Finite Group Databasebeta

← C23×C4×C7×C121 #72204170

Subgroups — C23×C4×C7×C121 #72204170

Order 27104.

708 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C7×C121C211yesmaximal, characteristic
2C22×C4×C7×C121C2114yesmaximal, direct product factor
3C23×C7×C121C418yesdirect product factor
4C23×C7×C121C2217yes—
5C2×C4×C7×C121C22128yesdirect product factor
6C23×C4×C121C711yesmaximal, characteristic, direct product factor, Hall
7C23×C4×C7×C11C1111yesmaximal, characteristic
8C22×C7×C121C2×C4128yesdirect product factor
9C22×C7×C121C2317yes—
10C4×C7×C121C2318yesdirect product factor
11C24×C121C2×C711yescharacteristic
12C22×C4×C121C2×C7114yesdirect product factor
13C2×C7×C121C22×C4114yesdirect product factor
14C2×C7×C121C2411yescharacteristic
15C24×C7×C11C2×C1111yescharacteristic
16C22×C4×C7×C11C2×C11114yes—
17C23×C121C4×C718yesdirect product factor
18C23×C121C22×C717yes—
19C2×C4×C121C22×C7128yesdirect product factor
20C7×C121C23×C411yescharacteristic, direct product factor, Hall
21C23×C7×C11C4×C1118yes—
22C23×C7×C11C22×C1117yes—
23C2×C4×C7×C11C22×C11128yes—
24C22×C121C2×C4×C7128yesdirect product factor
25C22×C121C23×C717yes—
26C4×C121C23×C718yesdirect product factor
27C23×C4×C11C7×C1111yescharacteristic
28C22×C7×C11C2×C4×C11128yes—
29C22×C7×C11C23×C1117yes—
30C4×C7×C11C23×C1118yes—
31C2×C121C22×C4×C7114yesdirect product factor
32C2×C121C24×C711yescharacteristic
33C23×C4×C7C12111yescharacteristic, direct product factor, Hall
34C24×C11C2×C7×C1111yescharacteristic
35C22×C4×C11C2×C7×C11114yes—
36C2×C7×C11C22×C4×C11114yes—
37C2×C7×C11C24×C1111yescharacteristic
38C121C23×C4×C711yescharacteristic, direct product factor, Sylow
39C24×C7C2×C12111yescharacteristic
40C22×C4×C7C2×C121114yesdirect product factor
41C23×C11C4×C7×C1118yes—
42C23×C11C22×C7×C1117yes—
43C2×C4×C11C22×C7×C11128yes—
44C7×C11C23×C4×C1111yescharacteristic
45C23×C7C4×C12118yesdirect product factor
46C23×C7C22×C12117yes—
47C2×C4×C7C22×C121128yesdirect product factor
48C22×C11C2×C4×C7×C11128yes—
49C22×C11C23×C7×C1117yes—
50C4×C11C23×C7×C1118yes—
51C23×C4C7×C12111yescharacteristic, direct product factor, Sylow
52C22×C7C2×C4×C121128yesdirect product factor
53C22×C7C23×C12117yes—
54C4×C7C23×C12118yesdirect product factor
55C2×C11C22×C4×C7×C11114yes—
56C2×C11C24×C7×C1111yescharacteristic
57C24C2×C7×C12111yescharacteristic
58C22×C4C2×C7×C121114yesdirect product factor
59C2×C7C22×C4×C121114yesdirect product factor
60C2×C7C24×C12111yescharacteristic
61C11C23×C4×C7×C1111yescharacteristic
62C23C4×C7×C12118yesdirect product factor
63C23C22×C7×C12117yes—
64C2×C4C22×C7×C121128yesdirect product factor
65C7C23×C4×C12111yescharacteristic, direct product factor, Sylow
66C22C2×C4×C7×C121128yesdirect product factor
67C22C23×C7×C12117yes—
68C4C23×C7×C12118yesdirect product factor
69C2C22×C4×C7×C121114yesdirect product factor
70C2C24×C7×C12111yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C11C24×C7×C11
socle, prime radicalC24×C7×C11C2×C11
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×C7×C121C2direct (N × H)
3. C23×C7×C121C4direct (N × H)
5. C2×C4×C7×C121C22direct (N × H)
6. C23×C4×C121C7direct (N × H)
8. C22×C7×C121C2×C4direct (N × H)
10. C4×C7×C121C23direct (N × H)
12. C22×C4×C121C2×C7direct (N × H)
13. C2×C7×C121C22×C4direct (N × H)
17. C23×C121C4×C7direct (N × H)
19. C2×C4×C121C22×C7direct (N × H)
20. C7×C121C23×C4direct (N × H)
24. C22×C121C2×C4×C7direct (N × H)
26. C4×C121C23×C7direct (N × H)
31. C2×C121C22×C4×C7direct (N × H)
33. C23×C4×C7C121direct (N × H)
38. C121C23×C4×C7direct (N × H)
40. C22×C4×C7C2×C121direct (N × H)
45. C23×C7C4×C121direct (N × H)
47. C2×C4×C7C22×C121direct (N × H)
51. C23×C4C7×C121direct (N × H)
52. C22×C7C2×C4×C121direct (N × H)
54. C4×C7C23×C121direct (N × H)
58. C22×C4C2×C7×C121direct (N × H)
59. C2×C7C22×C4×C121direct (N × H)
62. C23C4×C7×C121direct (N × H)
64. C2×C4C22×C7×C121direct (N × H)
65. C7C23×C4×C121direct (N × H)
66. C22C2×C4×C7×C121direct (N × H)
68. C4C23×C7×C121direct (N × H)
69. C2C22×C4×C7×C121direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C7×C1212
C22×C4×C7×C1212
C23×C4×C1217
C23×C4×C7×C1111