Finite Group Databasebeta

← #14543594

Subgroups — #14543594

Order 2176.

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#11763645C2112yesmaximal, complemented
2#11763644C2116yesmaximal, complemented
3C2×#10587691C2118yesmaximal, complemented
4H(5,𝔽2)×D17C2116yesmaximal, complemented
5C2×C17×H(5,𝔽2)C211yesmaximal, characteristic, complemented
6C17×H(5,𝔽2)C22116yescomplemented
7C2×#41×C17C2216yescomplemented
8C22×D4×C17C2219yescomplemented
9C22×C22⋊D17C22124yescomplemented
10#10587697C22148yescomplemented
11#10587696C22148yescomplemented
12#41×D17C22196yescomplemented
13#10587694C2218yescomplemented
14#10587691C221144yescomplemented
15#10587691C22172yescomplemented
16C2×#63254C22172yescomplemented
17C2×D4×D17C22172yescomplemented
18C2×#63252C22136yescomplemented
19C24×C17C2316yescomplemented
20C23×D17C23124yescomplemented
21#41×C17C23148yescomplemented
22C2×Q8×C17C2312yescomplemented
23C2×D4×C17C23172yescomplemented
24C2×D4×C17C23118yescomplemented
25C22×C4×C17C2319yescomplemented
26C2×C22⋊D17C23172yescomplemented
27C2×C22⋊D17C231144yescomplemented
28C22×Dic17C23124yescomplemented
29#63256C23132yescomplemented
30Q8×D17C23132yescomplemented
31#63254C231288yescomplemented
32D4×D17C231288yescomplemented
33#63252C23148yescomplemented
34#63252C231144yescomplemented
35C2×D68C23136yescomplemented
36C2×C4×D17C23172yescomplemented
37C2×Dic34C23136yescomplemented
38C23×C17—86no—
39C23×C17C24124yes—
40C23×C17C2419yes—
41C22×D17—824no—
42C22×D17C24172yes—
43Q8×C17C2418yes—
44D4×C17C24172yes—
45C2×C4×C17C24136yes—
46C2×C4×C17C2416yes—
47C22⋊D17C24116yes—
48C22⋊D17C241144yes—
49C2×Dic17C24172yes—
50D68C24148yes—
51C4×D17C24196yes—
52Dic34C24148yes—
53H(7,𝔽2)—171nomaximal, Sylow
54C22×C17—424no—
55C22×C17—49no—
56C22×C17C25118yes—
57C22×C17C2511yescharacteristic
58C2×D17—472no—
59C2×D17C25116yes—
60C4×C17C25112yes—
61Dic17C25116yes—
62#585—1716no—
63#585—1712no—
64C2×H(5,𝔽2)—1716no—
65C2×H(5,𝔽2)—1718no—
66C2×H(5,𝔽2)D1711yescharacteristic, complemented
67C2×C17—218no—
68C2×C17—21no—
69C2×C17C2611yescharacteristic
70D17—216no—
712−1+4—1748no—
722−1+4—178no—
73H(5,𝔽2)—1748no—
74H(5,𝔽2)—17144no—
75H(5,𝔽2)—1772no—
76H(5,𝔽2)C2×D17116yescomplemented
77C2×#41—1796no—
78C2×#41—1772no—
79C2×#41—1736no—
80C2×#41C2×D1716yescomplemented
81C22×D4—1724no—
82C22×D4—1772no—
83C22×D4C2×D1719yescomplemented
84C17H(7,𝔽2)11yescharacteristic, complemented, Sylow
85C24—1724no—
86C24C22×D1716yescomplemented
87#41—1732no—
88#41—17288no—
89#41—17144no—
90#41—1748no—
91#41C22×D17148yescomplemented
92C2×Q8—1732no—
93C2×Q8—1736no—
94C2×Q8C22×D1712yescomplemented
95C2×D4—17144no—
96C2×D4—1772no—
97C2×D4—17288no—
98C2×D4—1736no—
99C2×D4C22×D17172yescomplemented
100C2×D4C22×D17118yescomplemented
101C22×C4—1724no—
102C22×C4—1772no—
103C22×C4C22×D1719yescomplemented
104C23—86no—
105C23—1772no—
106C23—13624no—
107C23C23×D17124yes—
108C23C23×D1719yes—
109Q8—1748no—
110Q8C23×D1718yes—
111D4—17144no—
112D4—1716no—
113D4—1748no—
114D4C23×D17172yes—
115C2×C4—1772no—
116C2×C4—1796no—
117C2×C4C23×D17136yes—
118C2×C4C23×D1716yes—
119C22—49no—
120C22—424no—
121C22—1716no—
122C22—6872no—
123C22C24×D17118yes—
124C22C24×D1711yescharacteristic
125C4—1716no—
126C4C24×D17112yes—
127C2—21no—
128C2—218no—
129C2—3416no—
130C2C25×D1711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D17
commutator, socle, cosocleC2×C17C26
FittingC2×C17×H(5,𝔽2)C2
hypercenterC2×H(5,𝔽2)D17
hypocenterC17H(7,𝔽2)
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #11763645C2semidirect
2. #11763644C2semidirect
3. C2×#10587691C2semidirect
4. H(5,𝔽2)×D17C2semidirect
5. C2×C17×H(5,𝔽2)C2semidirect
6. C17×H(5,𝔽2)C22semidirect
7. C2×#41×C17C22semidirect
8. C22×D4×C17C22semidirect
9. C22×C22⋊D17C22semidirect
10. #10587697C22semidirect
11. #10587696C22semidirect
12. #41×D17C22semidirect
13. #10587694C22semidirect
14. #10587691C22semidirect
15. #10587691C22semidirect
16. C2×#63254C22semidirect
17. C2×D4×D17C22semidirect
18. C2×#63252C22semidirect
19. C24×C17C23semidirect
20. C23×D17C23semidirect
21. #41×C17C23semidirect
22. C2×Q8×C17C23semidirect
23. C2×D4×C17C23semidirect
24. C2×D4×C17C23semidirect
25. C22×C4×C17C23semidirect
26. C2×C22⋊D17C23semidirect
27. C2×C22⋊D17C23semidirect
28. C22×Dic17C23semidirect
29. #63256C23semidirect
30. Q8×D17C23semidirect
31. #63254C23semidirect
32. D4×D17C23semidirect
33. #63252C23semidirect
34. #63252C23semidirect
35. C2×D68C23semidirect
36. C2×C4×D17C23semidirect
37. C2×Dic34C23semidirect
66. C2×H(5,𝔽2)D17semidirect
76. H(5,𝔽2)C2×D17semidirect
80. C2×#41C2×D17semidirect
83. C22×D4C2×D17semidirect
84. C17H(7,𝔽2)semidirect
86. C24C22×D17semidirect
91. #41C22×D17semidirect
94. C2×Q8C22×D17semidirect
99. C2×D4C22×D17semidirect
100. C2×D4C22×D17semidirect
103. C22×C4C22×D17semidirect

Maximal subgroups

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

subgroupindex
#117636452
#117636442
C2×#105876912
H(5,𝔽2)×D172
C2×C17×H(5,𝔽2)2
H(7,𝔽2)17