Finite Group Databasebeta

← #27185943

Subgroups — #27185943

Order 12928.

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#20266014C2112yesmaximal, complemented
2#20266013C2115yesmaximal, complemented
3C2×#15692301C2115yesmaximal, complemented
4H(5,𝔽2)×D101C2120yesmaximal, complemented
5#585×C101C211yesmaximal, characteristic, complemented
6#15692303C22130yescomplemented
7#15692302C22120yescomplemented
8#15692301C221120yescomplemented
9#15692301C22160yescomplemented
10#15692300C22190yescomplemented
11C2×#13108075C22130yescomplemented
12C2×#13108074C22145yescomplemented
13#41×D101C221120yescomplemented
14C22×D404C22115yescomplemented
15C2×D4×D101C22190yescomplemented
162−1+4×C101C2216yescomplemented
17H(5,𝔽2)×C101C22110yescomplemented
18C2×#41×C101C22115yescomplemented
19C23×D101C23130yescomplemented
20#13108075C231120yescomplemented
21#13108074C231180yescomplemented
22#13108074C23160yescomplemented
23#13108073C231120yescomplemented
24C2×Dic202C23115yescomplemented
25Q8×D101C23140yescomplemented
26C2×D404C23190yescomplemented
27C2×D404C23145yescomplemented
28C2×C22⋊D101C23190yescomplemented
29D4×D101C231360yescomplemented
30C2×C4×D101C23190yescomplemented
31#41×C101C23160yescomplemented
32#41×C101C23120yescomplemented
33C2×Q8×C101C23115yescomplemented
34C2×D4×C101C23145yescomplemented
35C22×C4×C101C23115yescomplemented
36C23×C101C24115yes—
37C22×D101—830no—
38C22×D101C24190yes—
39Q8×C101C24120yes—
40D4×C101C24160yes—
41C2×C4×C101C24115yes—
42C2×C4×C101C24145yes—
43C22⋊D101C241120yes—
44C2×Dic101C24130yes—
45D404C24190yes—
46D404C24110yes—
47C4×D101C241120yes—
48Dic202C24130yes—
49Dic202C2416yes—
50C22×C101—415no—
51C22×C101C25115yes—
52C2×D101—490no—
53C2×D101C25120yes—
54C4×C101C2511yescharacteristic
55C4×C101C25115yes—
56Dic101C25112yes—
57C2×C101—215no—
58C2×C101C2611yescharacteristic
59D101—220no—
60H(7,𝔽2)—1011nomaximal, Sylow
61C101H(7,𝔽2)11yescharacteristic, complemented, Sylow
62#585—10112no—
63#585—10115no—
64#585D10111yescharacteristic, complemented
65C2×H(5,𝔽2)—10120no—
66C2×H(5,𝔽2)—10115no—
672−1+4—10120no—
682−1+4—10130no—
692−1+4C2×D10116yescomplemented
70H(5,𝔽2)—101120no—
71H(5,𝔽2)—10160no—
72H(5,𝔽2)—10190no—
73H(5,𝔽2)C2×D101110yescomplemented
74C2×#41—101120no—
75C2×#41—10130no—
76C2×#41—10145no—
77C2×#41C2×D101115yescomplemented
78C22×D4—10190no—
79C22×D4—10115no—
80C24—10130no—
81#41—101120no—
82#41—101120no—
83#41—101180no—
84#41—10160no—
85#41C22×D101160yescomplemented
86#41C22×D101120yescomplemented
87C2×Q8—10140no—
88C2×Q8—10115no—
89C2×Q8C22×D101115yescomplemented
90C2×D4—10190no—
91C2×D4—101360no—
92C2×D4—10190no—
93C2×D4—10145no—
94C2×D4C22×D101145yescomplemented
95C22×C4—10190no—
96C22×C4C22×D101115yescomplemented
97C23—80830no—
98C23—10190no—
99C23C23×D101115yes—
100Q8—10130no—
101Q8—1016no—
102Q8C23×D101120yes—
103D4—101120no—
104D4—10190no—
105D4—10110no—
106D4C23×D101160yes—
107C2×C4—10130no—
108C2×C4—101120no—
109C2×C4C23×D101145yes—
110C2×C4C23×D101115yes—
111C22—415no—
112C22—10120no—
113C22—40490no—
114C22C24×D101115yes—
115C4—10112no—
116C4C24×D101115yes—
117C4C24×D10111yescharacteristic
118C2—215no—
119C2—20220no—
120C2C25×D10111yescharacteristic
subgroupG/H
center, FrattiniC2C25×D101
commutator, socle, cosocleC2×C101C26
Fitting#585×C101C2
hypercenter#585D101
hypocenterC101H(7,𝔽2)
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20266014C2semidirect
2. #20266013C2semidirect
3. C2×#15692301C2semidirect
4. H(5,𝔽2)×D101C2semidirect
5. #585×C101C2semidirect
6. #15692303C22semidirect
7. #15692302C22semidirect
8. #15692301C22semidirect
9. #15692301C22semidirect
10. #15692300C22semidirect
11. C2×#13108075C22semidirect
12. C2×#13108074C22semidirect
13. #41×D101C22semidirect
14. C22×D404C22semidirect
15. C2×D4×D101C22semidirect
16. 2−1+4×C101C22semidirect
17. H(5,𝔽2)×C101C22semidirect
18. C2×#41×C101C22semidirect
19. C23×D101C23semidirect
20. #13108075C23semidirect
21. #13108074C23semidirect
22. #13108074C23semidirect
23. #13108073C23semidirect
24. C2×Dic202C23semidirect
25. Q8×D101C23semidirect
26. C2×D404C23semidirect
27. C2×D404C23semidirect
28. C2×C22⋊D101C23semidirect
29. D4×D101C23semidirect
30. C2×C4×D101C23semidirect
31. #41×C101C23semidirect
32. #41×C101C23semidirect
33. C2×Q8×C101C23semidirect
34. C2×D4×C101C23semidirect
35. C22×C4×C101C23semidirect
61. C101H(7,𝔽2)semidirect
64. #585D101semidirect
69. 2−1+4C2×D101semidirect
73. H(5,𝔽2)C2×D101semidirect
77. C2×#41C2×D101semidirect
85. #41C22×D101semidirect
86. #41C22×D101semidirect
89. C2×Q8C22×D101semidirect
94. C2×D4C22×D101semidirect
96. C22×C4C22×D101semidirect

Maximal subgroups

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

subgroupindex
#202660142
#202660132
C2×#156923012
H(5,𝔽2)×D1012
#585×C1012
H(7,𝔽2)101