Finite Group Databasebeta

← #36368373

Subgroups — #36368373

Order 22144.

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#26012327C2112yesmaximal, complemented
2#26012326C2116yesmaximal, complemented
3C2×#19149724C2118yesmaximal, complemented
4H(5,𝔽2)×D173C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C173C211yesmaximal, characteristic, complemented
6#19149727C2218yescomplemented
7#19149726C22148yescomplemented
8#19149725C22148yescomplemented
9#19149724C221144yescomplemented
10#19149724C22172yescomplemented
11C2×#14583826C22136yescomplemented
12C2×#14583825C22172yescomplemented
13#41×D173C22196yescomplemented
14C22×C22⋊D173C22124yescomplemented
15C2×D4×D173C22172yescomplemented
16H(5,𝔽2)×C173C22116yescomplemented
17C2×#41×C173C2216yescomplemented
18C22×D4×C173C2219yescomplemented
19C23×D173C23124yescomplemented
20#14583827C23132yescomplemented
21#14583826C231144yescomplemented
22#14583826C23148yescomplemented
23#14583825C231288yescomplemented
24C2×Dic346C23136yescomplemented
25Q8×D173C23132yescomplemented
26C2×D692C23136yescomplemented
27C2×C22⋊D173C231144yescomplemented
28C2×C22⋊D173C23172yescomplemented
29D4×D173C231288yescomplemented
30C22×Dic173C23124yescomplemented
31C2×C4×D173C23172yescomplemented
32C24×C173C2316yescomplemented
33#41×C173C23148yescomplemented
34C2×Q8×C173C2312yescomplemented
35C2×D4×C173C23118yescomplemented
36C2×D4×C173C23172yescomplemented
37C22×C4×C173C2319yescomplemented
38C22×D173—824no—
39C22×D173C24172yes—
40Dic346C24148yes—
41D692C24148yes—
42C22⋊D173C241144yes—
43C22⋊D173C24116yes—
44C2×Dic173C24172yes—
45C4×D173C24196yes—
46C23×C173—86no—
47C23×C173C24124yes—
48C23×C173C2419yes—
49Q8×C173C2418yes—
50D4×C173C24172yes—
51C2×C4×C173C24136yes—
52C2×C4×C173C2416yes—
53C22×C173—49no—
54C22×C173—424no—
55C22×C173C2511yescharacteristic
56C22×C173C25118yes—
57C2×D173—472no—
58C2×D173C25116yes—
59C4×C173C25112yes—
60Dic173C25116yes—
61C2×C173—218no—
62C2×C173—21no—
63C2×C173C2611yescharacteristic
64D173—216no—
65C173H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—1731nomaximal, Sylow
67#585—17316no—
68#585—17312no—
69C2×H(5,𝔽2)—17316no—
70C2×H(5,𝔽2)—17318no—
71C2×H(5,𝔽2)D17311yescharacteristic, complemented
722−1+4—17348no—
732−1+4—1738no—
74H(5,𝔽2)—17348no—
75H(5,𝔽2)—173144no—
76H(5,𝔽2)—17372no—
77H(5,𝔽2)C2×D173116yescomplemented
78C2×#41—17396no—
79C2×#41—17372no—
80C2×#41—17336no—
81C2×#41C2×D17316yescomplemented
82C22×D4—17324no—
83C22×D4—17372no—
84C22×D4C2×D17319yescomplemented
85C24—17324no—
86C24C22×D17316yescomplemented
87#41—17332no—
88#41—173288no—
89#41—173144no—
90#41—17348no—
91#41C22×D173148yescomplemented
92C2×Q8—17332no—
93C2×Q8—17336no—
94C2×Q8C22×D17312yescomplemented
95C2×D4—173144no—
96C2×D4—17372no—
97C2×D4—173288no—
98C2×D4—17336no—
99C2×D4C22×D173172yescomplemented
100C2×D4C22×D173118yescomplemented
101C22×C4—17324no—
102C22×C4—17372no—
103C22×C4C22×D17319yescomplemented
104C23—86no—
105C23—138424no—
106C23—17372no—
107C23C23×D173124yes—
108C23C23×D17319yes—
109Q8—17348no—
110Q8C23×D17318yes—
111D4—173144no—
112D4—17316no—
113D4—17348no—
114D4C23×D173172yes—
115C2×C4—17372no—
116C2×C4—17396no—
117C2×C4C23×D173136yes—
118C2×C4C23×D17316yes—
119C22—49no—
120C22—424no—
121C22—17316no—
122C22—69272no—
123C22C24×D173118yes—
124C22C24×D17311yescharacteristic
125C4—17316no—
126C4C24×D173112yes—
127C2—21no—
128C2—218no—
129C2—34616no—
130C2C25×D17311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D173
commutator, socle, cosocleC2×C173C26
FittingC2×H(5,𝔽2)×C173C2
hypercenterC2×H(5,𝔽2)D173
hypocenterC173H(7,𝔽2)
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012326C2semidirect
3. C2×#19149724C2semidirect
4. H(5,𝔽2)×D173C2semidirect
5. C2×H(5,𝔽2)×C173C2semidirect
6. #19149727C22semidirect
7. #19149726C22semidirect
8. #19149725C22semidirect
9. #19149724C22semidirect
10. #19149724C22semidirect
11. C2×#14583826C22semidirect
12. C2×#14583825C22semidirect
13. #41×D173C22semidirect
14. C22×C22⋊D173C22semidirect
15. C2×D4×D173C22semidirect
16. H(5,𝔽2)×C173C22semidirect
17. C2×#41×C173C22semidirect
18. C22×D4×C173C22semidirect
19. C23×D173C23semidirect
20. #14583827C23semidirect
21. #14583826C23semidirect
22. #14583826C23semidirect
23. #14583825C23semidirect
24. C2×Dic346C23semidirect
25. Q8×D173C23semidirect
26. C2×D692C23semidirect
27. C2×C22⋊D173C23semidirect
28. C2×C22⋊D173C23semidirect
29. D4×D173C23semidirect
30. C22×Dic173C23semidirect
31. C2×C4×D173C23semidirect
32. C24×C173C23semidirect
33. #41×C173C23semidirect
34. C2×Q8×C173C23semidirect
35. C2×D4×C173C23semidirect
36. C2×D4×C173C23semidirect
37. C22×C4×C173C23semidirect
65. C173H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D173semidirect
77. H(5,𝔽2)C2×D173semidirect
81. C2×#41C2×D173semidirect
84. C22×D4C2×D173semidirect
86. C24C22×D173semidirect
91. #41C22×D173semidirect
94. C2×Q8C22×D173semidirect
99. C2×D4C22×D173semidirect
100. C2×D4C22×D173semidirect
103. C22×C4C22×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260123262
C2×#191497242
H(5,𝔽2)×D1732
C2×H(5,𝔽2)×C1732
H(7,𝔽2)173