Finite Group Databasebeta

← C2×C89×H(7,𝔽2) #36440949

Subgroups — C2×C89×H(7,𝔽2) #36440949

Order 22784.

60570 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×H(7,𝔽2)C2164yesmaximal, direct product factor
2C2×#585×C89C2128yesmaximal, complemented
3C22×H(5,𝔽2)×C89C2135yesmaximal, complemented
4#585×C89C221896yescomplemented
5C2×2−1+4×C89C22156yescomplemented
6C2×H(5,𝔽2)×C89C221280yescomplemented
7C2×H(5,𝔽2)×C89C2211120yescomplemented
8C22×#41×C89C221210yescomplemented
9C23×D4×C89C221105yescomplemented
10C25×C89C23130yescomplemented
112−1+4×C89C231896yescomplemented
12H(5,𝔽2)×C89C2314480yescomplemented
13C2×#41×C89C231560yescomplemented
14C2×#41×C89C2313360yescomplemented
15C22×Q8×C89C23170yescomplemented
16C22×D4×C89C231630yescomplemented
17C22×D4×C89C2311680yescomplemented
18C23×C4×C89C231105yescomplemented
19C24×C89—860no—
20C24×C89C241105yes—
21C24×C89C241240yescomplemented
22#41×C89C2414480yescomplemented
23C2×Q8×C89C24156yes—
24C2×Q8×C89C241560yescomplemented
25C2×D4×C89C241280yes—
26C2×D4×C89C2415040yescomplemented
27C22×C4×C89C241210yes—
28C22×C4×C89C241840yescomplemented
29C23×C89—4210no—
30C23×C89—8240no—
31C23×C89C25135yes—
32C23×C89C251420yes—
33Q8×C89C251224yes—
34D4×C89C2511120yes—
35C2×C4×C89C251840yes—
36C2×C4×C89C25128yes—
37C22×C89—270no—
38C22×C89—4420no—
39C22×C89C26170yes—
40C22×C89C2611yescharacteristic
41C4×C89C26156yes—
42C2×H(7,𝔽2)C8911yesmaximal, characteristic, direct product factor, Sylow
43C2×C89—270no—
44C2×C89H(7,𝔽2)12yesdirect product factor
45C2×C89C2711yescharacteristic
46H(7,𝔽2)C2×C89164yesdirect product factor
47C2×#585C2×C89128yescomplemented
48C22×H(5,𝔽2)C2×C89135yescomplemented
49C89C2×H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
50#585C22×C891896yescomplemented
51C2×2−1+4C22×C89156yescomplemented
52C2×H(5,𝔽2)C22×C891280yescomplemented
53C2×H(5,𝔽2)C22×C8911120yescomplemented
54C22×#41C22×C891210yescomplemented
55C23×D4C22×C891105yescomplemented
56C25C23×C89130yescomplemented
572−1+4C23×C891896yescomplemented
58H(5,𝔽2)C23×C8914480yescomplemented
59C2×#41C23×C891560yescomplemented
60C2×#41C23×C8913360yescomplemented
61C22×Q8C23×C89170yescomplemented
62C22×D4C23×C891630yescomplemented
63C22×D4C23×C8911680yescomplemented
64C23×C4C23×C891105yescomplemented
65C24—860no—
66C24C24×C891105yes—
67C24C24×C891240yescomplemented
68#41C24×C8914480yescomplemented
69C2×Q8C24×C89156yes—
70C2×Q8C24×C891560yescomplemented
71C2×D4C24×C891280yes—
72C2×D4C24×C8915040yescomplemented
73C22×C4C24×C891210yes—
74C22×C4C24×C891840yescomplemented
75C23—4210no—
76C23—8240no—
77C23C25×C89135yes—
78C23C25×C891420yes—
79Q8C25×C891224yes—
80D4C25×C8911120yes—
81C2×C4C25×C89128yes—
82C2×C4C25×C891840yes—
83C22—270no—
84C22—4420no—
85C22C26×C89170yes—
86C22C26×C8911yescharacteristic
87C4C26×C89156yes—
88C2—270no—
89C2C89×H(7,𝔽2)12yesdirect product factor
90C2C27×C8911yescharacteristic
subgroupG/H
center, socleC22×C89C26
commutator, Frattini, cosocleC2C27×C89
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×H(7,𝔽2)C2direct (N × H)
2. C2×#585×C89C2semidirect
3. C22×H(5,𝔽2)×C89C2semidirect
4. #585×C89C22semidirect
5. C2×2−1+4×C89C22semidirect
6. C2×H(5,𝔽2)×C89C22semidirect
7. C2×H(5,𝔽2)×C89C22semidirect
8. C22×#41×C89C22semidirect
9. C23×D4×C89C22semidirect
10. C25×C89C23semidirect
11. 2−1+4×C89C23semidirect
12. H(5,𝔽2)×C89C23semidirect
13. C2×#41×C89C23semidirect
14. C2×#41×C89C23semidirect
15. C22×Q8×C89C23semidirect
16. C22×D4×C89C23semidirect
17. C22×D4×C89C23semidirect
18. C23×C4×C89C23semidirect
21. C24×C89C24semidirect
22. #41×C89C24semidirect
24. C2×Q8×C89C24semidirect
26. C2×D4×C89C24semidirect
28. C22×C4×C89C24semidirect
42. C2×H(7,𝔽2)C89direct (N × H)
44. C2×C89H(7,𝔽2)direct (N × H)
46. H(7,𝔽2)C2×C89direct (N × H)
47. C2×#585C2×C89semidirect
48. C22×H(5,𝔽2)C2×C89semidirect
49. C89C2×H(7,𝔽2)direct (N × H)
50. #585C22×C89semidirect
51. C2×2−1+4C22×C89semidirect
52. C2×H(5,𝔽2)C22×C89semidirect
53. C2×H(5,𝔽2)C22×C89semidirect
54. C22×#41C22×C89semidirect
55. C23×D4C22×C89semidirect
56. C25C23×C89semidirect
57. 2−1+4C23×C89semidirect
58. H(5,𝔽2)C23×C89semidirect
59. C2×#41C23×C89semidirect
60. C2×#41C23×C89semidirect
61. C22×Q8C23×C89semidirect
62. C22×D4C23×C89semidirect
63. C22×D4C23×C89semidirect
64. C23×C4C23×C89semidirect
67. C24C24×C89semidirect
68. #41C24×C89semidirect
70. C2×Q8C24×C89semidirect
72. C2×D4C24×C89semidirect
74. C22×C4C24×C89semidirect
89. C2C89×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
C89×H(7,𝔽2)2
C2×#585×C892
C22×H(5,𝔽2)×C892
C2×H(7,𝔽2)89