Finite Group Databasebeta

← C2×C53×H(7,𝔽2) #27270613

Subgroups — C2×C53×H(7,𝔽2) #27270613

Order 13568.

60570 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×H(7,𝔽2)C2164yesmaximal, direct product factor
2C2×C53×#585C2128yesmaximal, complemented
3C22×H(5,𝔽2)×C53C2135yesmaximal, complemented
4C53×#585C221896yescomplemented
5C2×2−1+4×C53C22156yescomplemented
6C2×H(5,𝔽2)×C53C221280yescomplemented
7C2×H(5,𝔽2)×C53C2211120yescomplemented
8C22×#41×C53C221210yescomplemented
9C23×D4×C53C221105yescomplemented
10C25×C53C23130yescomplemented
112−1+4×C53C231896yescomplemented
12H(5,𝔽2)×C53C2314480yescomplemented
13C2×#41×C53C231560yescomplemented
14C2×#41×C53C2313360yescomplemented
15C22×Q8×C53C23170yescomplemented
16C22×D4×C53C231630yescomplemented
17C22×D4×C53C2311680yescomplemented
18C23×C4×C53C231105yescomplemented
19C24×C53—860no—
20C24×C53C241105yes—
21C24×C53C241240yescomplemented
22#41×C53C2414480yescomplemented
23C2×Q8×C53C24156yes—
24C2×Q8×C53C241560yescomplemented
25C2×D4×C53C241280yes—
26C2×D4×C53C2415040yescomplemented
27C22×C4×C53C241210yes—
28C22×C4×C53C241840yescomplemented
29C23×C53—4210no—
30C23×C53—8240no—
31C23×C53C25135yes—
32C23×C53C251420yes—
33Q8×C53C251224yes—
34D4×C53C2511120yes—
35C2×C4×C53C251840yes—
36C2×C4×C53C25128yes—
37C2×H(7,𝔽2)C5311yesmaximal, characteristic, direct product factor, Sylow
38C22×C53—270no—
39C22×C53—4420no—
40C22×C53C26170yes—
41C22×C53C2611yescharacteristic
42C4×C53C26156yes—
43H(7,𝔽2)C2×C53164yesdirect product factor
44C2×#585C2×C53128yescomplemented
45C22×H(5,𝔽2)C2×C53135yescomplemented
46C2×C53—270no—
47C2×C53H(7,𝔽2)12yesdirect product factor
48C2×C53C2711yescharacteristic
49#585C22×C531896yescomplemented
50C2×2−1+4C22×C53156yescomplemented
51C2×H(5,𝔽2)C22×C531280yescomplemented
52C2×H(5,𝔽2)C22×C5311120yescomplemented
53C22×#41C22×C531210yescomplemented
54C23×D4C22×C531105yescomplemented
55C53C2×H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
56C25C23×C53130yescomplemented
572−1+4C23×C531896yescomplemented
58H(5,𝔽2)C23×C5314480yescomplemented
59C2×#41C23×C531560yescomplemented
60C2×#41C23×C5313360yescomplemented
61C22×Q8C23×C53170yescomplemented
62C22×D4C23×C531630yescomplemented
63C22×D4C23×C5311680yescomplemented
64C23×C4C23×C531105yescomplemented
65C24—860no—
66C24C24×C531105yes—
67C24C24×C531240yescomplemented
68#41C24×C5314480yescomplemented
69C2×Q8C24×C53156yes—
70C2×Q8C24×C531560yescomplemented
71C2×D4C24×C531280yes—
72C2×D4C24×C5315040yescomplemented
73C22×C4C24×C531210yes—
74C22×C4C24×C531840yescomplemented
75C23—4210no—
76C23—8240no—
77C23C25×C53135yes—
78C23C25×C531420yes—
79Q8C25×C531224yes—
80D4C25×C5311120yes—
81C2×C4C25×C53128yes—
82C2×C4C25×C531840yes—
83C22—270no—
84C22—4420no—
85C22C26×C53170yes—
86C22C26×C5311yescharacteristic
87C4C26×C53156yes—
88C2—270no—
89C2C53×H(7,𝔽2)12yesdirect product factor
90C2C27×C5311yescharacteristic
subgroupG/H
center, socleC22×C53C26
commutator, Frattini, cosocleC2C27×C53
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×H(7,𝔽2)C2direct (N × H)
2. C2×C53×#585C2semidirect
3. C22×H(5,𝔽2)×C53C2semidirect
4. C53×#585C22semidirect
5. C2×2−1+4×C53C22semidirect
6. C2×H(5,𝔽2)×C53C22semidirect
7. C2×H(5,𝔽2)×C53C22semidirect
8. C22×#41×C53C22semidirect
9. C23×D4×C53C22semidirect
10. C25×C53C23semidirect
11. 2−1+4×C53C23semidirect
12. H(5,𝔽2)×C53C23semidirect
13. C2×#41×C53C23semidirect
14. C2×#41×C53C23semidirect
15. C22×Q8×C53C23semidirect
16. C22×D4×C53C23semidirect
17. C22×D4×C53C23semidirect
18. C23×C4×C53C23semidirect
21. C24×C53C24semidirect
22. #41×C53C24semidirect
24. C2×Q8×C53C24semidirect
26. C2×D4×C53C24semidirect
28. C22×C4×C53C24semidirect
37. C2×H(7,𝔽2)C53direct (N × H)
43. H(7,𝔽2)C2×C53direct (N × H)
44. C2×#585C2×C53semidirect
45. C22×H(5,𝔽2)C2×C53semidirect
47. C2×C53H(7,𝔽2)direct (N × H)
49. #585C22×C53semidirect
50. C2×2−1+4C22×C53semidirect
51. C2×H(5,𝔽2)C22×C53semidirect
52. C2×H(5,𝔽2)C22×C53semidirect
53. C22×#41C22×C53semidirect
54. C23×D4C22×C53semidirect
55. C53C2×H(7,𝔽2)direct (N × H)
56. C25C23×C53semidirect
57. 2−1+4C23×C53semidirect
58. H(5,𝔽2)C23×C53semidirect
59. C2×#41C23×C53semidirect
60. C2×#41C23×C53semidirect
61. C22×Q8C23×C53semidirect
62. C22×D4C23×C53semidirect
63. C22×D4C23×C53semidirect
64. C23×C4C23×C53semidirect
67. C24C24×C53semidirect
68. #41C24×C53semidirect
70. C2×Q8C24×C53semidirect
72. C2×D4C24×C53semidirect
74. C22×C4C24×C53semidirect
89. C2C53×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
C53×H(7,𝔽2)2
C2×C53×#5852
C22×H(5,𝔽2)×C532
C2×H(7,𝔽2)53