Finite Group Databasebeta

← C22×#22575499 #46825687

Subgroups — C22×#22575499 #46825687

Order 34432.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#22575499C2148yesmaximal, direct product factor
2C22×#16864403C212yesmaximal, complemented
3C22×#16864402C213yesmaximal, complemented
4C23×D1076C213yesmaximal, complemented
5C22×D4×D269C216yesmaximal, complemented
6C22×#41×C269C211yesmaximal, characteristic, complemented
7C24×D269C2216yescomplemented
8#22575499C221256yesdirect product factor
9C2×#16864403C22148yescomplemented
10C2×#16864402C22172yescomplemented
11C22×Dic538C2211yescharacteristic, complemented
12C22×D1076C2216yescomplemented
13C22×D1076C2213yescomplemented
14C22×D1076C22172yescomplemented
15C22×C22⋊D269C2216yescomplemented
16C2×D4×D269C221144yescomplemented
17C22×C4×D269C2216yescomplemented
18C2×#41×C269C22124yescomplemented
19C22×Q8×C269C2211yescharacteristic, complemented
20C22×D4×C269C2213yescomplemented
21C23×C4×C269C2213yescomplemented
22C23×D269—424no—
23C23×D269C2316yes—
24C23×D269C23172yescomplemented
25#16864403C231128yescomplemented
26#16864402C231192yescomplemented
27C2×Dic538C23112yescomplemented
28C2×D1076C23172yescomplemented
29C2×D1076C23136yescomplemented
30C2×D1076C231192yescomplemented
31C2×C22⋊D269C23172yescomplemented
32D4×D269C231384yescomplemented
33C22×Dic269C2312yes—
34C2×C4×D269C23172yescomplemented
35C24×C269C2313yes—
36#41×C269C23164yescomplemented
37C2×Q8×C269C23112yescomplemented
38C2×D4×C269C23136yescomplemented
39C22×C4×C269C2313yes—
40C22×C4×C269C2311yescharacteristic
41C22×C4×C269C23136yescomplemented
42C22×D269—224no—
43C22×D269—4144no—
44C22×D269C24136yes—
45C22×D269C24196yescomplemented
46Dic538C24116yescomplemented
47D1076C24196yescomplemented
48D1076C24148yescomplemented
49C22⋊D269C24196yescomplemented
50C2×Dic269C24112yes—
51C4×D269C24196yescomplemented
52C23×C269—212no—
53C23×C269C24118yes—
54C23×C269C2411yescharacteristic
55Q8×C269C24116yescomplemented
56D4×C269C24148yescomplemented
57C2×C4×C269C2416yes—
58C2×C4×C269C24118yes—
59C2×C4×C269C24148yescomplemented
60C22×C269—236no—
61C22×C269H(5,𝔽2)14yescomplemented
62C22×C269C25112yes—
63C22×C269C2513yes—
64C2×D269—272no—
65C2×D269—496no—
66C2×D269C25124yes—
67C4×C269C2514yes—
68C4×C269C25112yes—
69Dic269C2518yes—
70C2×C269—212no—
71C2×C269C2×H(5,𝔽2)16yescomplemented
72C2×C269C2611yescharacteristic
73D269—224no—
74C269C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—2691nomaximal, Sylow
76C2×H(5,𝔽2)—26948no—
77C22×#41—2692no—
78C22×#41—2693no—
79C22×#41D26911yescharacteristic, complemented
80C23×D4—2696no—
81C23×D4—2693no—
82C25—2696no—
83H(5,𝔽2)—269256no—
84C2×#41—26948no—
85C2×#41—26972no—
86C2×#41C2×D269124yescomplemented
87C22×Q8—2691no—
88C22×Q8C2×D26911yescharacteristic, complemented
89C22×D4—2696no—
90C22×D4—269144no—
91C22×D4—2696no—
92C22×D4—2693no—
93C22×D4—26972no—
94C22×D4C2×D26913yescomplemented
95C23×C4—2696no—
96C23×C4C2×D26913yescomplemented
97C24—2696no—
98C24—26972no—
99C24—107624no—
100C24C22×D26913yes—
101#41—269128no—
102#41—269192no—
103#41C22×D269164yescomplemented
104C2×Q8—26912no—
105C2×Q8C22×D269112yescomplemented
106C2×D4—26972no—
107C2×D4—269384no—
108C2×D4—26972no—
109C2×D4—26936no—
110C2×D4—269192no—
111C2×D4C22×D269136yescomplemented
112C22×C4—2692no—
113C22×C4—26972no—
114C22×C4C22×D26913yes—
115C22×C4C22×D26911yescharacteristic
116C22×C4C22×D269136yescomplemented
117C23—212no—
118C23—26936no—
119C23—26996no—
120C23—53824no—
121C23—1076144no—
122C23C23×D269118yes—
123C23C23×D26911yescharacteristic
124Q8—26916no—
125Q8C23×D269116yescomplemented
126D4—26996no—
127D4—26996no—
128D4—26948no—
129D4C23×D269148yescomplemented
130C2×C4—26912no—
131C2×C4—26996no—
132C2×C4C23×D269118yes—
133C2×C4C23×D26916yes—
134C2×C4C23×D269148yescomplemented
135C22—236no—
136C22—26924no—
137C22—53872no—
138C22—107696no—
139C22#2257549914yesdirect product factor
140C22C24×D269112yes—
141C22C24×D26913yes—
142C4—2698no—
143C4C24×D269112yes—
144C4C24×D26914yes—
145C2—212no—
146C2—53824no—
147C2C2×#2257549916yesdirect product factor
148C2C25×D26911yescharacteristic
subgroupG/H
centerC23C23×D269
commutator, cosocleC2×C269C26
FrattiniC2C25×D269
FittingC22×#41×C269C2
socleC23×C269C24
hypercenterC22×#41D269
hypocenterC269C22×H(5,𝔽2)
LCSG ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2direct (N × H)
2. C22×#16864403C2semidirect
3. C22×#16864402C2semidirect
4. C23×D1076C2semidirect
5. C22×D4×D269C2semidirect
6. C22×#41×C269C2semidirect
7. C24×D269C22semidirect
8. #22575499C22direct (N × H)
9. C2×#16864403C22semidirect
10. C2×#16864402C22semidirect
11. C22×Dic538C22semidirect
12. C22×D1076C22semidirect
13. C22×D1076C22semidirect
14. C22×D1076C22semidirect
15. C22×C22⋊D269C22semidirect
16. C2×D4×D269C22semidirect
17. C22×C4×D269C22semidirect
18. C2×#41×C269C22semidirect
19. C22×Q8×C269C22semidirect
20. C22×D4×C269C22semidirect
21. C23×C4×C269C22semidirect
24. C23×D269C23semidirect
25. #16864403C23semidirect
26. #16864402C23semidirect
27. C2×Dic538C23semidirect
28. C2×D1076C23semidirect
29. C2×D1076C23semidirect
30. C2×D1076C23semidirect
31. C2×C22⋊D269C23semidirect
32. D4×D269C23semidirect
34. C2×C4×D269C23semidirect
36. #41×C269C23semidirect
37. C2×Q8×C269C23semidirect
38. C2×D4×C269C23semidirect
41. C22×C4×C269C23semidirect
45. C22×D269C24semidirect
46. Dic538C24semidirect
47. D1076C24semidirect
48. D1076C24semidirect
49. C22⋊D269C24semidirect
51. C4×D269C24semidirect
55. Q8×C269C24semidirect
56. D4×C269C24semidirect
59. C2×C4×C269C24semidirect
61. C22×C269H(5,𝔽2)semidirect
71. C2×C269C2×H(5,𝔽2)semidirect
74. C269C22×H(5,𝔽2)semidirect
79. C22×#41D269semidirect
86. C2×#41C2×D269semidirect
88. C22×Q8C2×D269semidirect
94. C22×D4C2×D269semidirect
96. C23×C4C2×D269semidirect
103. #41C22×D269semidirect
105. C2×Q8C22×D269semidirect
111. C2×D4C22×D269semidirect
116. C22×C4C22×D269semidirect
125. Q8C23×D269semidirect
129. D4C23×D269semidirect
134. C2×C4C23×D269semidirect
139. C22#22575499direct (N × H)
147. C2C2×#22575499direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225754992
C22×#168644032
C22×#168644022
C23×D10762
C22×D4×D2692
C22×#41×C2692
C22×H(5,𝔽2)269