Finite Group Databasebeta

← C2×#20259945 #27136090

Subgroups — C2×#20259945 #27136090

Order 12352.

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#20259945C2116yesmaximal, direct product factor
2C2×#15690556C212yesmaximal, complemented
3C2×#15690555C213yesmaximal, complemented
4C22×D772C213yesmaximal, complemented
5C2×D4×D193C216yesmaximal, complemented
6C2×#41×C193C211yesmaximal, characteristic, complemented
7C23×D193C2216yescomplemented
8#15690556C22116yescomplemented
9#15690555C22124yescomplemented
10C2×Dic386C2211yescharacteristic, complemented
11C2×D772C2216yescomplemented
12C2×D772C2213yescomplemented
13C2×D772C22124yescomplemented
14C2×C22⋊D193C2216yescomplemented
15D4×D193C22148yescomplemented
16C2×C4×D193C2216yescomplemented
17#41×C193C2218yescomplemented
18C2×Q8×C193C2211yescharacteristic, complemented
19C2×D4×C193C2213yescomplemented
20C22×C4×C193C2213yescomplemented
21C22×D193—412no—
22C22×D193C2316yes—
23C22×D193C23124yescomplemented
24Dic386C2314yescomplemented
25D772C23124yescomplemented
26D772C23112yescomplemented
27C22⋊D193C23124yescomplemented
28C2×Dic193C2312yes—
29C4×D193C23124yescomplemented
30C23×C193C2313yes—
31Q8×C193C2314yescomplemented
32D4×C193C23112yescomplemented
33C2×C4×C193C2311yescharacteristic
34C2×C4×C193C2313yes—
35C2×C4×C193C23112yescomplemented
36C22×C193—26no—
37C22×C193C2416yes—
38C22×C193C2411yescharacteristic
39C2×D193—212no—
40C2×D193—424no—
41C2×D193C24112yes—
42C4×C193C2412yes—
43C4×C193C2416yes—
44Dic193C2414yes—
45C2×C193—26no—
46C2×C193H(5,𝔽2)12yescomplemented
47C2×C193C2511yescharacteristic
48D193—212no—
49C193C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—1931nomaximal, Sylow
51H(5,𝔽2)—19316no—
52C2×#41—1932no—
53C2×#41—1933no—
54C2×#41D19311yescharacteristic, complemented
55C22×D4—1936no—
56C22×D4—1933no—
57C24—1936no—
58#41—19316no—
59#41—19324no—
60#41C2×D19318yescomplemented
61C2×Q8—1931no—
62C2×Q8C2×D19311yescharacteristic, complemented
63C2×D4—1936no—
64C2×D4—19348no—
65C2×D4—1936no—
66C2×D4—1933no—
67C2×D4—19324no—
68C2×D4C2×D19313yescomplemented
69C22×C4—1936no—
70C22×C4C2×D19313yescomplemented
71C23—77212no—
72C23—19324no—
73C23—1936no—
74C23C22×D19313yes—
75Q8—1934no—
76Q8C22×D19314yescomplemented
77D4—19324no—
78D4—19324no—
79D4—19312no—
80D4C22×D193112yescomplemented
81C2×C4—1932no—
82C2×C4—19324no—
83C2×C4C22×D19313yes—
84C2×C4C22×D19311yescharacteristic
85C2×C4C22×D193112yescomplemented
86C22—26no—
87C22—77224no—
88C22—38612no—
89C22—19312no—
90C22C23×D19316yes—
91C22C23×D19311yescharacteristic
92C4—1934no—
93C4C23×D19316yes—
94C4C23×D19312yes—
95C2—26no—
96C2—38612no—
97C2#2025994512yesdirect product factor
98C2C24×D19311yescharacteristic
subgroupG/H
centerC22C23×D193
commutator, cosocleC2×C193C25
FrattiniC2C24×D193
FittingC2×#41×C193C2
socleC22×C193C24
hypercenterC2×#41D193
hypocenterC193C2×H(5,𝔽2)
LCSG ▷ C2×C193 ▷ C193
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20259945C2direct (N × H)
2. C2×#15690556C2semidirect
3. C2×#15690555C2semidirect
4. C22×D772C2semidirect
5. C2×D4×D193C2semidirect
6. C2×#41×C193C2semidirect
7. C23×D193C22semidirect
8. #15690556C22semidirect
9. #15690555C22semidirect
10. C2×Dic386C22semidirect
11. C2×D772C22semidirect
12. C2×D772C22semidirect
13. C2×D772C22semidirect
14. C2×C22⋊D193C22semidirect
15. D4×D193C22semidirect
16. C2×C4×D193C22semidirect
17. #41×C193C22semidirect
18. C2×Q8×C193C22semidirect
19. C2×D4×C193C22semidirect
20. C22×C4×C193C22semidirect
23. C22×D193C23semidirect
24. Dic386C23semidirect
25. D772C23semidirect
26. D772C23semidirect
27. C22⋊D193C23semidirect
29. C4×D193C23semidirect
31. Q8×C193C23semidirect
32. D4×C193C23semidirect
35. C2×C4×C193C23semidirect
46. C2×C193H(5,𝔽2)semidirect
49. C193C2×H(5,𝔽2)semidirect
54. C2×#41D193semidirect
60. #41C2×D193semidirect
62. C2×Q8C2×D193semidirect
68. C2×D4C2×D193semidirect
70. C22×C4C2×D193semidirect
76. Q8C22×D193semidirect
80. D4C22×D193semidirect
85. C2×C4C22×D193semidirect
97. C2#20259945direct (N × H)

Maximal subgroups

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

subgroupindex
#202599452
C2×#156905562
C2×#156905552
C22×D7722
C2×D4×D1932
C2×#41×C1932
C2×H(5,𝔽2)193