Finite Group Databasebeta

← #14543595

Subgroups — #14543595

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#11763646C2112yesmaximal, complemented
2#11763645C2115yesmaximal, complemented
3C2×#10587696C2115yesmaximal, complemented
4H(5,𝔽2)×D17C2120yesmaximal, complemented
5C17×#585C211yesmaximal, characteristic, complemented
6C17×2−1+4C2216yescomplemented
7C17×H(5,𝔽2)C22110yescomplemented
8C2×#41×C17C22115yescomplemented
9#10587697C22120yescomplemented
10#10587696C221120yescomplemented
11#10587696C22160yescomplemented
12#41×D17C221120yescomplemented
13#10587694C22130yescomplemented
14C2×#63256C22130yescomplemented
15#10587691C22190yescomplemented
16C2×D4×D17C22190yescomplemented
17C2×#63252C22145yescomplemented
18C22×D68C22115yescomplemented
19C23×D17C23130yescomplemented
20#41×C17C23160yescomplemented
21#41×C17C23120yescomplemented
22C2×Q8×C17C23115yescomplemented
23C2×D4×C17C23145yescomplemented
24C22×C4×C17C23115yescomplemented
25C2×C22⋊D17C23190yescomplemented
26#63256C231120yescomplemented
27Q8×D17C23140yescomplemented
28#63254C231120yescomplemented
29D4×D17C231360yescomplemented
30#63252C23160yescomplemented
31#63252C231180yescomplemented
32C2×D68C23145yescomplemented
33C2×D68C23190yescomplemented
34C2×C4×D17C23190yescomplemented
35C2×Dic34C23115yescomplemented
36C23×C17C24115yes—
37C22×D17—830no—
38C22×D17C24190yes—
39Q8×C17C24120yes—
40D4×C17C24160yes—
41C2×C4×C17C24115yes—
42C2×C4×C17C24145yes—
43C22⋊D17C241120yes—
44C2×Dic17C24130yes—
45D68C24110yes—
46D68C24190yes—
47C4×D17C241120yes—
48Dic34C2416yes—
49Dic34C24130yes—
50H(7,𝔽2)—171nomaximal, Sylow
51C22×C17—415no—
52C22×C17C25115yes—
53C2×D17—490no—
54C2×D17C25120yes—
55C4×C17C2511yescharacteristic
56C4×C17C25115yes—
57Dic17C25112yes—
58#585—1712no—
59#585—1715no—
60#585D1711yescharacteristic, complemented
61C2×H(5,𝔽2)—1720no—
62C2×H(5,𝔽2)—1715no—
63C2×C17—215no—
64C2×C17C2611yescharacteristic
65D17—220no—
662−1+4—1720no—
672−1+4—1730no—
682−1+4C2×D1716yescomplemented
69H(5,𝔽2)—17120no—
70H(5,𝔽2)—1760no—
71H(5,𝔽2)—1790no—
72H(5,𝔽2)C2×D17110yescomplemented
73C2×#41—17120no—
74C2×#41—1730no—
75C2×#41—1745no—
76C2×#41C2×D17115yescomplemented
77C22×D4—1790no—
78C22×D4—1715no—
79C17H(7,𝔽2)11yescharacteristic, complemented, Sylow
80C24—1730no—
81#41—17120no—
82#41—17120no—
83#41—17180no—
84#41—1760no—
85#41C22×D17160yescomplemented
86#41C22×D17120yescomplemented
87C2×Q8—1740no—
88C2×Q8—1715no—
89C2×Q8C22×D17115yescomplemented
90C2×D4—1790no—
91C2×D4—17360no—
92C2×D4—1790no—
93C2×D4—1745no—
94C2×D4C22×D17145yescomplemented
95C22×C4—1790no—
96C22×C4C22×D17115yescomplemented
97C23—1790no—
98C23—13630no—
99C23C23×D17115yes—
100Q8—1730no—
101Q8—176no—
102Q8C23×D17120yes—
103D4—17120no—
104D4—1790no—
105D4—1710no—
106D4C23×D17160yes—
107C2×C4—1730no—
108C2×C4—17120no—
109C2×C4C23×D17145yes—
110C2×C4C23×D17115yes—
111C22—415no—
112C22—1720no—
113C22—6890no—
114C22C24×D17115yes—
115C4—1712no—
116C4C24×D17115yes—
117C4C24×D1711yescharacteristic
118C2—215no—
119C2—3420no—
120C2C25×D1711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D17
commutator, socle, cosocleC2×C17C26
FittingC17×#585C2
hypercenter#585D17
hypocenterC17H(7,𝔽2)
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #11763646C2semidirect
2. #11763645C2semidirect
3. C2×#10587696C2semidirect
4. H(5,𝔽2)×D17C2semidirect
5. C17×#585C2semidirect
6. C17×2−1+4C22semidirect
7. C17×H(5,𝔽2)C22semidirect
8. C2×#41×C17C22semidirect
9. #10587697C22semidirect
10. #10587696C22semidirect
11. #10587696C22semidirect
12. #41×D17C22semidirect
13. #10587694C22semidirect
14. C2×#63256C22semidirect
15. #10587691C22semidirect
16. C2×D4×D17C22semidirect
17. C2×#63252C22semidirect
18. C22×D68C22semidirect
19. C23×D17C23semidirect
20. #41×C17C23semidirect
21. #41×C17C23semidirect
22. C2×Q8×C17C23semidirect
23. C2×D4×C17C23semidirect
24. C22×C4×C17C23semidirect
25. C2×C22⋊D17C23semidirect
26. #63256C23semidirect
27. Q8×D17C23semidirect
28. #63254C23semidirect
29. D4×D17C23semidirect
30. #63252C23semidirect
31. #63252C23semidirect
32. C2×D68C23semidirect
33. C2×D68C23semidirect
34. C2×C4×D17C23semidirect
35. C2×Dic34C23semidirect
60. #585D17semidirect
68. 2−1+4C2×D17semidirect
72. H(5,𝔽2)C2×D17semidirect
76. C2×#41C2×D17semidirect
79. C17H(7,𝔽2)semidirect
85. #41C22×D17semidirect
86. #41C22×D17semidirect
89. C2×Q8C22×D17semidirect
94. C2×D4C22×D17semidirect
96. C22×C4C22×D17semidirect

Maximal subgroups

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

subgroupindex
#117636462
#117636452
C2×#105876962
H(5,𝔽2)×D172
C17×#5852
H(7,𝔽2)17