Finite Group Databasebeta

← C2×#27189763 #39910078

Subgroups — C2×#27189763 #39910078

Order 26176.

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#27189763C2116yesmaximal, direct product factor
2C2×#20266989C212yesmaximal, complemented
3C2×#20266988C213yesmaximal, complemented
4C22×D1636C213yesmaximal, complemented
5C2×D4×D409C216yesmaximal, complemented
6C2×#41×C409C211yesmaximal, characteristic, complemented
7C23×D409C2216yescomplemented
8#20266989C22116yescomplemented
9#20266988C22124yescomplemented
10C2×Dic818C2211yescharacteristic, complemented
11C2×D1636C2213yescomplemented
12C2×D1636C2216yescomplemented
13C2×D1636C22124yescomplemented
14C2×C22⋊D409C2216yescomplemented
15D4×D409C22148yescomplemented
16C2×C4×D409C2216yescomplemented
17#41×C409C2218yescomplemented
18C2×Q8×C409C2211yescharacteristic, complemented
19C2×D4×C409C2213yescomplemented
20C22×C4×C409C2213yescomplemented
21C22×D409—412no—
22C22×D409C2316yes—
23C22×D409C23124yescomplemented
24Dic818C2314yescomplemented
25D1636C23124yescomplemented
26D1636C23112yescomplemented
27C22⋊D409C23124yescomplemented
28C2×Dic409C2312yes—
29C4×D409C23124yescomplemented
30C23×C409C2313yes—
31Q8×C409C2314yescomplemented
32D4×C409C23112yescomplemented
33C2×C4×C409C2311yescharacteristic
34C2×C4×C409C2313yes—
35C2×C4×C409C23112yescomplemented
36C22×C409—26no—
37C22×C409C2416yes—
38C22×C409C2411yescharacteristic
39C2×D409—212no—
40C2×D409—424no—
41C2×D409C24112yes—
42C4×C409C2416yes—
43C4×C409C2412yes—
44Dic409C2414yes—
45C2×C409—26no—
46C2×C409H(5,𝔽2)12yescomplemented
47C2×C409C2511yescharacteristic
48D409—212no—
49C409C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—4091nomaximal, Sylow
51H(5,𝔽2)—40916no—
52C2×#41—4092no—
53C2×#41—4093no—
54C2×#41D40911yescharacteristic, complemented
55C22×D4—4096no—
56C22×D4—4093no—
57C24—4096no—
58#41—40916no—
59#41—40924no—
60#41C2×D40918yescomplemented
61C2×Q8—4091no—
62C2×Q8C2×D40911yescharacteristic, complemented
63C2×D4—4096no—
64C2×D4—40948no—
65C2×D4—4096no—
66C2×D4—4093no—
67C2×D4—40924no—
68C2×D4C2×D40913yescomplemented
69C22×C4—4096no—
70C22×C4C2×D40913yescomplemented
71C23—163612no—
72C23—4096no—
73C23—40924no—
74C23C22×D40913yes—
75Q8—4094no—
76Q8C22×D40914yescomplemented
77D4—40924no—
78D4—40924no—
79D4—40912no—
80D4C22×D409112yescomplemented
81C2×C4—4092no—
82C2×C4—40924no—
83C2×C4C22×D40913yes—
84C2×C4C22×D40911yescharacteristic
85C2×C4C22×D409112yescomplemented
86C22—26no—
87C22—81812no—
88C22—163624no—
89C22—40912no—
90C22C23×D40916yes—
91C22C23×D40911yescharacteristic
92C4—4094no—
93C4C23×D40916yes—
94C4C23×D40912yes—
95C2—26no—
96C2—81812no—
97C2#2718976312yesdirect product factor
98C2C24×D40911yescharacteristic
subgroupG/H
centerC22C23×D409
commutator, cosocleC2×C409C25
FrattiniC2C24×D409
FittingC2×#41×C409C2
socleC22×C409C24
hypercenterC2×#41D409
hypocenterC409C2×H(5,𝔽2)
LCSG ▷ C2×C409 ▷ C409
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27189763C2direct (N × H)
2. C2×#20266989C2semidirect
3. C2×#20266988C2semidirect
4. C22×D1636C2semidirect
5. C2×D4×D409C2semidirect
6. C2×#41×C409C2semidirect
7. C23×D409C22semidirect
8. #20266989C22semidirect
9. #20266988C22semidirect
10. C2×Dic818C22semidirect
11. C2×D1636C22semidirect
12. C2×D1636C22semidirect
13. C2×D1636C22semidirect
14. C2×C22⋊D409C22semidirect
15. D4×D409C22semidirect
16. C2×C4×D409C22semidirect
17. #41×C409C22semidirect
18. C2×Q8×C409C22semidirect
19. C2×D4×C409C22semidirect
20. C22×C4×C409C22semidirect
23. C22×D409C23semidirect
24. Dic818C23semidirect
25. D1636C23semidirect
26. D1636C23semidirect
27. C22⋊D409C23semidirect
29. C4×D409C23semidirect
31. Q8×C409C23semidirect
32. D4×C409C23semidirect
35. C2×C4×C409C23semidirect
46. C2×C409H(5,𝔽2)semidirect
49. C409C2×H(5,𝔽2)semidirect
54. C2×#41D409semidirect
60. #41C2×D409semidirect
62. C2×Q8C2×D409semidirect
68. C2×D4C2×D409semidirect
70. C22×C4C2×D409semidirect
76. Q8C22×D409semidirect
80. D4C22×D409semidirect
85. C2×C4C22×D409semidirect
97. C2#27189763direct (N × H)

Maximal subgroups

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

subgroupindex
#271897632
C2×#202669892
C2×#202669882
C22×D16362
C2×D4×D4092
C2×#41×C4092
C2×H(5,𝔽2)409