Finite Group Databasebeta

← #46825695

Subgroups — #46825695

Order 34432.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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#31771441C2112yesmaximal, complemented
2#31771440C2116yesmaximal, complemented
3C2×#22575498C2118yesmaximal, complemented
4H(5,𝔽2)×D269C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C269C211yesmaximal, characteristic, complemented
6#22575501C2218yescomplemented
7#22575500C22148yescomplemented
8#22575499C22148yescomplemented
9#22575498C221144yescomplemented
10#22575498C22172yescomplemented
11C2×#16864402C22136yescomplemented
12C2×#16864401C22172yescomplemented
13#41×D269C22196yescomplemented
14C22×C22⋊D269C22124yescomplemented
15C2×D4×D269C22172yescomplemented
16H(5,𝔽2)×C269C22116yescomplemented
17C2×#41×C269C2216yescomplemented
18C22×D4×C269C2219yescomplemented
19C23×D269C23124yescomplemented
20#16864403C23132yescomplemented
21#16864402C231144yescomplemented
22#16864402C23148yescomplemented
23#16864401C231288yescomplemented
24C2×Dic538C23136yescomplemented
25Q8×D269C23132yescomplemented
26C2×D1076C23136yescomplemented
27C2×C22⋊D269C231144yescomplemented
28C2×C22⋊D269C23172yescomplemented
29D4×D269C231288yescomplemented
30C22×Dic269C23124yescomplemented
31C2×C4×D269C23172yescomplemented
32C24×C269C2316yescomplemented
33#41×C269C23148yescomplemented
34C2×Q8×C269C2312yescomplemented
35C2×D4×C269C23172yescomplemented
36C2×D4×C269C23118yescomplemented
37C22×C4×C269C2319yescomplemented
38C22×D269—824no—
39C22×D269C24172yes—
40Dic538C24148yes—
41D1076C24148yes—
42C22⋊D269C241144yes—
43C22⋊D269C24116yes—
44C2×Dic269C24172yes—
45C4×D269C24196yes—
46C23×C269—86no—
47C23×C269C2419yes—
48C23×C269C24124yes—
49Q8×C269C2418yes—
50D4×C269C24172yes—
51C2×C4×C269C2416yes—
52C2×C4×C269C24136yes—
53C22×C269—49no—
54C22×C269—424no—
55C22×C269C2511yescharacteristic
56C22×C269C25118yes—
57C2×D269—472no—
58C2×D269C25116yes—
59C4×C269C25112yes—
60Dic269C25116yes—
61C2×C269—218no—
62C2×C269—21no—
63C2×C269C2611yescharacteristic
64D269—216no—
65C269H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—2691nomaximal, Sylow
67#585—26916no—
68#585—26912no—
69C2×H(5,𝔽2)—26916no—
70C2×H(5,𝔽2)—26918no—
71C2×H(5,𝔽2)D26911yescharacteristic, complemented
722−1+4—26948no—
732−1+4—2698no—
74H(5,𝔽2)—26948no—
75H(5,𝔽2)—269144no—
76H(5,𝔽2)—26972no—
77H(5,𝔽2)C2×D269116yescomplemented
78C2×#41—26996no—
79C2×#41—26972no—
80C2×#41—26936no—
81C2×#41C2×D26916yescomplemented
82C22×D4—26924no—
83C22×D4—26972no—
84C22×D4C2×D26919yescomplemented
85C24—26924no—
86C24C22×D26916yescomplemented
87#41—26932no—
88#41—269288no—
89#41—269144no—
90#41—26948no—
91#41C22×D269148yescomplemented
92C2×Q8—26932no—
93C2×Q8—26936no—
94C2×Q8C22×D26912yescomplemented
95C2×D4—269144no—
96C2×D4—26972no—
97C2×D4—269288no—
98C2×D4—26936no—
99C2×D4C22×D269172yescomplemented
100C2×D4C22×D269118yescomplemented
101C22×C4—26924no—
102C22×C4—26972no—
103C22×C4C22×D26919yescomplemented
104C23—86no—
105C23—26972no—
106C23—215224no—
107C23C23×D269124yes—
108C23C23×D26919yes—
109Q8—26948no—
110Q8C23×D26918yes—
111D4—269144no—
112D4—26916no—
113D4—26948no—
114D4C23×D269172yes—
115C2×C4—26972no—
116C2×C4—26996no—
117C2×C4C23×D269136yes—
118C2×C4C23×D26916yes—
119C22—49no—
120C22—424no—
121C22—26916no—
122C22—107672no—
123C22C24×D269118yes—
124C22C24×D26911yescharacteristic
125C4—26916no—
126C4C24×D269112yes—
127C2—21no—
128C2—218no—
129C2—53816no—
130C2C25×D26911yescharacteristic
subgroupG/H
center, FrattiniC2C25×D269
commutator, socle, cosocleC2×C269C26
FittingC2×H(5,𝔽2)×C269C2
hypercenterC2×H(5,𝔽2)D269
hypocenterC269H(7,𝔽2)
LCSG ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771440C2semidirect
3. C2×#22575498C2semidirect
4. H(5,𝔽2)×D269C2semidirect
5. C2×H(5,𝔽2)×C269C2semidirect
6. #22575501C22semidirect
7. #22575500C22semidirect
8. #22575499C22semidirect
9. #22575498C22semidirect
10. #22575498C22semidirect
11. C2×#16864402C22semidirect
12. C2×#16864401C22semidirect
13. #41×D269C22semidirect
14. C22×C22⋊D269C22semidirect
15. C2×D4×D269C22semidirect
16. H(5,𝔽2)×C269C22semidirect
17. C2×#41×C269C22semidirect
18. C22×D4×C269C22semidirect
19. C23×D269C23semidirect
20. #16864403C23semidirect
21. #16864402C23semidirect
22. #16864402C23semidirect
23. #16864401C23semidirect
24. C2×Dic538C23semidirect
25. Q8×D269C23semidirect
26. C2×D1076C23semidirect
27. C2×C22⋊D269C23semidirect
28. C2×C22⋊D269C23semidirect
29. D4×D269C23semidirect
30. C22×Dic269C23semidirect
31. C2×C4×D269C23semidirect
32. C24×C269C23semidirect
33. #41×C269C23semidirect
34. C2×Q8×C269C23semidirect
35. C2×D4×C269C23semidirect
36. C2×D4×C269C23semidirect
37. C22×C4×C269C23semidirect
65. C269H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D269semidirect
77. H(5,𝔽2)C2×D269semidirect
81. C2×#41C2×D269semidirect
84. C22×D4C2×D269semidirect
86. C24C22×D269semidirect
91. #41C22×D269semidirect
94. C2×Q8C22×D269semidirect
99. C2×D4C22×D269semidirect
100. C2×D4C22×D269semidirect
103. C22×C4C22×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317714402
C2×#225754982
H(5,𝔽2)×D2692
C2×H(5,𝔽2)×C2692
H(7,𝔽2)269