Finite Group Databasebeta

← C2×#41×D197 #27161904

Subgroups — C2×#41×D197 #27161904

Order 12608.

890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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×#15691359C211yesmaximal, characteristic, complemented
2C2×#15691358C213yesmaximal, complemented
3C2×#15691357C213yesmaximal, complemented
4#41×D197C2116yesmaximal, direct product factor
5C2×Q8×D197C211yesmaximal, characteristic, complemented
6C2×D4×D197C213yesmaximal, complemented
7C22×C4×D197C213yesmaximal, complemented
8C2×#41×C197C211yesmaximal, characteristic, complemented
9C23×D197C2213yes—
10#15691359C2218yescomplemented
11#15691358C22124yescomplemented
12#15691357C22124yescomplemented
13C2×Dic394C2213yescomplemented
14Q8×D197C2218yescomplemented
15C2×D788C2213yescomplemented
16C2×C22⋊D197C2216yescomplemented
17D4×D197C22124yescomplemented
18C22×Dic197C2213yescomplemented
19C2×C4×D197C2211yescharacteristic
20C2×C4×D197C2213yes—
21C2×C4×D197C2213yescomplemented
22C2×C4×D197C2213yescomplemented
23C2×C4×D197C22124yescomplemented
24#41×C197C2218yescomplemented
25C2×Q8×C197C2211yescharacteristic, complemented
26C2×D4×C197C2213yescomplemented
27C22×C4×C197C2213yescomplemented
28C22×D197—212no—
29C22×D197C23112yes—
30C22×D197C2313yes—
31C22×D197C2311yescharacteristic
32Dic394C23112yescomplemented
33D788C23112yescomplemented
34C22⋊D197C23124yescomplemented
35C2×Dic197C2311yescharacteristic
36C2×Dic197C2313yes—
37C2×Dic197C23112yescomplemented
38C4×D197C2314yes—
39C4×D197C23112yes—
40C4×D197C23112yescomplemented
41C4×D197C23112yescomplemented
42C23×C197C2313yes—
43Q8×C197C2314yescomplemented
44D4×C197C23112yescomplemented
45C2×C4×C197C2313yes—
46C2×C4×C197C2311yescharacteristic
47C2×C4×C197C23112yescomplemented
48C22×C197—26no—
49C22×C197C2416yes—
50C22×C197C2411yescharacteristic
51C2×D197—26no—
52C2×D197—224no—
53C2×D197#4114yesdirect product factor
54C2×D197C2416yes—
55C2×D197C2412yes—
56C4×C197C2412yes—
57C4×C197C2416yes—
58Dic197C2412yes—
59Dic197C2416yes—
60C2×C197—26no—
61C2×C197C2×#4112yescomplemented
62C2×C197C2511yescharacteristic
63D197—26no—
64D197C2×#4114yesdirect product factor
65C197C22×#4111yescharacteristic, complemented, Sylow
66C22×#41—1971nomaximal, Sylow
67C2×#41—19716no—
68C2×#41—1971no—
69C2×#41—1973no—
70C2×#41—1973no—
71C2×#41D19711yescharacteristic, direct product factor
72C22×Q8—1971no—
73C22×D4—1973no—
74C23×C4—1973no—
75C24—1973no—
76#41—1978no—
77#41—19724no—
78#41—19724no—
79#41C2×D19718yesdirect product factor
80C2×Q8—1978no—
81C2×Q8—1973no—
82C2×Q8C2×D19711yescharacteristic, complemented
83C2×D4—1976no—
84C2×D4—19724no—
85C2×D4—1973no—
86C2×D4C2×D19713yescomplemented
87C22×C4—1973no—
88C22×C4—1973no—
89C22×C4—1973no—
90C22×C4—1971no—
91C22×C4—1973no—
92C22×C4—19724no—
93C22×C4C2×D19713yescomplemented
94C23—39412no—
95C23—19712no—
96C23—1973no—
97C23—1971no—
98C23C22×D19713yes—
99Q8—19712no—
100Q8C22×D19714yescomplemented
101D4—19724no—
102D4—19712no—
103D4C22×D197112yescomplemented
104C2×C4—1971no—
105C2×C4—1973no—
106C2×C4—19712no—
107C2×C4—1974no—
108C2×C4—19712no—
109C2×C4—19712no—
110C2×C4—19712no—
111C2×C4C22×D19713yes—
112C2×C4C22×D19711yescharacteristic
113C2×C4C22×D197112yescomplemented
114C22—26no—
115C22—39424no—
116C22—3946no—
117C22—1976no—
118C22—1974no—
119C22—1972no—
120C22C23×D19716yes—
121C22C23×D19711yescharacteristic
122C4—1972no—
123C4—1976no—
124C4C23×D19712yes—
125C4C23×D19716yes—
126C2—26no—
127C2—3946no—
128C2—1974no—
129C2#41×D19712yesdirect product factor
130C2C24×D19711yescharacteristic
subgroupG/H
centerC2×C4C22×D197
commutator, cosocleC2×C197C25
FrattiniC2C24×D197
FittingC2×#41×C197C2
socleC22×C197C24
hypercenterC2×#41D197
hypocenterC197C22×#41
LCSG ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15691359C2semidirect
2. C2×#15691358C2semidirect
3. C2×#15691357C2semidirect
4. #41×D197C2direct (N × H)
5. C2×Q8×D197C2semidirect
6. C2×D4×D197C2semidirect
7. C22×C4×D197C2semidirect
8. C2×#41×C197C2semidirect
10. #15691359C22semidirect
11. #15691358C22semidirect
12. #15691357C22semidirect
13. C2×Dic394C22semidirect
14. Q8×D197C22semidirect
15. C2×D788C22semidirect
16. C2×C22⋊D197C22semidirect
17. D4×D197C22semidirect
18. C22×Dic197C22semidirect
21. C2×C4×D197C22semidirect
22. C2×C4×D197C22semidirect
23. C2×C4×D197C22semidirect
24. #41×C197C22semidirect
25. C2×Q8×C197C22semidirect
26. C2×D4×C197C22semidirect
27. C22×C4×C197C22semidirect
32. Dic394C23semidirect
33. D788C23semidirect
34. C22⋊D197C23semidirect
37. C2×Dic197C23semidirect
40. C4×D197C23semidirect
41. C4×D197C23semidirect
43. Q8×C197C23semidirect
44. D4×C197C23semidirect
47. C2×C4×C197C23semidirect
53. C2×D197#41direct (N × H)
61. C2×C197C2×#41semidirect
64. D197C2×#41direct (N × H)
65. C197C22×#41semidirect
71. C2×#41D197direct (N × H)
79. #41C2×D197direct (N × H)
82. C2×Q8C2×D197semidirect
86. C2×D4C2×D197semidirect
93. C22×C4C2×D197semidirect
100. Q8C22×D197semidirect
103. D4C22×D197semidirect
113. C2×C4C22×D197semidirect
129. C2#41×D197direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156913592
C2×#156913582
C2×#156913572
#41×D1972
C2×Q8×D1972
C2×D4×D1972
C22×C4×D1972
C2×#41×C1972
C22×#41197