Finite Group Databasebeta

← C2×#27211764 #41001192

Subgroups — C2×#27211764 #41001192

Order 26816.

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#27211764C2116yesmaximal, direct product factor
2C2×#20269412C212yesmaximal, complemented
3C2×#20269411C213yesmaximal, complemented
4C22×D1676C213yesmaximal, complemented
5C2×D4×D419C216yesmaximal, complemented
6C2×#41×C419C211yesmaximal, characteristic, complemented
7C23×D419C2216yescomplemented
8#20269412C22116yescomplemented
9#20269411C22124yescomplemented
10C2×Dic838C2211yescharacteristic, complemented
11C2×D1676C2216yescomplemented
12C2×D1676C2213yescomplemented
13C2×D1676C22124yescomplemented
14C2×C22⋊D419C2216yescomplemented
15D4×D419C22148yescomplemented
16C2×C4×D419C2216yescomplemented
17#41×C419C2218yescomplemented
18C2×Q8×C419C2211yescharacteristic, complemented
19C2×D4×C419C2213yescomplemented
20C22×C4×C419C2213yescomplemented
21C22×D419—412no—
22C22×D419C2316yes—
23C22×D419C23124yescomplemented
24Dic838C2314yescomplemented
25D1676C23124yescomplemented
26D1676C23112yescomplemented
27C22⋊D419C23124yescomplemented
28C2×Dic419C2312yes—
29C4×D419C23124yescomplemented
30C23×C419C2313yes—
31Q8×C419C2314yescomplemented
32D4×C419C23112yescomplemented
33C2×C4×C419C2311yescharacteristic
34C2×C4×C419C2313yes—
35C2×C4×C419C23112yescomplemented
36C22×C419—26no—
37C22×C419C2416yes—
38C22×C419C2411yescharacteristic
39C2×D419—212no—
40C2×D419—424no—
41C2×D419C24112yes—
42C4×C419C2412yes—
43C4×C419C2416yes—
44Dic419C2414yes—
45C2×C419—26no—
46C2×C419H(5,𝔽2)12yescomplemented
47C2×C419C2511yescharacteristic
48D419—212no—
49C419C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—4191nomaximal, Sylow
51H(5,𝔽2)—41916no—
52C2×#41—4192no—
53C2×#41—4193no—
54C2×#41D41911yescharacteristic, complemented
55C22×D4—4196no—
56C22×D4—4193no—
57C24—4196no—
58#41—41916no—
59#41—41924no—
60#41C2×D41918yescomplemented
61C2×Q8—4191no—
62C2×Q8C2×D41911yescharacteristic, complemented
63C2×D4—4196no—
64C2×D4—41948no—
65C2×D4—4196no—
66C2×D4—4193no—
67C2×D4—41924no—
68C2×D4C2×D41913yescomplemented
69C22×C4—4196no—
70C22×C4C2×D41913yescomplemented
71C23—167612no—
72C23—4196no—
73C23—41924no—
74C23C22×D41913yes—
75Q8—4194no—
76Q8C22×D41914yescomplemented
77D4—41924no—
78D4—41924no—
79D4—41912no—
80D4C22×D419112yescomplemented
81C2×C4—4192no—
82C2×C4—41924no—
83C2×C4C22×D41913yes—
84C2×C4C22×D41911yescharacteristic
85C2×C4C22×D419112yescomplemented
86C22—26no—
87C22—83812no—
88C22—167624no—
89C22—41912no—
90C22C23×D41916yes—
91C22C23×D41911yescharacteristic
92C4—4194no—
93C4C23×D41916yes—
94C4C23×D41912yes—
95C2—26no—
96C2—83812no—
97C2#2721176412yesdirect product factor
98C2C24×D41911yescharacteristic
subgroupG/H
centerC22C23×D419
commutator, cosocleC2×C419C25
FrattiniC2C24×D419
FittingC2×#41×C419C2
socleC22×C419C24
hypercenterC2×#41D419
hypocenterC419C2×H(5,𝔽2)
LCSG ▷ C2×C419 ▷ C419
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27211764C2direct (N × H)
2. C2×#20269412C2semidirect
3. C2×#20269411C2semidirect
4. C22×D1676C2semidirect
5. C2×D4×D419C2semidirect
6. C2×#41×C419C2semidirect
7. C23×D419C22semidirect
8. #20269412C22semidirect
9. #20269411C22semidirect
10. C2×Dic838C22semidirect
11. C2×D1676C22semidirect
12. C2×D1676C22semidirect
13. C2×D1676C22semidirect
14. C2×C22⋊D419C22semidirect
15. D4×D419C22semidirect
16. C2×C4×D419C22semidirect
17. #41×C419C22semidirect
18. C2×Q8×C419C22semidirect
19. C2×D4×C419C22semidirect
20. C22×C4×C419C22semidirect
23. C22×D419C23semidirect
24. Dic838C23semidirect
25. D1676C23semidirect
26. D1676C23semidirect
27. C22⋊D419C23semidirect
29. C4×D419C23semidirect
31. Q8×C419C23semidirect
32. D4×C419C23semidirect
35. C2×C4×C419C23semidirect
46. C2×C419H(5,𝔽2)semidirect
49. C419C2×H(5,𝔽2)semidirect
54. C2×#41D419semidirect
60. #41C2×D419semidirect
62. C2×Q8C2×D419semidirect
68. C2×D4C2×D419semidirect
70. C22×C4C2×D419semidirect
76. Q8C22×D419semidirect
80. D4C22×D419semidirect
85. C2×C4C22×D419semidirect
97. C2#27211764direct (N × H)

Maximal subgroups

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

subgroupindex
#272117642
C2×#202694122
C2×#202694112
C22×D16762
C2×D4×D4192
C2×#41×C4192
C2×H(5,𝔽2)419