Finite Group Databasebeta

← C2×#20259368 #27133494

Subgroups — C2×#20259368 #27133494

Order 12224.

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#20259368C2116yesmaximal, direct product factor
2C2×#15690288C212yesmaximal, complemented
3C2×#15690287C213yesmaximal, complemented
4C22×D764C213yesmaximal, complemented
5C2×D4×D191C216yesmaximal, complemented
6C2×#41×C191C211yesmaximal, characteristic, complemented
7C23×D191C2216yescomplemented
8#15690288C22116yescomplemented
9#15690287C22124yescomplemented
10C2×Dic382C2211yescharacteristic, complemented
11C2×D764C2216yescomplemented
12C2×D764C2213yescomplemented
13C2×D764C22124yescomplemented
14C2×C22⋊D191C2216yescomplemented
15D4×D191C22148yescomplemented
16C2×C4×D191C2216yescomplemented
17#41×C191C2218yescomplemented
18C2×Q8×C191C2211yescharacteristic, complemented
19C2×D4×C191C2213yescomplemented
20C22×C4×C191C2213yescomplemented
21C22×D191—412no—
22C22×D191C2316yes—
23C22×D191C23124yescomplemented
24Dic382C2314yescomplemented
25D764C23124yescomplemented
26D764C23112yescomplemented
27C22⋊D191C23124yescomplemented
28C2×Dic191C2312yes—
29C4×D191C23124yescomplemented
30C23×C191C2313yes—
31Q8×C191C2314yescomplemented
32D4×C191C23112yescomplemented
33C2×C4×C191C2311yescharacteristic
34C2×C4×C191C2313yes—
35C2×C4×C191C23112yescomplemented
36C22×C191—26no—
37C22×C191C2416yes—
38C22×C191C2411yescharacteristic
39C2×D191—212no—
40C2×D191—424no—
41C2×D191C24112yes—
42C4×C191C2412yes—
43C4×C191C2416yes—
44Dic191C2414yes—
45C2×C191—26no—
46C2×C191H(5,𝔽2)12yescomplemented
47C2×C191C2511yescharacteristic
48D191—212no—
49C191C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—1911nomaximal, Sylow
51H(5,𝔽2)—19116no—
52C2×#41—1912no—
53C2×#41—1913no—
54C2×#41D19111yescharacteristic, complemented
55C22×D4—1916no—
56C22×D4—1913no—
57C24—1916no—
58#41—19116no—
59#41—19124no—
60#41C2×D19118yescomplemented
61C2×Q8—1911no—
62C2×Q8C2×D19111yescharacteristic, complemented
63C2×D4—1916no—
64C2×D4—19148no—
65C2×D4—1916no—
66C2×D4—1913no—
67C2×D4—19124no—
68C2×D4C2×D19113yescomplemented
69C22×C4—1916no—
70C22×C4C2×D19113yescomplemented
71C23—1916no—
72C23—19124no—
73C23—76412no—
74C23C22×D19113yes—
75Q8—1914no—
76Q8C22×D19114yescomplemented
77D4—19124no—
78D4—19124no—
79D4—19112no—
80D4C22×D191112yescomplemented
81C2×C4—1912no—
82C2×C4—19124no—
83C2×C4C22×D19113yes—
84C2×C4C22×D19111yescharacteristic
85C2×C4C22×D191112yescomplemented
86C22—26no—
87C22—38212no—
88C22—19112no—
89C22—76424no—
90C22C23×D19116yes—
91C22C23×D19111yescharacteristic
92C4—1914no—
93C4C23×D19116yes—
94C4C23×D19112yes—
95C2—26no—
96C2—38212no—
97C2#2025936812yesdirect product factor
98C2C24×D19111yescharacteristic
subgroupG/H
centerC22C23×D191
commutator, cosocleC2×C191C25
FrattiniC2C24×D191
FittingC2×#41×C191C2
socleC22×C191C24
hypercenterC2×#41D191
hypocenterC191C2×H(5,𝔽2)
LCSG ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20259368C2direct (N × H)
2. C2×#15690288C2semidirect
3. C2×#15690287C2semidirect
4. C22×D764C2semidirect
5. C2×D4×D191C2semidirect
6. C2×#41×C191C2semidirect
7. C23×D191C22semidirect
8. #15690288C22semidirect
9. #15690287C22semidirect
10. C2×Dic382C22semidirect
11. C2×D764C22semidirect
12. C2×D764C22semidirect
13. C2×D764C22semidirect
14. C2×C22⋊D191C22semidirect
15. D4×D191C22semidirect
16. C2×C4×D191C22semidirect
17. #41×C191C22semidirect
18. C2×Q8×C191C22semidirect
19. C2×D4×C191C22semidirect
20. C22×C4×C191C22semidirect
23. C22×D191C23semidirect
24. Dic382C23semidirect
25. D764C23semidirect
26. D764C23semidirect
27. C22⋊D191C23semidirect
29. C4×D191C23semidirect
31. Q8×C191C23semidirect
32. D4×C191C23semidirect
35. C2×C4×C191C23semidirect
46. C2×C191H(5,𝔽2)semidirect
49. C191C2×H(5,𝔽2)semidirect
54. C2×#41D191semidirect
60. #41C2×D191semidirect
62. C2×Q8C2×D191semidirect
68. C2×D4C2×D191semidirect
70. C22×C4C2×D191semidirect
76. Q8C22×D191semidirect
80. D4C22×D191semidirect
85. C2×C4C22×D191semidirect
97. C2#20259368direct (N × H)

Maximal subgroups

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

subgroupindex
#202593682
C2×#156902882
C2×#156902872
C22×D7642
C2×D4×D1912
C2×#41×C1912
C2×H(5,𝔽2)191