Finite Group Databasebeta

← C2×C61×H(7,𝔽2) #29566534

Subgroups — C2×C61×H(7,𝔽2) #29566534

Order 15616.

60570 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×H(7,𝔽2)C2164yesmaximal, direct product factor
2C2×C61×#585C2128yesmaximal, complemented
3C22×H(5,𝔽2)×C61C2135yesmaximal, complemented
4C61×#585C221896yescomplemented
5C2×2−1+4×C61C22156yescomplemented
6C2×H(5,𝔽2)×C61C221280yescomplemented
7C2×H(5,𝔽2)×C61C2211120yescomplemented
8C22×#41×C61C221210yescomplemented
9C23×D4×C61C221105yescomplemented
10C25×C61C23130yescomplemented
112−1+4×C61C231896yescomplemented
12H(5,𝔽2)×C61C2314480yescomplemented
13C2×#41×C61C231560yescomplemented
14C2×#41×C61C2313360yescomplemented
15C22×Q8×C61C23170yescomplemented
16C22×D4×C61C231630yescomplemented
17C22×D4×C61C2311680yescomplemented
18C23×C4×C61C231105yescomplemented
19C24×C61—860no—
20C24×C61C241105yes—
21C24×C61C241240yescomplemented
22#41×C61C2414480yescomplemented
23C2×Q8×C61C24156yes—
24C2×Q8×C61C241560yescomplemented
25C2×D4×C61C241280yes—
26C2×D4×C61C2415040yescomplemented
27C22×C4×C61C241210yes—
28C22×C4×C61C241840yescomplemented
29C23×C61—4210no—
30C23×C61—8240no—
31C23×C61C25135yes—
32C23×C61C251420yes—
33Q8×C61C251224yes—
34D4×C61C2511120yes—
35C2×C4×C61C251840yes—
36C2×C4×C61C25128yes—
37C2×H(7,𝔽2)C6111yesmaximal, characteristic, direct product factor, Sylow
38C22×C61—270no—
39C22×C61—4420no—
40C22×C61C26170yes—
41C22×C61C2611yescharacteristic
42C4×C61C26156yes—
43H(7,𝔽2)C2×C61164yesdirect product factor
44C2×#585C2×C61128yescomplemented
45C22×H(5,𝔽2)C2×C61135yescomplemented
46C2×C61—270no—
47C2×C61H(7,𝔽2)12yesdirect product factor
48C2×C61C2711yescharacteristic
49#585C22×C611896yescomplemented
50C2×2−1+4C22×C61156yescomplemented
51C2×H(5,𝔽2)C22×C611280yescomplemented
52C2×H(5,𝔽2)C22×C6111120yescomplemented
53C22×#41C22×C611210yescomplemented
54C23×D4C22×C611105yescomplemented
55C61C2×H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
56C25C23×C61130yescomplemented
572−1+4C23×C611896yescomplemented
58H(5,𝔽2)C23×C6114480yescomplemented
59C2×#41C23×C611560yescomplemented
60C2×#41C23×C6113360yescomplemented
61C22×Q8C23×C61170yescomplemented
62C22×D4C23×C611630yescomplemented
63C22×D4C23×C6111680yescomplemented
64C23×C4C23×C611105yescomplemented
65C24—860no—
66C24C24×C611105yes—
67C24C24×C611240yescomplemented
68#41C24×C6114480yescomplemented
69C2×Q8C24×C61156yes—
70C2×Q8C24×C611560yescomplemented
71C2×D4C24×C611280yes—
72C2×D4C24×C6115040yescomplemented
73C22×C4C24×C611210yes—
74C22×C4C24×C611840yescomplemented
75C23—4210no—
76C23—8240no—
77C23C25×C61135yes—
78C23C25×C611420yes—
79Q8C25×C611224yes—
80D4C25×C6111120yes—
81C2×C4C25×C61128yes—
82C2×C4C25×C611840yes—
83C22—270no—
84C22—4420no—
85C22C26×C61170yes—
86C22C26×C6111yescharacteristic
87C4C26×C61156yes—
88C2—270no—
89C2C61×H(7,𝔽2)12yesdirect product factor
90C2C27×C6111yescharacteristic
subgroupG/H
center, socleC22×C61C26
commutator, Frattini, cosocleC2C27×C61
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×H(7,𝔽2)C2direct (N × H)
2. C2×C61×#585C2semidirect
3. C22×H(5,𝔽2)×C61C2semidirect
4. C61×#585C22semidirect
5. C2×2−1+4×C61C22semidirect
6. C2×H(5,𝔽2)×C61C22semidirect
7. C2×H(5,𝔽2)×C61C22semidirect
8. C22×#41×C61C22semidirect
9. C23×D4×C61C22semidirect
10. C25×C61C23semidirect
11. 2−1+4×C61C23semidirect
12. H(5,𝔽2)×C61C23semidirect
13. C2×#41×C61C23semidirect
14. C2×#41×C61C23semidirect
15. C22×Q8×C61C23semidirect
16. C22×D4×C61C23semidirect
17. C22×D4×C61C23semidirect
18. C23×C4×C61C23semidirect
21. C24×C61C24semidirect
22. #41×C61C24semidirect
24. C2×Q8×C61C24semidirect
26. C2×D4×C61C24semidirect
28. C22×C4×C61C24semidirect
37. C2×H(7,𝔽2)C61direct (N × H)
43. H(7,𝔽2)C2×C61direct (N × H)
44. C2×#585C2×C61semidirect
45. C22×H(5,𝔽2)C2×C61semidirect
47. C2×C61H(7,𝔽2)direct (N × H)
49. #585C22×C61semidirect
50. C2×2−1+4C22×C61semidirect
51. C2×H(5,𝔽2)C22×C61semidirect
52. C2×H(5,𝔽2)C22×C61semidirect
53. C22×#41C22×C61semidirect
54. C23×D4C22×C61semidirect
55. C61C2×H(7,𝔽2)direct (N × H)
56. C25C23×C61semidirect
57. 2−1+4C23×C61semidirect
58. H(5,𝔽2)C23×C61semidirect
59. C2×#41C23×C61semidirect
60. C2×#41C23×C61semidirect
61. C22×Q8C23×C61semidirect
62. C22×D4C23×C61semidirect
63. C22×D4C23×C61semidirect
64. C23×C4C23×C61semidirect
67. C24C24×C61semidirect
68. #41C24×C61semidirect
70. C2×Q8C24×C61semidirect
72. C2×D4C24×C61semidirect
74. C22×C4C24×C61semidirect
89. C2C61×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
C61×H(7,𝔽2)2
C2×C61×#5852
C22×H(5,𝔽2)×C612
C2×H(7,𝔽2)61