Finite Group Databasebeta

← #72274291

Subgroups — #72274291

Order 4096.

8935217648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72258201C211023yesmaximal, complemented
2C22×#63103C22186955yescomplemented
32−1+10C21496yesmaximal, complemented
4H(11,𝔽2)C21528yesmaximal, complemented
5#72258201C22187296yescomplemented
6#72258201C221261888yescomplemented
7C2×2−1+8C221122760yescomplemented
8C2×H(9,𝔽2)C221139128yescomplemented
92−1+8C23110475520yescomplemented
10H(9,𝔽2)C23111872256yescomplemented
11C2×#63103C2315565120yescomplemented
12C2×#63103C23116695360yescomplemented
13C22×2−1+6C2312434740yescomplemented
14C22×H(7,𝔽2)C2313130380yescomplemented
15C23×#585C231782595yescomplemented
16#63103C24123744512yescomplemented
17#63103C241356167680yescomplemented
18C2×2−1+6C241155823360yescomplemented
19C2×H(7,𝔽2)C241200344320yescomplemented
20C22×#585C24129216880yescomplemented
21C22×#585C24187650640yescomplemented
22C23×2−1+4C2414695570yescomplemented
23C23×H(5,𝔽2)C2417825950yescomplemented
24C24×#41C241782595yescomplemented
252−1+6C251664846336yescomplemented
26H(7,𝔽2)C251854802432yescomplemented
27C2×#585C25193494016yescomplemented
28C2×#585C2511402410240yescomplemented
29C22×2−1+4C251175301280yescomplemented
30C22×H(5,𝔽2)C251292168800yescomplemented
31C23×#41C25115651900yescomplemented
32C23×#41C25146955700yescomplemented
33C24×Q8C251782595yescomplemented
34C24×D4C2512347785yescomplemented
35C25×C4C25175735yescomplemented
36C26C26175735yes—
37#585C26123744512yes—
38#585C2611495904256yes—
39C2×2−1+4C261560964096yes—
40C2×H(5,𝔽2)C261934940160yes—
41C22×#41C26129216880yes—
42C22×#41C261438253200yes—
43C23×Q8C26115651900yes—
44C23×D4C26146955700yes—
45C24×C4C261782595yes—
46C24×C4C2612347785yes—
47C25—3275735no—
48C25C271782595yes—
492−1+4C271142467072yes—
50H(5,𝔽2)C271237445120yes—
51C2×#41C2715565120yes—
52C2×#41C271350602560yes—
53C22×Q8C27129216880yes—
54C22×D4C27187650640yes—
55C23×C4C271782595yes—
56C23×C4C27111738925yes—
57C24—16782595no—
58C24C281782595yes—
59#41C28187296yes—
60#41C28122260480yes—
61C2×Q8C2815565120yes—
62C2×D4C28116695360yes—
63C22×C4C28186955yes—
64C22×C4C2815478165yes—
65C23—8782595no—
66C23C29186955yes—
67Q8C29187296yes—
68D4C291261888yes—
69C2×C4C291260865yes—
70C2×C4C2911023yes—
71C22—486955no—
72C22C21011023yes—
73C4C21011023yes—
74C4C21011yescharacteristic
75C2—21023no—
76C2C21111yescharacteristic
subgroupG/H
center, socleC4C210
commutator, Frattini, cosocleC2C211
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ G
LECSG ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#72258201C2semidirect
2. C22×#63103C22semidirect
3. 2−1+10C2semidirect
4. H(11,𝔽2)C2semidirect
5. #72258201C22semidirect
6. #72258201C22semidirect
7. C2×2−1+8C22semidirect
8. C2×H(9,𝔽2)C22semidirect
9. 2−1+8C23semidirect
10. H(9,𝔽2)C23semidirect
11. C2×#63103C23semidirect
12. C2×#63103C23semidirect
13. C22×2−1+6C23semidirect
14. C22×H(7,𝔽2)C23semidirect
15. C23×#585C23semidirect
16. #63103C24semidirect
17. #63103C24semidirect
18. C2×2−1+6C24semidirect
19. C2×H(7,𝔽2)C24semidirect
20. C22×#585C24semidirect
21. C22×#585C24semidirect
22. C23×2−1+4C24semidirect
23. C23×H(5,𝔽2)C24semidirect
24. C24×#41C24semidirect
25. 2−1+6C25semidirect
26. H(7,𝔽2)C25semidirect
27. C2×#585C25semidirect
28. C2×#585C25semidirect
29. C22×2−1+4C25semidirect
30. C22×H(5,𝔽2)C25semidirect
31. C23×#41C25semidirect
32. C23×#41C25semidirect
33. C24×Q8C25semidirect
34. C24×D4C25semidirect
35. C25×C4C25semidirect

Maximal subgroups

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

subgroupindex
C2×#722582012
2−1+102
H(11,𝔽2)2