Finite Group Databasebeta

← C2×#72215767 #72341523

Subgroups — C2×#72215767 #72341523

Order 3072.

Only the 712 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 122 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11715831C211yesmaximal, characteristic, complemented
2C2×#11715829C213yesmaximal, complemented
3C2×#11715830C211yesmaximal, characteristic, complemented
4C22×#87520C213yesmaximal, complemented
5C2×#11715827C213yesmaximal, complemented
6C2×#11710937C211yesmaximal, characteristic, complemented
7#72215767C2116yesmaximal, direct product factor
8C2×#11715828C213yesmaximal, complemented
9#11715831C2218yescomplemented
10#11715830C2218yescomplemented
11#11715829C22124yescomplemented
12#11715828C22124yescomplemented
13#11715827C22124yescomplemented
14C2×#87518C2213yescomplemented
15C2×#87522C2213yescomplemented
16C2×#87516C2216yescomplemented
17C2×#87520C2213yescomplemented
18C2×#87520C2213yescomplemented
19C2×#87520C2213yescomplemented
20C2×#87520C2211yescharacteristic, complemented
21C2×#87520C22124yescomplemented
22C22×#5933C2213yescomplemented
23C22×#5935C2213yescomplemented
24#11710937C2218yescomplemented
25C2×#87548C2211yescharacteristic, complemented
26C2×#87547C2213yescomplemented
27C22×#5950C2213yescomplemented
28C2×#63103D311yescharacteristic, complemented
29#87548C2314yescomplemented
30#87547C23112yescomplemented
31C2×#5950C2311yescharacteristic
32C2×#5950C2313yes—
33C2×#5950C23112yescomplemented
34C22×#997C2313yes—
35#87522C23112yescomplemented
36#87520C2314yescomplemented
37#87520C23112yescomplemented
38#87520C23112yescomplemented
39#87520C23112yescomplemented
40#87518C23112yescomplemented
41#87516C23124yescomplemented
42C2×#5935C2311yescharacteristic, complemented
43C2×#5935C2313yescomplemented
44C2×#5935C23112yescomplemented
45C2×#5933C2311yescharacteristic, complemented
46C2×#5933C2313yescomplemented
47C2×#5933C23112yescomplemented
48#63103C2×D318yescomplemented
49C2×2−1+6C2×D311yescharacteristic, complemented
50C2×H(7,𝔽2)C2×D313yescomplemented
51C22×#585C2×D313yescomplemented
52#5950C2416yes—
53#5950C2412yes—
54C2×#997C2411yescharacteristic
55C2×#997C2416yes—
56#5935C2412yescomplemented
57#5935C2416yescomplemented
58#5933C2412yescomplemented
59#5933C2416yescomplemented
602−1+6C22×D314yescomplemented
61H(7,𝔽2)C22×D3112yescomplemented
62C2×#585S412yescomplemented
63C2×#585C22×D311yescharacteristic
64C2×#585C22×D313yes—
65C2×#585C22×D3112yescomplemented
66C22×H(5,𝔽2)C22×D313yes—
67C23×#41S411yescharacteristic
68#997C2511yescharacteristic
69#585C2×S4116yescomplemented
70#585C23×D312yes—
71#585C23×D316yes—
72C2×2−1+4C2×S416yescomplemented
73C2×H(5,𝔽2)C2×S412yescomplemented
74C2×H(5,𝔽2)C23×D311yescharacteristic
75C2×H(5,𝔽2)C23×D316yes—
76C22×#41C2×S418yes—
77C22×#41C2×S416yescomplemented
78C23×Q8C2×S411yescharacteristic
79C23×D4C2×S413yes—
80C24×C4C2×S413yes—
81C25C22×S413yes—
822−1+4C22×S4124yescomplemented
83H(5,𝔽2)C22×S418yescomplemented
84H(5,𝔽2)C24×D311yescharacteristic
85C2×#41C22×S416yes—
86C2×#41C22×S412yes—
87C2×#41C22×S4124yescomplemented
88C2×#41#102011yescharacteristic
89C22×Q8C22×S414yes—
90C22×Q8C22×S416yes—
91C22×D4C22×S4112yes—
92C23×C4C22×S413yes—
93C23×C4C22×S4112yes—
94C23×C4C22×S411yescharacteristic
95C24C23×S411yescharacteristic
96C24C23×S416yes—
97#41C23×S414yes—
98#41C23×S4112yes—
99#41C2×#102018yes—
100C2×Q8C23×S412yes—
101C2×Q8C23×S4112yes—
102C2×Q8C2×#102011yescharacteristic
103C2×D4C2×#102013yes—
104C22×C4C23×S412yes—
105C22×C4C23×S416yes—
106C22×C4C2×#102013yes—
107C23C24×S411yescharacteristic
108C23C22×#102013yes—
109Q8C24×S412yes—
110Q8C22×#102014yes—
111D4C22×#1020112yes—
112C2×C4C22×#102013yes—
113C2×C4C22×#1020112yes—
114C2×C4C22×#102011yescharacteristic
115C22C23×#102016yes—
116C22C23×#102011yescharacteristic
117C4C23×#102012yes—
118C4C23×#102016yes—
119C2C24×#102011yescharacteristic
120C2#7221576712yesdirect product factor
subgroupG/H
centerC2×C4C22×#1020
commutator, cosocle, hypocenter#997C25
FrattiniC2C24×#1020
FittingC2×#63103D3
socleC22C23×#1020
hypercenterC2×#41#1020
LCSG ▷ #997
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11715831C2semidirect
2. C2×#11715829C2semidirect
3. C2×#11715830C2semidirect
4. C22×#87520C2semidirect
5. C2×#11715827C2semidirect
6. C2×#11710937C2semidirect
7. #72215767C2direct (N × H)
8. C2×#11715828C2semidirect
9. #11715831C22semidirect
10. #11715830C22semidirect
11. #11715829C22semidirect
12. #11715828C22semidirect
13. #11715827C22semidirect
14. C2×#87518C22semidirect
15. C2×#87522C22semidirect
16. C2×#87516C22semidirect
17. C2×#87520C22semidirect
18. C2×#87520C22semidirect
19. C2×#87520C22semidirect
20. C2×#87520C22semidirect
21. C2×#87520C22semidirect
22. C22×#5933C22semidirect
23. C22×#5935C22semidirect
24. #11710937C22semidirect
25. C2×#87548C22semidirect
26. C2×#87547C22semidirect
27. C22×#5950C22semidirect
28. C2×#63103D3semidirect
29. #87548C23semidirect
30. #87547C23semidirect
33. C2×#5950C23semidirect
35. #87522C23semidirect
36. #87520C23semidirect
37. #87520C23semidirect
38. #87520C23semidirect
39. #87520C23semidirect
40. #87518C23semidirect
41. #87516C23semidirect
42. C2×#5935C23semidirect
43. C2×#5935C23semidirect
44. C2×#5935C23semidirect
45. C2×#5933C23semidirect
46. C2×#5933C23semidirect
47. C2×#5933C23semidirect
48. #63103C2×D3semidirect
49. C2×2−1+6C2×D3semidirect
50. C2×H(7,𝔽2)C2×D3semidirect
51. C22×#585C2×D3semidirect
56. #5935C24semidirect
57. #5935C24semidirect
58. #5933C24semidirect
59. #5933C24semidirect
60. 2−1+6C22×D3semidirect
61. H(7,𝔽2)C22×D3semidirect
62. C2×#585S4semidirect
65. C2×#585C22×D3semidirect
69. #585C2×S4semidirect
72. C2×2−1+4C2×S4semidirect
73. C2×H(5,𝔽2)C2×S4semidirect
77. C22×#41C2×S4semidirect
82. 2−1+4C22×S4semidirect
83. H(5,𝔽2)C22×S4semidirect
87. C2×#41C22×S4semidirect
120. C2#72215767direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117158312
C2×#117158292
C2×#117158302
C22×#875202
C2×#105748543
C2×#117158272
C2×#117109372
#722157672
C2×#117158282
C2×#874974
C2×#41×S44