Finite Group Databasebeta

← C2×#27165783 #38786945

Subgroups — C2×#27165783 #38786945

Order 25664.

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#27165783C2116yesmaximal, direct product factor
2C2×#20264367C212yesmaximal, complemented
3C2×#20264366C213yesmaximal, complemented
4C22×D1604C213yesmaximal, complemented
5C2×D4×D401C216yesmaximal, complemented
6C2×#41×C401C211yesmaximal, characteristic, complemented
7C23×D401C2216yescomplemented
8#20264367C22116yescomplemented
9#20264366C22124yescomplemented
10C2×Dic802C2211yescharacteristic, complemented
11C2×D1604C2216yescomplemented
12C2×D1604C2213yescomplemented
13C2×D1604C22124yescomplemented
14C2×C22⋊D401C2216yescomplemented
15D4×D401C22148yescomplemented
16C2×C4×D401C2216yescomplemented
17#41×C401C2218yescomplemented
18C2×Q8×C401C2211yescharacteristic, complemented
19C2×D4×C401C2213yescomplemented
20C22×C4×C401C2213yescomplemented
21C22×D401—412no—
22C22×D401C2316yes—
23C22×D401C23124yescomplemented
24Dic802C2314yescomplemented
25D1604C23124yescomplemented
26D1604C23112yescomplemented
27C22⋊D401C23124yescomplemented
28C2×Dic401C2312yes—
29C4×D401C23124yescomplemented
30C23×C401C2313yes—
31Q8×C401C2314yescomplemented
32D4×C401C23112yescomplemented
33C2×C4×C401C2311yescharacteristic
34C2×C4×C401C2313yes—
35C2×C4×C401C23112yescomplemented
36C22×C401—26no—
37C22×C401C2416yes—
38C22×C401C2411yescharacteristic
39C2×D401—212no—
40C2×D401—424no—
41C2×D401C24112yes—
42C4×C401C2412yes—
43C4×C401C2416yes—
44Dic401C2414yes—
45C2×C401—26no—
46C2×C401H(5,𝔽2)12yescomplemented
47C2×C401C2511yescharacteristic
48D401—212no—
49C401C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—4011nomaximal, Sylow
51H(5,𝔽2)—40116no—
52C2×#41—4012no—
53C2×#41—4013no—
54C2×#41D40111yescharacteristic, complemented
55C22×D4—4016no—
56C22×D4—4013no—
57C24—4016no—
58#41—40116no—
59#41—40124no—
60#41C2×D40118yescomplemented
61C2×Q8—4011no—
62C2×Q8C2×D40111yescharacteristic, complemented
63C2×D4—4016no—
64C2×D4—40148no—
65C2×D4—4016no—
66C2×D4—4013no—
67C2×D4—40124no—
68C2×D4C2×D40113yescomplemented
69C22×C4—4016no—
70C22×C4C2×D40113yescomplemented
71C23—160412no—
72C23—4016no—
73C23—40124no—
74C23C22×D40113yes—
75Q8—4014no—
76Q8C22×D40114yescomplemented
77D4—40124no—
78D4—40124no—
79D4—40112no—
80D4C22×D401112yescomplemented
81C2×C4—4012no—
82C2×C4—40124no—
83C2×C4C22×D40113yes—
84C2×C4C22×D40111yescharacteristic
85C2×C4C22×D401112yescomplemented
86C22—26no—
87C22—80212no—
88C22—160424no—
89C22—40112no—
90C22C23×D40116yes—
91C22C23×D40111yescharacteristic
92C4—4014no—
93C4C23×D40116yes—
94C4C23×D40112yes—
95C2—26no—
96C2—80212no—
97C2#2716578312yesdirect product factor
98C2C24×D40111yescharacteristic
subgroupG/H
centerC22C23×D401
commutator, cosocleC2×C401C25
FrattiniC2C24×D401
FittingC2×#41×C401C2
socleC22×C401C24
hypercenterC2×#41D401
hypocenterC401C2×H(5,𝔽2)
LCSG ▷ C2×C401 ▷ C401
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27165783C2direct (N × H)
2. C2×#20264367C2semidirect
3. C2×#20264366C2semidirect
4. C22×D1604C2semidirect
5. C2×D4×D401C2semidirect
6. C2×#41×C401C2semidirect
7. C23×D401C22semidirect
8. #20264367C22semidirect
9. #20264366C22semidirect
10. C2×Dic802C22semidirect
11. C2×D1604C22semidirect
12. C2×D1604C22semidirect
13. C2×D1604C22semidirect
14. C2×C22⋊D401C22semidirect
15. D4×D401C22semidirect
16. C2×C4×D401C22semidirect
17. #41×C401C22semidirect
18. C2×Q8×C401C22semidirect
19. C2×D4×C401C22semidirect
20. C22×C4×C401C22semidirect
23. C22×D401C23semidirect
24. Dic802C23semidirect
25. D1604C23semidirect
26. D1604C23semidirect
27. C22⋊D401C23semidirect
29. C4×D401C23semidirect
31. Q8×C401C23semidirect
32. D4×C401C23semidirect
35. C2×C4×C401C23semidirect
46. C2×C401H(5,𝔽2)semidirect
49. C401C2×H(5,𝔽2)semidirect
54. C2×#41D401semidirect
60. #41C2×D401semidirect
62. C2×Q8C2×D401semidirect
68. C2×D4C2×D401semidirect
70. C22×C4C2×D401semidirect
76. Q8C22×D401semidirect
80. D4C22×D401semidirect
85. C2×C4C22×D401semidirect
97. C2#27165783direct (N × H)

Maximal subgroups

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

subgroupindex
#271657832
C2×#202643672
C2×#202643662
C22×D16042
C2×D4×D4012
C2×#41×C4012
C2×H(5,𝔽2)401