemented
25Dic446C2318yescomplemented
26D892C2318yescomplemented
27C22⋊D223C23132yescomplemented
28C22⋊D223C23116yescomplemented
29C2×Dic223C2314yes—
30C2×Dic223C23116yescomplemented
31C4×D223C23116yescomplemented
32C23×C223—44no—
33C23×C223C2311yescharacteristic
34C23×C223C2314yes—
35C23×C223C2318yescomplemented
36D4×C223C23116yescomplemented
37C2×C4×C223C2312yes—
38C2×C4×C223C2314yescomplemented
39C22×C223—22no—
40C22×C223—28no—
41C22×C223—48no—
42C22×C223C2412yes—
43C22×C223C2418yes—
44C22×C223C2411yescharacteristic
45C2×D223—28no—
46C2×D223—416no—
47C2×D223C2418yes—
48C4×C223C2414yes—
49Dic223C2418yes—
50C2×C223—28no—
51C2×C223—22no—
52C2×C223H(5,𝔽2)12yescomplemented
53C2×C223C2511yescharacteristic
54D223—28no—
55C223C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
56C2×H(5,𝔽2)—2231nomaximal, Sylow
57H(5,𝔽2)—22316no—
58C2×#41—2234no—
59C2×#41—2232no—
60C22×D4—2234no—
61C22×D4—2234no—
62C22×D4D22311yescharacteristic, complemented
63C24—2234no—
64C24C2×D22312yescomplemented
65#41—22332no—
66#41—22316no—
67C2×Q8—2232no—
68C2×D4—2238no—
69C2×D4—2234no—
70C2×D4—22332no—
71C2×D4—22332no—
72C2×D4—2232no—
73C2×D4C2×D22314yescomplemented
74C2×D4C2×D22318yescomplemented
75C22×C4—2234no—
76C22×C4—2234no—
77C22×C4C2×D22311yescharacteristic, complemented
78C23—44no—
79C23—8928no—
80C23—22316no—
81C23—2234no—
82C23C22×D22314yes—
83C23C22×D22311yescharacteristic
84C23C22×D22318yescomplemented
85Q8—2238no—
86D4—22332no—
87D4—22316no—
88D4—2238no—
89D4C22×D223116yescomplemented
90C2×C4—22316no—
91C2×C4—2234no—
92C2×C4—22316no—
93C2×C4C22×D22312yes—
94C2×C4C22×D22314yescomplemented
95C22—22no—
96C22—28no—
97C22—48no—
98C22—89216no—
99C22—4468no—
100C22—2238no—
101C22C23×D22318yes—
102C22C23×D22312yes—
103C22C23×D22311yescharacteristic
104C4—2238no—
105C4C23×D22314yes—
106C2—22no—
107C2—28no—
108C2—4468no—
109C2#2029734912yesdirect product factor
110C2C24×D22311yescharacteristic
subgroupG/H
centerC22C23×D223
commutator, cosocleC2×C223C25
FrattiniC2C24×D223
FittingC22×D4×C223C2
socleC22×C223C24
hypercenterC22×D4D223
hypocenterC223C2×H(5,𝔽2)
LCSG ▷ C2×C223 ▷ C223
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20297349C2direct (N × H)
2. C2×#16813170C2semidirect
3. C2×#16813169C2semidirect
4. C22×C22⋊D223C2semidirect
5. C2×D4×D223C2semidirect
6. C22×D4×C223C2semidirect
7. C23×D223C22semidirect
8. #16813170C22semidirect
9. #16813169C22semidirect
10. C2×Dic446C22semidirect
11. C2×D892C22semidirect
12. C2×C22⋊D223C22semidirect
13. C2×C22⋊D223C22semidirect
14. C2×C22⋊D223C22semidirect
15. D4×D223C22semidirect
16. C22×Dic223C22semidirect
17. C2×C4×D223C22semidirect
18. C24×C223C22semidirect
19. C2×D4×C223C22semidirect
20. C2×D4×C223C22semidirect
21. C22×C4×C223C22semidirect
24. C22×D223C23semidirect
25. Dic446C23semidirect
26. D892C23semidirect
27. C22⋊D223C23semidirect
28. C22⋊D223C23semidirect
30. C2×Dic223C23semidirect
31. C4×D223C23semidirect
35. C23×C223C23semidirect
36. D4×C223C23semidirect
38. C2×C4×C223C23semidirect
52. C2×C223H(5,𝔽2)semidirect
55. C223C2×H(5,𝔽2)semidirect
62. C22×D4D223semidirect
64. C24C2×D223semidirect
73. C2×D4C2×D223semidirect
74. C2×D4C2×D223semidirect
77. C22×C4C2×D223semidirect
84. C23C22×D223semidirect
89. D4C22×D223semidirect
94. C2×C4C22×D223semidirect
109. C2#20297349direct (N × H)

Maximal subgroups

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

subgroupindex
#202973492
C2×#168131702
C2×#168131692
C22×C22⋊D2232
C2×D4×D2232
C22×D4×C2232
C2×H(5,𝔽2)223