Finite Group Databasebeta

← C2×C79×H(7,𝔽2) #34174790

Subgroups — C2×C79×H(7,𝔽2) #34174790

Order 20224.

60570 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C79×H(7,𝔽2)C2164yesmaximal, direct product factor
2C2×#585×C79C2128yesmaximal, complemented
3C22×H(5,𝔽2)×C79C2135yesmaximal, complemented
4#585×C79C221896yescomplemented
5C2×2−1+4×C79C22156yescomplemented
6C2×H(5,𝔽2)×C79C221280yescomplemented
7C2×H(5,𝔽2)×C79C2211120yescomplemented
8C22×#41×C79C221210yescomplemented
9C23×D4×C79C221105yescomplemented
10C25×C79C23130yescomplemented
112−1+4×C79C231896yescomplemented
12H(5,𝔽2)×C79C2314480yescomplemented
13C2×#41×C79C231560yescomplemented
14C2×#41×C79C2313360yescomplemented
15C22×Q8×C79C23170yescomplemented
16C22×D4×C79C231630yescomplemented
17C22×D4×C79C2311680yescomplemented
18C23×C4×C79C231105yescomplemented
19C24×C79—860no—
20C24×C79C241105yes—
21C24×C79C241240yescomplemented
22#41×C79C2414480yescomplemented
23C2×Q8×C79C24156yes—
24C2×Q8×C79C241560yescomplemented
25C2×D4×C79C241280yes—
26C2×D4×C79C2415040yescomplemented
27C22×C4×C79C241210yes—
28C22×C4×C79C241840yescomplemented
29C23×C79—4210no—
30C23×C79—8240no—
31C23×C79C25135yes—
32C23×C79C251420yes—
33Q8×C79C251224yes—
34D4×C79C2511120yes—
35C2×C4×C79C251840yes—
36C2×C4×C79C25128yes—
37C22×C79—270no—
38C22×C79—4420no—
39C22×C79C26170yes—
40C22×C79C2611yescharacteristic
41C4×C79C26156yes—
42C2×H(7,𝔽2)C7911yesmaximal, characteristic, direct product factor, Sylow
43C2×C79—270no—
44C2×C79H(7,𝔽2)12yesdirect product factor
45C2×C79C2711yescharacteristic
46H(7,𝔽2)C2×C79164yesdirect product factor
47C2×#585C2×C79128yescomplemented
48C22×H(5,𝔽2)C2×C79135yescomplemented
49C79C2×H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
50#585C22×C791896yescomplemented
51C2×2−1+4C22×C79156yescomplemented
52C2×H(5,𝔽2)C22×C791280yescomplemented
53C2×H(5,𝔽2)C22×C7911120yescomplemented
54C22×#41C22×C791210yescomplemented
55C23×D4C22×C791105yescomplemented
56C25C23×C79130yescomplemented
572−1+4C23×C791896yescomplemented
58H(5,𝔽2)C23×C7914480yescomplemented
59C2×#41C23×C791560yescomplemented
60C2×#41C23×C7913360yescomplemented
61C22×Q8C23×C79170yescomplemented
62C22×D4C23×C791630yescomplemented
63C22×D4C23×C7911680yescomplemented
64C23×C4C23×C791105yescomplemented
65C24—860no—
66C24C24×C791105yes—
67C24C24×C791240yescomplemented
68#41C24×C7914480yescomplemented
69C2×Q8C24×C79156yes—
70C2×Q8C24×C791560yescomplemented
71C2×D4C24×C791280yes—
72C2×D4C24×C7915040yescomplemented
73C22×C4C24×C791210yes—
74C22×C4C24×C791840yescomplemented
75C23—4210no—
76C23—8240no—
77C23C25×C79135yes—
78C23C25×C791420yes—
79Q8C25×C791224yes—
80D4C25×C7911120yes—
81C2×C4C25×C79128yes—
82C2×C4C25×C791840yes—
83C22—270no—
84C22—4420no—
85C22C26×C79170yes—
86C22C26×C7911yescharacteristic
87C4C26×C79156yes—
88C2—270no—
89C2C79×H(7,𝔽2)12yesdirect product factor
90C2C27×C7911yescharacteristic
subgroupG/H
center, socleC22×C79C26
commutator, Frattini, cosocleC2C27×C79
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C79 ◁ G

Complements

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

NH ≅ G/Ntype
1. C79×H(7,𝔽2)C2direct (N × H)
2. C2×#585×C79C2semidirect
3. C22×H(5,𝔽2)×C79C2semidirect
4. #585×C79C22semidirect
5. C2×2−1+4×C79C22semidirect
6. C2×H(5,𝔽2)×C79C22semidirect
7. C2×H(5,𝔽2)×C79C22semidirect
8. C22×#41×C79C22semidirect
9. C23×D4×C79C22semidirect
10. C25×C79C23semidirect
11. 2−1+4×C79C23semidirect
12. H(5,𝔽2)×C79C23semidirect
13. C2×#41×C79C23semidirect
14. C2×#41×C79C23semidirect
15. C22×Q8×C79C23semidirect
16. C22×D4×C79C23semidirect
17. C22×D4×C79C23semidirect
18. C23×C4×C79C23semidirect
21. C24×C79C24semidirect
22. #41×C79C24semidirect
24. C2×Q8×C79C24semidirect
26. C2×D4×C79C24semidirect
28. C22×C4×C79C24semidirect
42. C2×H(7,𝔽2)C79direct (N × H)
44. C2×C79H(7,𝔽2)direct (N × H)
46. H(7,𝔽2)C2×C79direct (N × H)
47. C2×#585C2×C79semidirect
48. C22×H(5,𝔽2)C2×C79semidirect
49. C79C2×H(7,𝔽2)direct (N × H)
50. #585C22×C79semidirect
51. C2×2−1+4C22×C79semidirect
52. C2×H(5,𝔽2)C22×C79semidirect
53. C2×H(5,𝔽2)C22×C79semidirect
54. C22×#41C22×C79semidirect
55. C23×D4C22×C79semidirect
56. C25C23×C79semidirect
57. 2−1+4C23×C79semidirect
58. H(5,𝔽2)C23×C79semidirect
59. C2×#41C23×C79semidirect
60. C2×#41C23×C79semidirect
61. C22×Q8C23×C79semidirect
62. C22×D4C23×C79semidirect
63. C22×D4C23×C79semidirect
64. C23×C4C23×C79semidirect
67. C24C24×C79semidirect
68. #41C24×C79semidirect
70. C2×Q8C24×C79semidirect
72. C2×D4C24×C79semidirect
74. C22×C4C24×C79semidirect
89. C2C79×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
C79×H(7,𝔽2)2
C2×#585×C792
C22×H(5,𝔽2)×C792
C2×H(7,𝔽2)79