Finite Group Databasebeta

← C22×C4×C72×C31 #72201600

Subgroups — C22×C4×C72×C31 #72201600

Order 24304.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C72×C31C211yesmaximal, characteristic
2C2×C4×C72×C31C216yesmaximal, direct product factor
3C22×C4×C7×C31C718yesmaximal, direct product factor
4C22×C72×C31C414yesdirect product factor
5C22×C72×C31C2213yes—
6C4×C72×C31C2214yesdirect product factor
7C2×C72×C31C2×C416yesdirect product factor
8C2×C72×C31C2311yescharacteristic
9C23×C7×C31C2×C718yes—
10C2×C4×C7×C31C2×C7148yesdirect product factor
11C72×C31C22×C411yescharacteristic, direct product factor, Hall
12C22×C7×C31C4×C7132yesdirect product factor
13C22×C7×C31C22×C7124yes—
14C4×C7×C31C22×C7132yesdirect product factor
15C22×C4×C72C3111yesmaximal, characteristic, direct product factor, Hall
16C22×C4×C31C7211yescharacteristic, direct product factor, Hall
17C2×C7×C31C2×C4×C7148yesdirect product factor
18C2×C7×C31C23×C718yes—
19C23×C72C2×C3111yescharacteristic
20C2×C4×C72C2×C3116yesdirect product factor
21C23×C31C2×C7211yescharacteristic
22C2×C4×C31C2×C7216yesdirect product factor
23C7×C31C22×C4×C718yesdirect product factor
24C22×C72C4×C3114yesdirect product factor
25C22×C72C22×C3113yes—
26C4×C72C22×C3114yesdirect product factor
27C22×C31C4×C7214yesdirect product factor
28C22×C31C22×C7213yes—
29C4×C31C22×C7214yesdirect product factor
30C22×C4×C7C7×C3118yesdirect product factor
31C2×C72C2×C4×C3116yesdirect product factor
32C2×C72C23×C3111yescharacteristic
33C2×C31C2×C4×C7216yesdirect product factor
34C2×C31C23×C7211yescharacteristic
35C23×C7C2×C7×C3118yes—
36C2×C4×C7C2×C7×C31148yesdirect product factor
37C72C22×C4×C3111yescharacteristic, direct product factor, Sylow
38C31C22×C4×C7211yescharacteristic, direct product factor, Sylow
39C22×C7C4×C7×C31132yesdirect product factor
40C22×C7C22×C7×C31124yes—
41C4×C7C22×C7×C31132yesdirect product factor
42C22×C4C72×C3111yescharacteristic, direct product factor, Sylow
43C2×C7C2×C4×C7×C31148yesdirect product factor
44C2×C7C23×C7×C3118yes—
45C23C2×C72×C3111yescharacteristic
46C2×C4C2×C72×C3116yesdirect product factor
47C7C22×C4×C7×C3118yesdirect product factor
48C22C4×C72×C3114yesdirect product factor
49C22C22×C72×C3113yes—
50C4C22×C72×C3114yesdirect product factor
51C2C2×C4×C72×C3116yesdirect product factor
52C2C23×C72×C3111yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C23×C72×C31
socle, prime radicalC23×C72×C31C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×C72×C31C2direct (N × H)
3. C22×C4×C7×C31C7direct (N × H)
4. C22×C72×C31C4direct (N × H)
6. C4×C72×C31C22direct (N × H)
7. C2×C72×C31C2×C4direct (N × H)
10. C2×C4×C7×C31C2×C7direct (N × H)
11. C72×C31C22×C4direct (N × H)
12. C22×C7×C31C4×C7direct (N × H)
14. C4×C7×C31C22×C7direct (N × H)
15. C22×C4×C72C31direct (N × H)
16. C22×C4×C31C72direct (N × H)
17. C2×C7×C31C2×C4×C7direct (N × H)
20. C2×C4×C72C2×C31direct (N × H)
22. C2×C4×C31C2×C72direct (N × H)
23. C7×C31C22×C4×C7direct (N × H)
24. C22×C72C4×C31direct (N × H)
26. C4×C72C22×C31direct (N × H)
27. C22×C31C4×C72direct (N × H)
29. C4×C31C22×C72direct (N × H)
30. C22×C4×C7C7×C31direct (N × H)
31. C2×C72C2×C4×C31direct (N × H)
33. C2×C31C2×C4×C72direct (N × H)
36. C2×C4×C7C2×C7×C31direct (N × H)
37. C72C22×C4×C31direct (N × H)
38. C31C22×C4×C72direct (N × H)
39. C22×C7C4×C7×C31direct (N × H)
41. C4×C7C22×C7×C31direct (N × H)
42. C22×C4C72×C31direct (N × H)
43. C2×C7C2×C4×C7×C31direct (N × H)
46. C2×C4C2×C72×C31direct (N × H)
47. C7C22×C4×C7×C31direct (N × H)
48. C22C4×C72×C31direct (N × H)
50. C4C22×C72×C31direct (N × H)
51. C2C2×C4×C72×C31direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C72×C312
C2×C4×C72×C312
C22×C4×C7×C317
C22×C4×C7231