Finite Group Databasebeta

← #31795600

Subgroups — #31795600

Order 17536.

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#22578240C2112yesmaximal, complemented
2#22578239C2115yesmaximal, complemented
3C2×#17984134C2115yesmaximal, complemented
4H(5,𝔽2)×D137C2120yesmaximal, complemented
5#585×C137C211yesmaximal, characteristic, complemented
6#17984136C22130yescomplemented
7#17984135C22120yescomplemented
8#17984134C221120yescomplemented
9#17984134C22160yescomplemented
10#17984133C22190yescomplemented
11C2×#14555511C22130yescomplemented
12C2×#14555510C22145yescomplemented
13#41×D137C221120yescomplemented
14C22×D548C22115yescomplemented
15C2×D4×D137C22190yescomplemented
162−1+4×C137C2216yescomplemented
17H(5,𝔽2)×C137C22110yescomplemented
18C2×#41×C137C22115yescomplemented
19C23×D137C23130yescomplemented
20#14555511C231120yescomplemented
21#14555510C231180yescomplemented
22#14555510C23160yescomplemented
23#14555509C231120yescomplemented
24C2×Dic274C23115yescomplemented
25Q8×D137C23140yescomplemented
26C2×D548C23190yescomplemented
27C2×D548C23145yescomplemented
28C2×C22⋊D137C23190yescomplemented
29D4×D137C231360yescomplemented
30C2×C4×D137C23190yescomplemented
31#41×C137C23160yescomplemented
32#41×C137C23120yescomplemented
33C2×Q8×C137C23115yescomplemented
34C2×D4×C137C23145yescomplemented
35C22×C4×C137C23115yescomplemented
36C22×D137—830no—
37C22×D137C24190yes—
38Dic274C24130yes—
39Dic274C2416yes—
40D548C24190yes—
41D548C24110yes—
42C22⋊D137C241120yes—
43C2×Dic137C24130yes—
44C4×D137C241120yes—
45C23×C137C24115yes—
46Q8×C137C24120yes—
47D4×C137C24160yes—
48C2×C4×C137C24115yes—
49C2×C4×C137C24145yes—
50C22×C137—415no—
51C22×C137C25115yes—
52C2×D137—490no—
53C2×D137C25120yes—
54C4×C137C2511yescharacteristic
55C4×C137C25115yes—
56Dic137C25112yes—
57C2×C137—215no—
58C2×C137C2611yescharacteristic
59D137—220no—
60C137H(7,𝔽2)11yescharacteristic, complemented, Sylow
61H(7,𝔽2)—1371nomaximal, Sylow
62#585—13712no—
63#585—13715no—
64#585D13711yescharacteristic, complemented
65C2×H(5,𝔽2)—13720no—
66C2×H(5,𝔽2)—13715no—
672−1+4—13720no—
682−1+4—13730no—
692−1+4C2×D13716yescomplemented
70H(5,𝔽2)—137120no—
71H(5,𝔽2)—13760no—
72H(5,𝔽2)—13790no—
73H(5,𝔽2)C2×D137110yescomplemented
74C2×#41—137120no—
75C2×#41—13730no—
76C2×#41—13745no—
77C2×#41C2×D137115yescomplemented
78C22×D4—13790no—
79C22×D4—13715no—
80C24—13730no—
81#41—137120no—
82#41—137120no—
83#41—137180no—
84#41—13760no—
85#41C22×D137160yescomplemented
86#41C22×D137120yescomplemented
87C2×Q8—13740no—
88C2×Q8—13715no—
89C2×Q8C22×D137115yescomplemented
90C2×D4—13790no—
91C2×D4—137360no—
92C2×D4—13790no—
93C2×D4—13745no—
94C2×D4C22×D137145yescomplemented
95C22×C4—13790no—
96C22×C4C22×D137115yescomplemented
97C23—109630no—
98C23—13790no—
99C23C23×D137115yes—
100Q8—13730no—
101Q8—1376no—
102Q8C23×D137120yes—
103D4—137120no—
104D4—13790no—
105D4—13710no—
106D4C23×D137160yes—
107C2×C4—13730no—
108C2×C4—137120no—
109C2×C4C23×D137145yes—
110C2×C4C23×D137115yes—
111C22—415no—
112C22—54890no—
113C22—13720no—
114C22C24×D137115yes—
115C4—13712no—
116C4C24×D137115yes—
117C4C24×D13711yescharacteristic
118C2—215no—
119C2—27420no—
120C2C25×D13711yescharacteristic
subgroupG/H
center, FrattiniC2C25×D137
commutator, socle, cosocleC2×C137C26
Fitting#585×C137C2
hypercenter#585D137
hypocenterC137H(7,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22578240C2semidirect
2. #22578239C2semidirect
3. C2×#17984134C2semidirect
4. H(5,𝔽2)×D137C2semidirect
5. #585×C137C2semidirect
6. #17984136C22semidirect
7. #17984135C22semidirect
8. #17984134C22semidirect
9. #17984134C22semidirect
10. #17984133C22semidirect
11. C2×#14555511C22semidirect
12. C2×#14555510C22semidirect
13. #41×D137C22semidirect
14. C22×D548C22semidirect
15. C2×D4×D137C22semidirect
16. 2−1+4×C137C22semidirect
17. H(5,𝔽2)×C137C22semidirect
18. C2×#41×C137C22semidirect
19. C23×D137C23semidirect
20. #14555511C23semidirect
21. #14555510C23semidirect
22. #14555510C23semidirect
23. #14555509C23semidirect
24. C2×Dic274C23semidirect
25. Q8×D137C23semidirect
26. C2×D548C23semidirect
27. C2×D548C23semidirect
28. C2×C22⋊D137C23semidirect
29. D4×D137C23semidirect
30. C2×C4×D137C23semidirect
31. #41×C137C23semidirect
32. #41×C137C23semidirect
33. C2×Q8×C137C23semidirect
34. C2×D4×C137C23semidirect
35. C22×C4×C137C23semidirect
60. C137H(7,𝔽2)semidirect
64. #585D137semidirect
69. 2−1+4C2×D137semidirect
73. H(5,𝔽2)C2×D137semidirect
77. C2×#41C2×D137semidirect
85. #41C22×D137semidirect
86. #41C22×D137semidirect
89. C2×Q8C22×D137semidirect
94. C2×D4C22×D137semidirect
96. C22×C4C22×D137semidirect

Maximal subgroups

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

subgroupindex
#225782402
#225782392
C2×#179841342
H(5,𝔽2)×D1372
#585×C1372
H(7,𝔽2)137