Finite Group Databasebeta

← C2×#72341130 #72341252

Subgroups — C2×#72341130 #72341252

Order 6144.

At least 10245 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has at least 10245 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72341129C211yesmaximal, characteristic, complemented
2C2×#72261513C212yesmaximal, complemented
3C2×C4×#85559C211yesmaximal, characteristic, complemented
4C2×#11714271C2211yescharacteristic, complemented
5#72341130C218yesmaximal, direct product factor
6#72341129C2214yescomplemented
7C2×#72340507C211yesmaximal, characteristic, complemented
8C2×C4×#5464C2211yescharacteristic, complemented
9#72340507C2214yescomplemented
10C22×#11715753C212yesmaximal, complemented
11C22×#3532⋊C3C2212yescomplemented
12C22×#85559C2211yescharacteristic
13C22×#85559C2212yescomplemented
14C2×#8062881D311yescharacteristic, complemented
15#72261513C2218yescomplemented
16C4×#85559C2214yescomplemented
17C2×#11715753C2218yescomplemented
18C2×#85559D414yescomplemented
19C2×#85559C2312yes—
20C2×#85559C2314yescomplemented
21#11714271C2312yescomplemented
22C2×#3532⋊C3C2314yescomplemented
23C22×#5464C2311yescharacteristic
24C4×#5464C2312yescomplemented
25C22×#3532C2×D312yescomplemented
26#8062881C2×D314yescomplemented
27C2×C4×H(3,𝔽4)C2×D311yescharacteristic, complemented
28C2×#5464C2×D412yescomplemented
29C2×#5464C2411yescharacteristic
30#85559C2×D414yescomplemented
31C25×D4S411yescharacteristic, complemented
32C2×#3532C22×D314yescomplemented
33C22×H(3,𝔽4)C22×D311yescharacteristic
34C2×#2867S411yescharacteristic, complemented
35C2×#2847S411yescharacteristic, complemented
36C4×H(3,𝔽4)C22×D312yescomplemented
37#5464C22×D411yescharacteristic, complemented
38C27C2×S412yes—
39C24×D4C2×S414yescomplemented
40C25×C4C2×S411yescharacteristic
41C25×C4C2×S411yescharacteristic, complemented
42C2×H(3,𝔽4)D3×D412yescomplemented
43C2×H(3,𝔽4)C23×D311yescharacteristic
44C22×#127C2×S412yescomplemented
45C22×#120C2×S412yescomplemented
46#2867C2×S414yescomplemented
47#2847C2×S414yescomplemented
48C2×C43C2×S411yescharacteristic, complemented
49C26C22×S414yes—
50C26C22×S411yescharacteristic
51C26C22×S411yescharacteristic
52C23×D4#102011yescharacteristic
53C24×C4C22×S412yes—
54C24×C4C22×S412yescomplemented
55H(3,𝔽4)C2×D3×D411yescharacteristic, complemented
56C2×#127C22×S414yescomplemented
57C2×#120C22×S414yescomplemented
58C22×C42C22×S411yescharacteristic
59C43C22×S412yescomplemented
60C25D4×S412yes—
61C25D4×S412yescomplemented
62C25C23×S411yescharacteristic
63C25C23×S411yescharacteristic
64C25C2×#102012yes—
65C22×D4C2×#102014yes—
66C23×C4C2×#102011yescharacteristic
67C2×C42D4×S412yescomplemented
68C2×C42C23×S411yescharacteristic
69C24C2×D4×S411yescharacteristic
70C24C2×D4×S411yescharacteristic, complemented
71C24C22×#102014yes—
72C24C22×#102011yescharacteristic
73C22×C4C22×#102012yes—
74C42C2×D4×S411yescharacteristic, complemented
75C23D4×#102012yes—
76C23C23×#102011yescharacteristic
77C2×C4#1171575311yescharacteristic, complemented
78C22C2×#1171575311yescharacteristic
79C22C2×D4×#102011yescharacteristic
80C4C2×#1171575312yescomplemented
81C2C22×#1171575311yescharacteristic
82C2#7234113012yesdirect product factor
subgroupG/H
centerC22C2×#11715753
commutator, cosocleC2×#5464C24
FrattiniC23C23×#1020
FittingC2×#8062881D3
socleC24C22×#1020
hypercenterC2×C4#11715753
hypocenter#5464C22×D4
LCSG ▷ C2×#5464 ▷ #5464
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#72341129C2semidirect
2. C2×#72261513C2semidirect
3. C2×C4×#85559C2semidirect
4. C2×#11714271C22semidirect
5. #72341130C2direct (N × H)
6. #72341129C22semidirect
7. C2×#72340507C2semidirect
8. C2×C4×#5464C22semidirect
9. #72340507C22semidirect
10. C22×#11715753C2semidirect
11. C22×#3532⋊C3C22semidirect
13. C22×#85559C22semidirect
14. C2×#8062881D3semidirect
15. #72261513C22semidirect
16. C4×#85559C22semidirect
17. C2×#11715753C22semidirect
18. C2×#85559D4semidirect
20. C2×#85559C23semidirect
21. #11714271C23semidirect
22. C2×#3532⋊C3C23semidirect
24. C4×#5464C23semidirect
25. C22×#3532C2×D3semidirect
26. #8062881C2×D3semidirect
27. C2×C4×H(3,𝔽4)C2×D3semidirect
28. C2×#5464C2×D4semidirect
30. #85559C2×D4semidirect
31. C25×D4S4semidirect
32. C2×#3532C22×D3semidirect
34. C2×#2867S4semidirect
35. C2×#2847S4semidirect
36. C4×H(3,𝔽4)C22×D3semidirect
37. #5464C22×D4semidirect
39. C24×D4C2×S4semidirect
41. C25×C4C2×S4semidirect
42. C2×H(3,𝔽4)D3×D4semidirect
44. C22×#127C2×S4semidirect
45. C22×#120C2×S4semidirect
46. #2867C2×S4semidirect
47. #2847C2×S4semidirect
48. C2×C43C2×S4semidirect
54. C24×C4C22×S4semidirect
55. H(3,𝔽4)C2×D3×D4semidirect
56. C2×#127C22×S4semidirect
57. C2×#120C22×S4semidirect
59. C43C22×S4semidirect
61. C25D4×S4semidirect
67. C2×C42D4×S4semidirect
70. C24C2×D4×S4semidirect
74. C42C2×D4×S4semidirect
77. C2×C4#11715753semidirect
80. C4C2×#11715753semidirect
82. C2#72341130direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#723411292
C2×#722615132
C2×C4×#855592
#723411302
C2×#723405072
C2×#723114873
C22×#117157532
C2×#117132714
C2×#117136284
C2×D4×#10204