Finite Group Databasebeta

← C2×C3×C4×C72×C23 #72204113

Subgroups — C2×C3×C4×C72×C23 #72204113

Order 27048.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C72×C23C311yesmaximal, characteristic, direct product factor, Hall
2C2×C3×C4×C7×C23C718yesmaximal, direct product factor
3C22×C3×C72×C23C211yesmaximal, characteristic
4C3×C4×C72×C23C212yesmaximal, direct product factor
5C2×C3×C72×C23C412yesdirect product factor
6C2×C3×C72×C23C2211yescharacteristic
7C22×C72×C23C2×C311yescharacteristic
8C4×C72×C23C2×C312yesdirect product factor
9C3×C72×C23C2×C411yescharacteristic, direct product factor, Hall
10C2×C72×C23C3×C412yesdirect product factor
11C2×C72×C23C22×C311yescharacteristic
12C22×C3×C7×C23C2×C718yes—
13C3×C4×C7×C23C2×C7116yesdirect product factor
14C2×C4×C7×C23C3×C718yesdirect product factor
15C2×C3×C4×C72C2311yesmaximal, characteristic, direct product factor, Hall
16C72×C23C2×C3×C411yescharacteristic, direct product factor, Hall
17C2×C3×C7×C23C4×C7116yesdirect product factor
18C2×C3×C7×C23C22×C718yes—
19C22×C7×C23C2×C3×C718yes—
20C4×C7×C23C2×C3×C7116yesdirect product factor
21C22×C3×C72C2×C2311yescharacteristic
22C3×C4×C72C2×C2312yesdirect product factor
23C2×C3×C4×C23C7211yescharacteristic, direct product factor, Hall
24C3×C7×C23C2×C4×C718yesdirect product factor
25C2×C4×C72C3×C2311yescharacteristic, direct product factor, Hall
26C2×C7×C23C3×C4×C7116yesdirect product factor
27C2×C7×C23C22×C3×C718yes—
28C2×C3×C72C4×C2312yesdirect product factor
29C2×C3×C72C22×C2311yescharacteristic
30C22×C3×C23C2×C7211yescharacteristic
31C3×C4×C23C2×C7212yesdirect product factor
32C22×C72C2×C3×C2311yescharacteristic
33C4×C72C2×C3×C2312yesdirect product factor
34C2×C4×C23C3×C7211yescharacteristic, direct product factor, Hall
35C2×C3×C4×C7C7×C2318yesdirect product factor
36C7×C23C2×C3×C4×C718yesdirect product factor
37C3×C72C2×C4×C2311yescharacteristic, direct product factor, Hall
38C2×C3×C23C4×C7212yesdirect product factor
39C2×C3×C23C22×C7211yescharacteristic
40C2×C72C3×C4×C2312yesdirect product factor
41C2×C72C22×C3×C2311yescharacteristic
42C22×C23C2×C3×C7211yescharacteristic
43C4×C23C2×C3×C7212yesdirect product factor
44C22×C3×C7C2×C7×C2318yes—
45C3×C4×C7C2×C7×C23116yesdirect product factor
46C3×C23C2×C4×C7211yescharacteristic, direct product factor, Hall
47C2×C4×C7C3×C7×C2318yesdirect product factor
48C72C2×C3×C4×C2311yescharacteristic, direct product factor, Sylow
49C2×C23C3×C4×C7212yesdirect product factor
50C2×C23C22×C3×C7211yescharacteristic
51C2×C3×C7C4×C7×C23116yesdirect product factor
52C2×C3×C7C22×C7×C2318yes—
53C22×C7C2×C3×C7×C2318yes—
54C4×C7C2×C3×C7×C23116yesdirect product factor
55C2×C3×C4C72×C2311yescharacteristic, direct product factor, Hall
56C23C2×C3×C4×C7211yescharacteristic, direct product factor, Sylow
57C3×C7C2×C4×C7×C2318yesdirect product factor
58C2×C7C3×C4×C7×C23116yesdirect product factor
59C2×C7C22×C3×C7×C2318yes—
60C22×C3C2×C72×C2311yescharacteristic
61C3×C4C2×C72×C2312yesdirect product factor
62C2×C4C3×C72×C2311yescharacteristic, direct product factor, Sylow
63C7C2×C3×C4×C7×C2318yesdirect product factor
64C2×C3C4×C72×C2312yesdirect product factor
65C2×C3C22×C72×C2311yescharacteristic
66C22C2×C3×C72×C2311yescharacteristic
67C4C2×C3×C72×C2312yesdirect product factor
68C3C2×C4×C72×C2311yescharacteristic, direct product factor, Sylow
69C2C3×C4×C72×C2312yesdirect product factor
70C2C22×C3×C72×C2311yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C22×C3×C72×C23
socle, prime radicalC22×C3×C72×C23C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×C72×C23C3direct (N × H)
2. C2×C3×C4×C7×C23C7direct (N × H)
4. C3×C4×C72×C23C2direct (N × H)
5. C2×C3×C72×C23C4direct (N × H)
8. C4×C72×C23C2×C3direct (N × H)
9. C3×C72×C23C2×C4direct (N × H)
10. C2×C72×C23C3×C4direct (N × H)
13. C3×C4×C7×C23C2×C7direct (N × H)
14. C2×C4×C7×C23C3×C7direct (N × H)
15. C2×C3×C4×C72C23direct (N × H)
16. C72×C23C2×C3×C4direct (N × H)
17. C2×C3×C7×C23C4×C7direct (N × H)
20. C4×C7×C23C2×C3×C7direct (N × H)
22. C3×C4×C72C2×C23direct (N × H)
23. C2×C3×C4×C23C72direct (N × H)
24. C3×C7×C23C2×C4×C7direct (N × H)
25. C2×C4×C72C3×C23direct (N × H)
26. C2×C7×C23C3×C4×C7direct (N × H)
28. C2×C3×C72C4×C23direct (N × H)
31. C3×C4×C23C2×C72direct (N × H)
33. C4×C72C2×C3×C23direct (N × H)
34. C2×C4×C23C3×C72direct (N × H)
35. C2×C3×C4×C7C7×C23direct (N × H)
36. C7×C23C2×C3×C4×C7direct (N × H)
37. C3×C72C2×C4×C23direct (N × H)
38. C2×C3×C23C4×C72direct (N × H)
40. C2×C72C3×C4×C23direct (N × H)
43. C4×C23C2×C3×C72direct (N × H)
45. C3×C4×C7C2×C7×C23direct (N × H)
46. C3×C23C2×C4×C72direct (N × H)
47. C2×C4×C7C3×C7×C23direct (N × H)
48. C72C2×C3×C4×C23direct (N × H)
49. C2×C23C3×C4×C72direct (N × H)
51. C2×C3×C7C4×C7×C23direct (N × H)
54. C4×C7C2×C3×C7×C23direct (N × H)
55. C2×C3×C4C72×C23direct (N × H)
56. C23C2×C3×C4×C72direct (N × H)
57. C3×C7C2×C4×C7×C23direct (N × H)
58. C2×C7C3×C4×C7×C23direct (N × H)
61. C3×C4C2×C72×C23direct (N × H)
62. C2×C4C3×C72×C23direct (N × H)
63. C7C2×C3×C4×C7×C23direct (N × H)
64. C2×C3C4×C72×C23direct (N × H)
67. C4C2×C3×C72×C23direct (N × H)
68. C3C2×C4×C72×C23direct (N × H)
69. C2C3×C4×C72×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C72×C233
C2×C3×C4×C7×C237
C22×C3×C72×C232
C3×C4×C72×C232
C2×C3×C4×C7223