Finite Group Databasebeta

← C2×#26040061 #37527274

Subgroups — C2×#26040061 #37527274

Order 22976.

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#26040061C2116yesmaximal, direct product factor
2C2×#19153342C212yesmaximal, complemented
3C2×#19153341C213yesmaximal, complemented
4C22×D1436C213yesmaximal, complemented
5C2×D4×D359C216yesmaximal, complemented
6C2×#41×C359C211yesmaximal, characteristic, complemented
7C23×D359C2216yescomplemented
8#19153342C22116yescomplemented
9#19153341C22124yescomplemented
10C2×Dic718C2211yescharacteristic, complemented
11C2×D1436C2216yescomplemented
12C2×D1436C2213yescomplemented
13C2×D1436C22124yescomplemented
14C2×C22⋊D359C2216yescomplemented
15D4×D359C22148yescomplemented
16C2×C4×D359C2216yescomplemented
17#41×C359C2218yescomplemented
18C2×Q8×C359C2211yescharacteristic, complemented
19C2×D4×C359C2213yescomplemented
20C22×C4×C359C2213yescomplemented
21C22×D359—412no—
22C22×D359C2316yes—
23C22×D359C23124yescomplemented
24Dic718C2314yescomplemented
25D1436C23124yescomplemented
26D1436C23112yescomplemented
27C22⋊D359C23124yescomplemented
28C2×Dic359C2312yes—
29C4×D359C23124yescomplemented
30C23×C359C2313yes—
31Q8×C359C2314yescomplemented
32D4×C359C23112yescomplemented
33C2×C4×C359C2311yescharacteristic
34C2×C4×C359C2313yes—
35C2×C4×C359C23112yescomplemented
36C22×C359—26no—
37C22×C359C2416yes—
38C22×C359C2411yescharacteristic
39C2×D359—212no—
40C2×D359—424no—
41C2×D359C24112yes—
42C4×C359C2412yes—
43C4×C359C2416yes—
44Dic359C2414yes—
45C2×C359—26no—
46C2×C359H(5,𝔽2)12yescomplemented
47C2×C359C2511yescharacteristic
48D359—212no—
49C359C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—3591nomaximal, Sylow
51H(5,𝔽2)—35916no—
52C2×#41—3592no—
53C2×#41—3593no—
54C2×#41D35911yescharacteristic, complemented
55C22×D4—3596no—
56C22×D4—3593no—
57C24—3596no—
58#41—35916no—
59#41—35924no—
60#41C2×D35918yescomplemented
61C2×Q8—3591no—
62C2×Q8C2×D35911yescharacteristic, complemented
63C2×D4—3596no—
64C2×D4—35948no—
65C2×D4—3596no—
66C2×D4—3593no—
67C2×D4—35924no—
68C2×D4C2×D35913yescomplemented
69C22×C4—3596no—
70C22×C4C2×D35913yescomplemented
71C23—35924no—
72C23—3596no—
73C23—143612no—
74C23C22×D35913yes—
75Q8—3594no—
76Q8C22×D35914yescomplemented
77D4—35924no—
78D4—35924no—
79D4—35912no—
80D4C22×D359112yescomplemented
81C2×C4—3592no—
82C2×C4—35924no—
83C2×C4C22×D35913yes—
84C2×C4C22×D35911yescharacteristic
85C2×C4C22×D359112yescomplemented
86C22—26no—
87C22—35912no—
88C22—143624no—
89C22—71812no—
90C22C23×D35916yes—
91C22C23×D35911yescharacteristic
92C4—3594no—
93C4C23×D35916yes—
94C4C23×D35912yes—
95C2—26no—
96C2—71812no—
97C2#2604006112yesdirect product factor
98C2C24×D35911yescharacteristic
subgroupG/H
centerC22C23×D359
commutator, cosocleC2×C359C25
FrattiniC2C24×D359
FittingC2×#41×C359C2
socleC22×C359C24
hypercenterC2×#41D359
hypocenterC359C2×H(5,𝔽2)
LCSG ▷ C2×C359 ▷ C359
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26040061C2direct (N × H)
2. C2×#19153342C2semidirect
3. C2×#19153341C2semidirect
4. C22×D1436C2semidirect
5. C2×D4×D359C2semidirect
6. C2×#41×C359C2semidirect
7. C23×D359C22semidirect
8. #19153342C22semidirect
9. #19153341C22semidirect
10. C2×Dic718C22semidirect
11. C2×D1436C22semidirect
12. C2×D1436C22semidirect
13. C2×D1436C22semidirect
14. C2×C22⋊D359C22semidirect
15. D4×D359C22semidirect
16. C2×C4×D359C22semidirect
17. #41×C359C22semidirect
18. C2×Q8×C359C22semidirect
19. C2×D4×C359C22semidirect
20. C22×C4×C359C22semidirect
23. C22×D359C23semidirect
24. Dic718C23semidirect
25. D1436C23semidirect
26. D1436C23semidirect
27. C22⋊D359C23semidirect
29. C4×D359C23semidirect
31. Q8×C359C23semidirect
32. D4×C359C23semidirect
35. C2×C4×C359C23semidirect
46. C2×C359H(5,𝔽2)semidirect
49. C359C2×H(5,𝔽2)semidirect
54. C2×#41D359semidirect
60. #41C2×D359semidirect
62. C2×Q8C2×D359semidirect
68. C2×D4C2×D359semidirect
70. C22×C4C2×D359semidirect
76. Q8C22×D359semidirect
80. D4C22×D359semidirect
85. C2×C4C22×D359semidirect
97. C2#26040061direct (N × H)

Maximal subgroups

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

subgroupindex
#260400612
C2×#191533422
C2×#191533412
C22×D14362
C2×D4×D3592
C2×#41×C3592
C2×H(5,𝔽2)359