Finite Group Databasebeta

← C2×#72215704 #72343050

Subgroups — C2×#72215704 #72343050

Order 3072.

At least 7062 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has at least 7062 conjugacy classes of subgroups — too many to draw. Only its 176 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×#11715682C211yesmaximal, characteristic, complemented
2C2×#11715710C211yesmaximal, characteristic, complemented
3C2×#11715830C211yesmaximal, characteristic, complemented
4C2×#11715708C211yesmaximal, characteristic, complemented
5C2×#11710899C211yesmaximal, characteristic, complemented
6C2×#11715739C211yesmaximal, characteristic, complemented
7#72215704C218yesmaximal, direct product factor
8C2×#11715828C211yesmaximal, characteristic, complemented
9#11715830C2214yescomplemented
10#11715828C2214yescomplemented
11C2×#87522C2211yescharacteristic, complemented
12C2×#87522C2211yescharacteristic, complemented
13C2×#87520C2211yescharacteristic
14#11715739C2214yescomplemented
15#11715710C2214yescomplemented
16#11715708C2214yescomplemented
17#11715682C2214yescomplemented
18C2×#85567C2211yescharacteristic, complemented
19C2×#87548C2211yescharacteristic, complemented
20C2×#87547C2211yescharacteristic, complemented
21#11710899C2214yescomplemented
22C2×#85666C2211yescharacteristic, complemented
23C2×#63092D311yescharacteristic, complemented
24#87548C2312yescomplemented
25#87547C2312yescomplemented
26C2×#5950C2311yescharacteristic
27#87522C2312yescomplemented
28#87522C2312yescomplemented
29#87520D412yescomplemented
30#87520C2312yes—
31C2×#5935D411yescharacteristic, complemented
32C2×#5933D411yescharacteristic, complemented
33#85666C2312yescomplemented
34#85567C2312yescomplemented
35C2×2−1+6C2×D311yescharacteristic, complemented
36C2×H(7,𝔽2)C2×D311yescharacteristic
37#63092C2×D314yescomplemented
38C2×#3573C2×D311yescharacteristic, complemented
39#5950C2×D411yescharacteristic
40#5950C2411yescharacteristic
41C2×#997C2×D411yescharacteristic
42#5935C2×D412yescomplemented
43#5933C2×D412yescomplemented
442−1+6C22×D312yescomplemented
45H(7,𝔽2)C22×D312yes—
46C2×#585C22×D311yescharacteristic
47C2×#577S412yescomplemented
48C23×SD16S411yescharacteristic
49#3573C22×D312yescomplemented
50#997C22×D411yescharacteristic
51#585D3×D411yescharacteristic
52#585C23×D311yescharacteristic
53C2×2−1+4C2×S412yes—
54C2×H(5,𝔽2)D3×D411yescharacteristic
55C2×H(5,𝔽2)C2×S412yescomplemented
56C23×Q8C2×S411yescharacteristic
57C23×D4C2×S411yescharacteristic
58#577C2×S418yescomplemented
59C22×SD16C2×S414yes—
60C2×#131C2×S412yescomplemented
61C23×C8C2×S411yescharacteristic
622−1+4C22×S414yes—
63H(5,𝔽2)C2×D3×D411yescharacteristic
64H(5,𝔽2)C22×S414yescomplemented
65C2×#41C22×S412yes—
66C22×Q8C22×S412yes—
67C22×D4C22×S412yes—
68C23×C4C22×S411yescharacteristic
69C2×SD16#102011yescharacteristic
70#131C22×S414yescomplemented
71C22×C8C22×S412yes—
72C24D4×S411yescharacteristic
73#41D4×S412yes—
74#41C23×S412yes—
75C2×Q8D4×S412yes—
76C2×Q8C2×#102011yescharacteristic
77C2×D4C2×#102011yescharacteristic
78C22×C4D4×S411yescharacteristic
79C22×C4C23×S411yescharacteristic
80SD16C2×#102014yes—
81C2×C8C2×#102011yescharacteristic
82C23C2×D4×S411yescharacteristic
83Q8C2×D4×S412yes—
84Q8C22×#102012yes—
85D4C22×#102012yes—
86C2×C4C22×#102011yescharacteristic
87C8C22×#102012yes—
88C22D4×#102011yescharacteristic
89C4D4×#102011yescharacteristic
90C4C23×#102011yescharacteristic
91C2#7221570412yesdirect product factor
92C2C2×D4×#102011yescharacteristic
subgroupG/H
center, socleC22D4×#1020
commutator, cosocle#5950C24
FrattiniC4C23×#1020
FittingC2×#63092D3
hypercenterC2×SD16#1020
hypocenter#997C22×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#11715682C2semidirect
2. C2×#11715710C2semidirect
3. C2×#11715830C2semidirect
4. C2×#11715708C2semidirect
5. C2×#11710899C2semidirect
6. C2×#11715739C2semidirect
7. #72215704C2direct (N × H)
8. C2×#11715828C2semidirect
9. #11715830C22semidirect
10. #11715828C22semidirect
11. C2×#87522C22semidirect
12. C2×#87522C22semidirect
14. #11715739C22semidirect
15. #11715710C22semidirect
16. #11715708C22semidirect
17. #11715682C22semidirect
18. C2×#85567C22semidirect
19. C2×#87548C22semidirect
20. C2×#87547C22semidirect
21. #11710899C22semidirect
22. C2×#85666C22semidirect
23. C2×#63092D3semidirect
24. #87548C23semidirect
25. #87547C23semidirect
27. #87522C23semidirect
28. #87522C23semidirect
29. #87520D4semidirect
31. C2×#5935D4semidirect
32. C2×#5933D4semidirect
33. #85666C23semidirect
34. #85567C23semidirect
35. C2×2−1+6C2×D3semidirect
37. #63092C2×D3semidirect
38. C2×#3573C2×D3semidirect
42. #5935C2×D4semidirect
43. #5933C2×D4semidirect
44. 2−1+6C22×D3semidirect
47. C2×#577S4semidirect
49. #3573C22×D3semidirect
55. C2×H(5,𝔽2)C2×S4semidirect
58. #577C2×S4semidirect
60. C2×#131C2×S4semidirect
64. H(5,𝔽2)C22×S4semidirect
70. #131C22×S4semidirect
91. C2#72215704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117156822
C2×#117157102
C2×#117158302
C2×#117157082
C2×#117108992
C2×#65736993
C2×#117157392
#722157042
C2×#117158282
C2×#854704
C2×#854604