Finite Group Databasebeta

← C23×#41×C67 #22558021

Subgroups — C23×#41×C67 #22558021

Order 8576.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#41×C67C2156yesmaximal, direct product factor
2C23×Q8×C67C211yesmaximal, characteristic, complemented
3C23×D4×C67C213yesmaximal, complemented
4C24×C4×C67C213yesmaximal, complemented
5C25×C67C2213yes—
6C2×#41×C67C221448yesdirect product factor
7C22×Q8×C67C22128yescomplemented
8C22×D4×C67C22184yescomplemented
9C23×C4×C67C2213yes—
10C23×C4×C67C2211yescharacteristic
11C23×C4×C67C22184yescomplemented
12C24×C67—224no—
13C24×C67C23142yes—
14C24×C67C2311yescharacteristic
15#41×C67C231512yesdirect product factor
16C2×Q8×C67C231112yescomplemented
17C2×D4×C67C231336yescomplemented
18C22×C4×C67C23142yes—
19C22×C4×C67C23114yes—
20C22×C4×C67C231336yescomplemented
21C23×C67—2168no—
22C23×C67#4118yesdirect product factor
23C23×C67C24184yes—
24C23×C67C2417yes—
25Q8×C67C24164yescomplemented
26D4×C67C241192yescomplemented
27C2×C4×C67C24128yes—
28C2×C4×C67C24184yes—
29C2×C4×C67C241192yescomplemented
30C22×C67—2168no—
31C22×C67C2×#41128yesdirect product factor
32C22×C67C25124yes—
33C22×C67C2517yes—
34C4×C67C2518yes—
35C4×C67C25124yes—
36C2×C67—224no—
37C2×C67C22×#41114yesdirect product factor
38C2×C67C2611yescharacteristic
39C23×#41C6711yesmaximal, characteristic, direct product factor, Sylow
40C67C23×#4111yescharacteristic, direct product factor, Sylow
41C22×#41C2×C67156yesdirect product factor
42C23×Q8C2×C6711yescharacteristic, complemented
43C23×D4C2×C6713yescomplemented
44C24×C4C2×C6713yescomplemented
45C25C22×C6713yes—
46C2×#41C22×C671448yesdirect product factor
47C22×Q8C22×C67128yescomplemented
48C22×D4C22×C67184yescomplemented
49C23×C4C22×C6713yes—
50C23×C4C22×C6711yescharacteristic
51C23×C4C22×C67184yescomplemented
52C24—224no—
53C24C23×C67142yes—
54C24C23×C6711yescharacteristic
55#41C23×C671512yesdirect product factor
56C2×Q8C23×C671112yescomplemented
57C2×D4C23×C671336yescomplemented
58C22×C4C23×C67114yes—
59C22×C4C23×C67142yes—
60C22×C4C23×C671336yescomplemented
61C23—2168no—
62C23#41×C6718yesdirect product factor
63C23C24×C67184yes—
64C23C24×C6717yes—
65Q8C24×C67164yescomplemented
66D4C24×C671192yescomplemented
67C2×C4C24×C67128yes—
68C2×C4C24×C67184yes—
69C2×C4C24×C671192yescomplemented
70C22—2168no—
71C22C2×#41×C67128yesdirect product factor
72C22C25×C67124yes—
73C22C25×C6717yes—
74C4C25×C6718yes—
75C4C25×C67124yes—
76C2—224no—
77C2C22×#41×C67114yesdirect product factor
78C2C26×C6711yescharacteristic
subgroupG/H
centerC23×C4×C67C22
commutator, Frattini, cosocleC2C26×C67
socleC24×C67C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#41×C67C2direct (N × H)
2. C23×Q8×C67C2semidirect
3. C23×D4×C67C2semidirect
4. C24×C4×C67C2semidirect
6. C2×#41×C67C22direct (N × H)
7. C22×Q8×C67C22semidirect
8. C22×D4×C67C22semidirect
11. C23×C4×C67C22semidirect
15. #41×C67C23direct (N × H)
16. C2×Q8×C67C23semidirect
17. C2×D4×C67C23semidirect
20. C22×C4×C67C23semidirect
22. C23×C67#41direct (N × H)
25. Q8×C67C24semidirect
26. D4×C67C24semidirect
29. C2×C4×C67C24semidirect
31. C22×C67C2×#41direct (N × H)
37. C2×C67C22×#41direct (N × H)
39. C23×#41C67direct (N × H)
40. C67C23×#41direct (N × H)
41. C22×#41C2×C67direct (N × H)
42. C23×Q8C2×C67semidirect
43. C23×D4C2×C67semidirect
44. C24×C4C2×C67semidirect
46. C2×#41C22×C67direct (N × H)
47. C22×Q8C22×C67semidirect
48. C22×D4C22×C67semidirect
51. C23×C4C22×C67semidirect
55. #41C23×C67direct (N × H)
56. C2×Q8C23×C67semidirect
57. C2×D4C23×C67semidirect
60. C22×C4C23×C67semidirect
62. C23#41×C67direct (N × H)
65. Q8C24×C67semidirect
66. D4C24×C67semidirect
69. C2×C4C24×C67semidirect
71. C22C2×#41×C67direct (N × H)
77. C2C22×#41×C67direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#41×C672
C23×Q8×C672
C23×D4×C672
C24×C4×C672
C23×#4167