Finite Group Databasebeta

← #11744566

Subgroups — #11744566

Order 896.

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
1C7×#585C211yesmaximal, characteristic, complemented
2#90620C2112yesmaximal, complemented
3D7×H(5,𝔽2)C2120yesmaximal, complemented
4#90616C2115yesmaximal, complemented
5C2×#6553C2115yesmaximal, complemented
6C7×2−1+4C2216yescomplemented
7C7×H(5,𝔽2)C22110yescomplemented
8C2×C7×#41C22115yescomplemented
9#6554C22120yescomplemented
10#6553C22160yescomplemented
11#6553C221120yescomplemented
12D7×#41C221120yescomplemented
13#6551C22130yescomplemented
14C2×#1161C22130yescomplemented
15#6548C22190yescomplemented
16C2×D4×D7C22190yescomplemented
17C2×#1157C22145yescomplemented
18C22×D28C22115yescomplemented
19H(7,𝔽2)—71nomaximal, Sylow
20C23×D7C23130yescomplemented
21C7×#41C23160yescomplemented
22C7×#41C23120yescomplemented
23C2×C7×Q8C23115yescomplemented
24C2×C7×D4C23145yescomplemented
25C22×C4×C7C23115yescomplemented
26C2×C22⋊D7C23190yescomplemented
27#1161C231120yescomplemented
28Q8×D7C23140yescomplemented
29#1159C231120yescomplemented
30D4×D7C231360yescomplemented
31#1157C23160yescomplemented
32#1157C231180yescomplemented
33C2×D28C23145yescomplemented
34C2×D28C23190yescomplemented
35C2×C4×D7C23190yescomplemented
36C2×Dic14C23115yescomplemented
37#585—712no—
38#585—715no—
39#585D711yescharacteristic, complemented
40C2×H(5,𝔽2)—720no—
41C2×H(5,𝔽2)—715no—
42C23×C7C24115yes—
43C22×D7—830no—
44C22×D7C24190yes—
45C7×Q8C24120yes—
46C7×D4C24160yes—
47C2×C4×C7C24115yes—
48C2×C4×C7C24145yes—
49C22⋊D7C241120yes—
50C2×Dic7C24130yes—
51D28C24110yes—
52D28C24190yes—
53C4×D7C241120yes—
54Dic14C2416yes—
55Dic14C24130yes—
562−1+4—720no—
572−1+4—730no—
582−1+4C2×D716yescomplemented
59H(5,𝔽2)—7120no—
60H(5,𝔽2)—760no—
61H(5,𝔽2)—790no—
62H(5,𝔽2)C2×D7110yescomplemented
63C2×#41—7120no—
64C2×#41—730no—
65C2×#41—745no—
66C2×#41C2×D7115yescomplemented
67C22×D4—790no—
68C22×D4—715no—
69C22×C7—415no—
70C22×C7C25115yes—
71C2×D7—490no—
72C2×D7C25120yes—
73C4×C7C2511yescharacteristic
74C4×C7C25115yes—
75Dic7C25112yes—
76C24—730no—
77#41—7120no—
78#41—7120no—
79#41—7180no—
80#41—760no—
81#41C22×D7160yescomplemented
82#41C22×D7120yescomplemented
83C2×Q8—740no—
84C2×Q8—715no—
85C2×Q8C22×D7115yescomplemented
86C2×D4—790no—
87C2×D4—7360no—
88C2×D4—790no—
89C2×D4—745no—
90C2×D4C22×D7145yescomplemented
91C22×C4—790no—
92C22×C4C22×D7115yescomplemented
93C2×C7—215no—
94C2×C7C2611yescharacteristic
95D7—220no—
96C23—790no—
97C23—5630no—
98C23C23×D7115yes—
99Q8—730no—
100Q8—76no—
101Q8C23×D7120yes—
102D4—7120no—
103D4—790no—
104D4—710no—
105D4C23×D7160yes—
106C2×C4—730no—
107C2×C4—7120no—
108C2×C4C23×D7145yes—
109C2×C4C23×D7115yes—
110C7H(7,𝔽2)11yescharacteristic, complemented, Sylow
111C22—415no—
112C22—720no—
113C22—2890no—
114C22C24×D7115yes—
115C4—712no—
116C4C24×D7115yes—
117C4C24×D711yescharacteristic
118C2—215no—
119C2—1420no—
120C2C25×D711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D7
commutator, socle, cosocleC2×C7C26
FittingC7×#585C2
hypercenter#585D7
hypocenterC7H(7,𝔽2)
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C7×#585C2semidirect
2. #90620C2semidirect
3. D7×H(5,𝔽2)C2semidirect
4. #90616C2semidirect
5. C2×#6553C2semidirect
6. C7×2−1+4C22semidirect
7. C7×H(5,𝔽2)C22semidirect
8. C2×C7×#41C22semidirect
9. #6554C22semidirect
10. #6553C22semidirect
11. #6553C22semidirect
12. D7×#41C22semidirect
13. #6551C22semidirect
14. C2×#1161C22semidirect
15. #6548C22semidirect
16. C2×D4×D7C22semidirect
17. C2×#1157C22semidirect
18. C22×D28C22semidirect
20. C23×D7C23semidirect
21. C7×#41C23semidirect
22. C7×#41C23semidirect
23. C2×C7×Q8C23semidirect
24. C2×C7×D4C23semidirect
25. C22×C4×C7C23semidirect
26. C2×C22⋊D7C23semidirect
27. #1161C23semidirect
28. Q8×D7C23semidirect
29. #1159C23semidirect
30. D4×D7C23semidirect
31. #1157C23semidirect
32. #1157C23semidirect
33. C2×D28C23semidirect
34. C2×D28C23semidirect
35. C2×C4×D7C23semidirect
36. C2×Dic14C23semidirect
39. #585D7semidirect
58. 2−1+4C2×D7semidirect
62. H(5,𝔽2)C2×D7semidirect
66. C2×#41C2×D7semidirect
81. #41C22×D7semidirect
82. #41C22×D7semidirect
85. C2×Q8C22×D7semidirect
90. C2×D4C22×D7semidirect
92. C22×C4C22×D7semidirect
110. C7H(7,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C7×#5852
#906202
D7×H(5,𝔽2)2
#906162
C2×#65532
H(7,𝔽2)7