Finite Group Databasebeta

← C2×#585×C137 #31778604

Subgroups — C2×#585×C137 #31778604

Order 17536.

5956 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C137C2132yesmaximal, direct product factor
2C2×2−1+4×C137C216yesmaximal, complemented
3C2×H(5,𝔽2)×C137C2110yesmaximal, complemented
4C22×#41×C137C2115yesmaximal, complemented
52−1+4×C137C22196yescomplemented
6H(5,𝔽2)×C137C221160yescomplemented
7C2×#41×C137C22120yescomplemented
8C2×#41×C137C22160yescomplemented
9C2×#41×C137C221240yescomplemented
10C22×Q8×C137C22115yescomplemented
11C22×D4×C137C22145yescomplemented
12C23×C4×C137C22115yescomplemented
13C24×C137C23115yes—
14#41×C137C231160yescomplemented
15#41×C137C231480yescomplemented
16C2×Q8×C137C23120yes—
17C2×Q8×C137C231120yescomplemented
18C2×D4×C137C23160yes—
19C2×D4×C137C231360yescomplemented
20C22×C4×C137C23145yes—
21C22×C4×C137C23115yes—
22C22×C4×C137C231120yescomplemented
23C23×C137—430no—
24C23×C137C24115yes—
25C23×C137C24160yes—
26Q8×C137C24180yes—
27D4×C137C241240yes—
28C2×C4×C137C24115yes—
29C2×C4×C137C24160yes—
30C2×C4×C137C241180yes—
31C2×C4×C137C2411yescharacteristic
32C22×C137—230no—
33C22×C137—460no—
34C22×C137C25130yes—
35C22×C137C2511yescharacteristic
36C4×C137C2512yes—
37C4×C137C25130yes—
38C2×C137—230no—
39C2×C137#58512yesdirect product factor
40C2×C137C2611yescharacteristic
41C137C2×#58511yescharacteristic, direct product factor, Sylow
42C2×#585C13711yesmaximal, characteristic, direct product factor, Sylow
43#585C2×C137132yesdirect product factor
44C2×2−1+4C2×C13716yescomplemented
45C2×H(5,𝔽2)C2×C137110yescomplemented
46C22×#41C2×C137115yescomplemented
472−1+4C22×C137196yescomplemented
48H(5,𝔽2)C22×C1371160yescomplemented
49C2×#41C22×C137120yescomplemented
50C2×#41C22×C137160yescomplemented
51C2×#41C22×C1371240yescomplemented
52C22×Q8C22×C137115yescomplemented
53C22×D4C22×C137145yescomplemented
54C23×C4C22×C137115yescomplemented
55C24C23×C137115yes—
56#41C23×C1371160yescomplemented
57#41C23×C1371480yescomplemented
58C2×Q8C23×C137120yes—
59C2×Q8C23×C1371120yescomplemented
60C2×D4C23×C137160yes—
61C2×D4C23×C1371360yescomplemented
62C22×C4C23×C137115yes—
63C22×C4C23×C137145yes—
64C22×C4C23×C1371120yescomplemented
65C23—430no—
66C23C24×C137115yes—
67C23C24×C137160yes—
68Q8C24×C137180yes—
69D4C24×C1371240yes—
70C2×C4C24×C13711yescharacteristic
71C2×C4C24×C137115yes—
72C2×C4C24×C137160yes—
73C2×C4C24×C1371180yes—
74C22—230no—
75C22—460no—
76C22C25×C137130yes—
77C22C25×C13711yescharacteristic
78C4C25×C13712yes—
79C4C25×C137130yes—
80C2—230no—
81C2#585×C13712yesdirect product factor
82C2C26×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137C24
commutator, Frattini, cosocleC2C26×C137
socleC22×C137C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C137C2direct (N × H)
2. C2×2−1+4×C137C2semidirect
3. C2×H(5,𝔽2)×C137C2semidirect
4. C22×#41×C137C2semidirect
5. 2−1+4×C137C22semidirect
6. H(5,𝔽2)×C137C22semidirect
7. C2×#41×C137C22semidirect
8. C2×#41×C137C22semidirect
9. C2×#41×C137C22semidirect
10. C22×Q8×C137C22semidirect
11. C22×D4×C137C22semidirect
12. C23×C4×C137C22semidirect
14. #41×C137C23semidirect
15. #41×C137C23semidirect
17. C2×Q8×C137C23semidirect
19. C2×D4×C137C23semidirect
22. C22×C4×C137C23semidirect
39. C2×C137#585direct (N × H)
41. C137C2×#585direct (N × H)
42. C2×#585C137direct (N × H)
43. #585C2×C137direct (N × H)
44. C2×2−1+4C2×C137semidirect
45. C2×H(5,𝔽2)C2×C137semidirect
46. C22×#41C2×C137semidirect
47. 2−1+4C22×C137semidirect
48. H(5,𝔽2)C22×C137semidirect
49. C2×#41C22×C137semidirect
50. C2×#41C22×C137semidirect
51. C2×#41C22×C137semidirect
52. C22×Q8C22×C137semidirect
53. C22×D4C22×C137semidirect
54. C23×C4C22×C137semidirect
56. #41C23×C137semidirect
57. #41C23×C137semidirect
59. C2×Q8C23×C137semidirect
61. C2×D4C23×C137semidirect
64. C22×C4C23×C137semidirect
81. C2#585×C137direct (N × H)

Maximal subgroups

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

subgroupindex
#585×C1372
C2×2−1+4×C1372
C2×H(5,𝔽2)×C1372
C22×#41×C1372
C2×#585137