Finite Group Databasebeta

← C2×C47×H(7,𝔽2) #26102799

Subgroups — C2×C47×H(7,𝔽2) #26102799

Order 12032.

60570 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×H(7,𝔽2)C2164yesmaximal, direct product factor
2C2×C47×#585C2128yesmaximal, complemented
3C22×H(5,𝔽2)×C47C2135yesmaximal, complemented
4C47×#585C221896yescomplemented
5C2×2−1+4×C47C22156yescomplemented
6C2×H(5,𝔽2)×C47C221280yescomplemented
7C2×H(5,𝔽2)×C47C2211120yescomplemented
8C22×#41×C47C221210yescomplemented
9C23×D4×C47C221105yescomplemented
10C25×C47C23130yescomplemented
112−1+4×C47C231896yescomplemented
12H(5,𝔽2)×C47C2314480yescomplemented
13C2×#41×C47C231560yescomplemented
14C2×#41×C47C2313360yescomplemented
15C22×Q8×C47C23170yescomplemented
16C22×D4×C47C231630yescomplemented
17C22×D4×C47C2311680yescomplemented
18C23×C4×C47C231105yescomplemented
19C24×C47—860no—
20C24×C47C241105yes—
21C24×C47C241240yescomplemented
22#41×C47C2414480yescomplemented
23C2×Q8×C47C24156yes—
24C2×Q8×C47C241560yescomplemented
25C2×D4×C47C241280yes—
26C2×D4×C47C2415040yescomplemented
27C22×C4×C47C241210yes—
28C22×C4×C47C241840yescomplemented
29C23×C47—4210no—
30C23×C47—8240no—
31C23×C47C25135yes—
32C23×C47C251420yes—
33Q8×C47C251224yes—
34D4×C47C2511120yes—
35C2×C4×C47C251840yes—
36C2×C4×C47C25128yes—
37C2×H(7,𝔽2)C4711yesmaximal, characteristic, direct product factor, Sylow
38C22×C47—270no—
39C22×C47—4420no—
40C22×C47C26170yes—
41C22×C47C2611yescharacteristic
42C4×C47C26156yes—
43H(7,𝔽2)C2×C47164yesdirect product factor
44C2×#585C2×C47128yescomplemented
45C22×H(5,𝔽2)C2×C47135yescomplemented
46C2×C47—270no—
47C2×C47H(7,𝔽2)12yesdirect product factor
48C2×C47C2711yescharacteristic
49#585C22×C471896yescomplemented
50C2×2−1+4C22×C47156yescomplemented
51C2×H(5,𝔽2)C22×C471280yescomplemented
52C2×H(5,𝔽2)C22×C4711120yescomplemented
53C22×#41C22×C471210yescomplemented
54C23×D4C22×C471105yescomplemented
55C47C2×H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
56C25C23×C47130yescomplemented
572−1+4C23×C471896yescomplemented
58H(5,𝔽2)C23×C4714480yescomplemented
59C2×#41C23×C471560yescomplemented
60C2×#41C23×C4713360yescomplemented
61C22×Q8C23×C47170yescomplemented
62C22×D4C23×C471630yescomplemented
63C22×D4C23×C4711680yescomplemented
64C23×C4C23×C471105yescomplemented
65C24—860no—
66C24C24×C471105yes—
67C24C24×C471240yescomplemented
68#41C24×C4714480yescomplemented
69C2×Q8C24×C47156yes—
70C2×Q8C24×C471560yescomplemented
71C2×D4C24×C471280yes—
72C2×D4C24×C4715040yescomplemented
73C22×C4C24×C471210yes—
74C22×C4C24×C471840yescomplemented
75C23—4210no—
76C23—8240no—
77C23C25×C47135yes—
78C23C25×C471420yes—
79Q8C25×C471224yes—
80D4C25×C4711120yes—
81C2×C4C25×C47128yes—
82C2×C4C25×C471840yes—
83C22—270no—
84C22—4420no—
85C22C26×C47170yes—
86C22C26×C4711yescharacteristic
87C4C26×C47156yes—
88C2—270no—
89C2C47×H(7,𝔽2)12yesdirect product factor
90C2C27×C4711yescharacteristic
subgroupG/H
center, socleC22×C47C26
commutator, Frattini, cosocleC2C27×C47
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×H(7,𝔽2)C2direct (N × H)
2. C2×C47×#585C2semidirect
3. C22×H(5,𝔽2)×C47C2semidirect
4. C47×#585C22semidirect
5. C2×2−1+4×C47C22semidirect
6. C2×H(5,𝔽2)×C47C22semidirect
7. C2×H(5,𝔽2)×C47C22semidirect
8. C22×#41×C47C22semidirect
9. C23×D4×C47C22semidirect
10. C25×C47C23semidirect
11. 2−1+4×C47C23semidirect
12. H(5,𝔽2)×C47C23semidirect
13. C2×#41×C47C23semidirect
14. C2×#41×C47C23semidirect
15. C22×Q8×C47C23semidirect
16. C22×D4×C47C23semidirect
17. C22×D4×C47C23semidirect
18. C23×C4×C47C23semidirect
21. C24×C47C24semidirect
22. #41×C47C24semidirect
24. C2×Q8×C47C24semidirect
26. C2×D4×C47C24semidirect
28. C22×C4×C47C24semidirect
37. C2×H(7,𝔽2)C47direct (N × H)
43. H(7,𝔽2)C2×C47direct (N × H)
44. C2×#585C2×C47semidirect
45. C22×H(5,𝔽2)C2×C47semidirect
47. C2×C47H(7,𝔽2)direct (N × H)
49. #585C22×C47semidirect
50. C2×2−1+4C22×C47semidirect
51. C2×H(5,𝔽2)C22×C47semidirect
52. C2×H(5,𝔽2)C22×C47semidirect
53. C22×#41C22×C47semidirect
54. C23×D4C22×C47semidirect
55. C47C2×H(7,𝔽2)direct (N × H)
56. C25C23×C47semidirect
57. 2−1+4C23×C47semidirect
58. H(5,𝔽2)C23×C47semidirect
59. C2×#41C23×C47semidirect
60. C2×#41C23×C47semidirect
61. C22×Q8C23×C47semidirect
62. C22×D4C23×C47semidirect
63. C22×D4C23×C47semidirect
64. C23×C4C23×C47semidirect
67. C24C24×C47semidirect
68. #41C24×C47semidirect
70. C2×Q8C24×C47semidirect
72. C2×D4C24×C47semidirect
74. C22×C4C24×C47semidirect
89. C2C47×H(7,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C47×H(7,𝔽2)2
C2×C47×#5852
C22×H(5,𝔽2)×C472
C2×H(7,𝔽2)47