Finite Group Databasebeta

← C2×#20264001 #27164849

Subgroups — C2×#20264001 #27164849

Order 12736.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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
1#20264001C2116yesmaximal, direct product factor
2C2×#15691887C212yesmaximal, complemented
3C2×#15691886C213yesmaximal, complemented
4C22×D796C213yesmaximal, complemented
5C2×D4×D199C216yesmaximal, complemented
6C2×#41×C199C211yesmaximal, characteristic, complemented
7C23×D199C2216yescomplemented
8#15691887C22116yescomplemented
9#15691886C22124yescomplemented
10C2×Dic398C2211yescharacteristic, complemented
11C2×D796C2216yescomplemented
12C2×D796C2213yescomplemented
13C2×D796C22124yescomplemented
14C2×C22⋊D199C2216yescomplemented
15D4×D199C22148yescomplemented
16C2×C4×D199C2216yescomplemented
17#41×C199C2218yescomplemented
18C2×Q8×C199C2211yescharacteristic, complemented
19C2×D4×C199C2213yescomplemented
20C22×C4×C199C2213yescomplemented
21C22×D199—412no—
22C22×D199C2316yes—
23C22×D199C23124yescomplemented
24Dic398C2314yescomplemented
25D796C23124yescomplemented
26D796C23112yescomplemented
27C22⋊D199C23124yescomplemented
28C2×Dic199C2312yes—
29C4×D199C23124yescomplemented
30C23×C199C2313yes—
31Q8×C199C2314yescomplemented
32D4×C199C23112yescomplemented
33C2×C4×C199C2311yescharacteristic
34C2×C4×C199C2313yes—
35C2×C4×C199C23112yescomplemented
36C22×C199—26no—
37C22×C199C2416yes—
38C22×C199C2411yescharacteristic
39C2×D199—212no—
40C2×D199—424no—
41C2×D199C24112yes—
42C4×C199C2412yes—
43C4×C199C2416yes—
44Dic199C2414yes—
45C2×C199—26no—
46C2×C199H(5,𝔽2)12yescomplemented
47C2×C199C2511yescharacteristic
48D199—212no—
49C199C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—1991nomaximal, Sylow
51H(5,𝔽2)—19916no—
52C2×#41—1992no—
53C2×#41—1993no—
54C2×#41D19911yescharacteristic, complemented
55C22×D4—1996no—
56C22×D4—1993no—
57C24—1996no—
58#41—19916no—
59#41—19924no—
60#41C2×D19918yescomplemented
61C2×Q8—1991no—
62C2×Q8C2×D19911yescharacteristic, complemented
63C2×D4—1996no—
64C2×D4—19948no—
65C2×D4—1996no—
66C2×D4—1993no—
67C2×D4—19924no—
68C2×D4C2×D19913yescomplemented
69C22×C4—1996no—
70C22×C4C2×D19913yescomplemented
71C23—79612no—
72C23—19924no—
73C23—1996no—
74C23C22×D19913yes—
75Q8—1994no—
76Q8C22×D19914yescomplemented
77D4—19924no—
78D4—19924no—
79D4—19912no—
80D4C22×D199112yescomplemented
81C2×C4—1992no—
82C2×C4—19924no—
83C2×C4C22×D19913yes—
84C2×C4C22×D19911yescharacteristic
85C2×C4C22×D199112yescomplemented
86C22—26no—
87C22—79624no—
88C22—39812no—
89C22—19912no—
90C22C23×D19916yes—
91C22C23×D19911yescharacteristic
92C4—1994no—
93C4C23×D19916yes—
94C4C23×D19912yes—
95C2—26no—
96C2—39812no—
97C2#2026400112yesdirect product factor
98C2C24×D19911yescharacteristic
subgroupG/H
centerC22C23×D199
commutator, cosocleC2×C199C25
FrattiniC2C24×D199
FittingC2×#41×C199C2
socleC22×C199C24
hypercenterC2×#41D199
hypocenterC199C2×H(5,𝔽2)
LCSG ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20264001C2direct (N × H)
2. C2×#15691887C2semidirect
3. C2×#15691886C2semidirect
4. C22×D796C2semidirect
5. C2×D4×D199C2semidirect
6. C2×#41×C199C2semidirect
7. C23×D199C22semidirect
8. #15691887C22semidirect
9. #15691886C22semidirect
10. C2×Dic398C22semidirect
11. C2×D796C22semidirect
12. C2×D796C22semidirect
13. C2×D796C22semidirect
14. C2×C22⋊D199C22semidirect
15. D4×D199C22semidirect
16. C2×C4×D199C22semidirect
17. #41×C199C22semidirect
18. C2×Q8×C199C22semidirect
19. C2×D4×C199C22semidirect
20. C22×C4×C199C22semidirect
23. C22×D199C23semidirect
24. Dic398C23semidirect
25. D796C23semidirect
26. D796C23semidirect
27. C22⋊D199C23semidirect
29. C4×D199C23semidirect
31. Q8×C199C23semidirect
32. D4×C199C23semidirect
35. C2×C4×C199C23semidirect
46. C2×C199H(5,𝔽2)semidirect
49. C199C2×H(5,𝔽2)semidirect
54. C2×#41D199semidirect
60. #41C2×D199semidirect
62. C2×Q8C2×D199semidirect
68. C2×D4C2×D199semidirect
70. C22×C4C2×D199semidirect
76. Q8C22×D199semidirect
80. D4C22×D199semidirect
85. C2×C4C22×D199semidirect
97. C2#20264001direct (N × H)

Maximal subgroups

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

subgroupindex
#202640012
C2×#156918872
C2×#156918862
C22×D7962
C2×D4×D1992
C2×#41×C1992
C2×H(5,𝔽2)199