Finite Group Databasebeta

← #22603150

Subgroups — #22603150

Order 9088.

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#17986835C2112yesmaximal, complemented
2#17986834C2116yesmaximal, complemented
3C2×#14556173C2118yesmaximal, complemented
4H(5,𝔽2)×D71C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C71C211yesmaximal, characteristic, complemented
6#14556176C2218yescomplemented
7#14556175C22148yescomplemented
8#14556174C22148yescomplemented
9#14556173C221144yescomplemented
10#14556173C22172yescomplemented
11C2×#11765865C22136yescomplemented
12C2×#11765864C22172yescomplemented
13#41×D71C22196yescomplemented
14C22×C22⋊D71C22124yescomplemented
15C2×D4×D71C22172yescomplemented
16H(5,𝔽2)×C71C22116yescomplemented
17C2×#41×C71C2216yescomplemented
18C22×D4×C71C2219yescomplemented
19C23×D71C23124yescomplemented
20#11765866C23132yescomplemented
21#11765865C231144yescomplemented
22#11765865C23148yescomplemented
23#11765864C231288yescomplemented
24C2×Dic142C23136yescomplemented
25Q8×D71C23132yescomplemented
26C2×D284C23136yescomplemented
27C2×C22⋊D71C231144yescomplemented
28C2×C22⋊D71C23172yescomplemented
29D4×D71C231288yescomplemented
30C22×Dic71C23124yescomplemented
31C2×C4×D71C23172yescomplemented
32C24×C71C2316yescomplemented
33#41×C71C23148yescomplemented
34C2×Q8×C71C2312yescomplemented
35C2×D4×C71C23172yescomplemented
36C2×D4×C71C23118yescomplemented
37C22×C4×C71C2319yescomplemented
38C23×C71—86no—
39C23×C71C2419yes—
40C23×C71C24124yes—
41C22×D71—824no—
42C22×D71C24172yes—
43Q8×C71C2418yes—
44D4×C71C24172yes—
45C2×C4×C71C2416yes—
46C2×C4×C71C24136yes—
47C22⋊D71C241144yes—
48C22⋊D71C24116yes—
49C2×Dic71C24172yes—
50D284C24148yes—
51C4×D71C24196yes—
52Dic142C24148yes—
53C22×C71—424no—
54C22×C71—49no—
55C22×C71C25118yes—
56C22×C71C2511yescharacteristic
57C2×D71—472no—
58C2×D71C25116yes—
59C4×C71C25112yes—
60Dic71C25116yes—
61C2×C71—218no—
62C2×C71—21no—
63C2×C71C2611yescharacteristic
64D71—216no—
65H(7,𝔽2)—711nomaximal, Sylow
66C71H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—7116no—
68#585—7112no—
69C2×H(5,𝔽2)—7116no—
70C2×H(5,𝔽2)—7118no—
71C2×H(5,𝔽2)D7111yescharacteristic, complemented
722−1+4—7148no—
732−1+4—718no—
74H(5,𝔽2)—7148no—
75H(5,𝔽2)—71144no—
76H(5,𝔽2)—7172no—
77H(5,𝔽2)C2×D71116yescomplemented
78C2×#41—7196no—
79C2×#41—7172no—
80C2×#41—7136no—
81C2×#41C2×D7116yescomplemented
82C22×D4—7124no—
83C22×D4—7172no—
84C22×D4C2×D7119yescomplemented
85C24—7124no—
86C24C22×D7116yescomplemented
87#41—7132no—
88#41—71288no—
89#41—71144no—
90#41—7148no—
91#41C22×D71148yescomplemented
92C2×Q8—7132no—
93C2×Q8—7136no—
94C2×Q8C22×D7112yescomplemented
95C2×D4—71144no—
96C2×D4—7172no—
97C2×D4—71288no—
98C2×D4—7136no—
99C2×D4C22×D71172yescomplemented
100C2×D4C22×D71118yescomplemented
101C22×C4—7124no—
102C22×C4—7172no—
103C22×C4C22×D7119yescomplemented
104C23—86no—
105C23—56824no—
106C23—7172no—
107C23C23×D71124yes—
108C23C23×D7119yes—
109Q8—7148no—
110Q8C23×D7118yes—
111D4—71144no—
112D4—7116no—
113D4—7148no—
114D4C23×D71172yes—
115C2×C4—7172no—
116C2×C4—7196no—
117C2×C4C23×D71136yes—
118C2×C4C23×D7116yes—
119C22—49no—
120C22—424no—
121C22—28472no—
122C22—7116no—
123C22C24×D71118yes—
124C22C24×D7111yescharacteristic
125C4—7116no—
126C4C24×D71112yes—
127C2—21no—
128C2—218no—
129C2—14216no—
130C2C25×D7111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D71
commutator, socleC2×C71C26
FittingC2×H(5,𝔽2)×C71C2
hypercenterC2×H(5,𝔽2)D71
hypocenterC71H(7,𝔽2)
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #17986835C2semidirect
2. #17986834C2semidirect
3. C2×#14556173C2semidirect
4. H(5,𝔽2)×D71C2semidirect
5. C2×H(5,𝔽2)×C71C2semidirect
6. #14556176C22semidirect
7. #14556175C22semidirect
8. #14556174C22semidirect
9. #14556173C22semidirect
10. #14556173C22semidirect
11. C2×#11765865C22semidirect
12. C2×#11765864C22semidirect
13. #41×D71C22semidirect
14. C22×C22⋊D71C22semidirect
15. C2×D4×D71C22semidirect
16. H(5,𝔽2)×C71C22semidirect
17. C2×#41×C71C22semidirect
18. C22×D4×C71C22semidirect
19. C23×D71C23semidirect
20. #11765866C23semidirect
21. #11765865C23semidirect
22. #11765865C23semidirect
23. #11765864C23semidirect
24. C2×Dic142C23semidirect
25. Q8×D71C23semidirect
26. C2×D284C23semidirect
27. C2×C22⋊D71C23semidirect
28. C2×C22⋊D71C23semidirect
29. D4×D71C23semidirect
30. C22×Dic71C23semidirect
31. C2×C4×D71C23semidirect
32. C24×C71C23semidirect
33. #41×C71C23semidirect
34. C2×Q8×C71C23semidirect
35. C2×D4×C71C23semidirect
36. C2×D4×C71C23semidirect
37. C22×C4×C71C23semidirect
66. C71H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D71semidirect
77. H(5,𝔽2)C2×D71semidirect
81. C2×#41C2×D71semidirect
84. C22×D4C2×D71semidirect
86. C24C22×D71semidirect
91. #41C22×D71semidirect
94. C2×Q8C22×D71semidirect
99. C2×D4C22×D71semidirect
100. C2×D4C22×D71semidirect
103. C22×C4C22×D71semidirect