Finite Group Databasebeta

← #31821214

Subgroups — #31821214

Order 17792.

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#22581789C2112yesmaximal, complemented
2#22581788C2116yesmaximal, complemented
3C2×#17984887C2118yesmaximal, complemented
4H(5,𝔽2)×D139C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C139C211yesmaximal, characteristic, complemented
6#17984890C2218yescomplemented
7#17984889C22148yescomplemented
8#17984888C22148yescomplemented
9#17984887C221144yescomplemented
10#17984887C22172yescomplemented
11C2×#14555785C22136yescomplemented
12C2×#14555784C22172yescomplemented
13#41×D139C22196yescomplemented
14C22×C22⋊D139C22124yescomplemented
15C2×D4×D139C22172yescomplemented
16H(5,𝔽2)×C139C22116yescomplemented
17C2×#41×C139C2216yescomplemented
18C22×D4×C139C2219yescomplemented
19C23×D139C23124yescomplemented
20#14555786C23132yescomplemented
21#14555785C231144yescomplemented
22#14555785C23148yescomplemented
23#14555784C231288yescomplemented
24C2×Dic278C23136yescomplemented
25Q8×D139C23132yescomplemented
26C2×D556C23136yescomplemented
27C2×C22⋊D139C231144yescomplemented
28C2×C22⋊D139C23172yescomplemented
29D4×D139C231288yescomplemented
30C22×Dic139C23124yescomplemented
31C2×C4×D139C23172yescomplemented
32C24×C139C2316yescomplemented
33#41×C139C23148yescomplemented
34C2×Q8×C139C2312yescomplemented
35C2×D4×C139C23172yescomplemented
36C2×D4×C139C23118yescomplemented
37C22×C4×C139C2319yescomplemented
38C22×D139—824no—
39C22×D139C24172yes—
40Dic278C24148yes—
41D556C24148yes—
42C22⋊D139C241144yes—
43C22⋊D139C24116yes—
44C2×Dic139C24172yes—
45C4×D139C24196yes—
46C23×C139—86no—
47C23×C139C2419yes—
48C23×C139C24124yes—
49Q8×C139C2418yes—
50D4×C139C24172yes—
51C2×C4×C139C2416yes—
52C2×C4×C139C24136yes—
53C22×C139—49no—
54C22×C139—424no—
55C22×C139C2511yescharacteristic
56C22×C139C25118yes—
57C2×D139—472no—
58C2×D139C25116yes—
59C4×C139C25112yes—
60Dic139C25116yes—
61C2×C139—218no—
62C2×C139—21no—
63C2×C139C2611yescharacteristic
64D139—216no—
65C139H(7,𝔽2)11yescharacteristic, complemented, Sylow
66H(7,𝔽2)—1391nomaximal, Sylow
67#585—13916no—
68#585—13912no—
69C2×H(5,𝔽2)—13916no—
70C2×H(5,𝔽2)—13918no—
71C2×H(5,𝔽2)D13911yescharacteristic, complemented
722−1+4—13948no—
732−1+4—1398no—
74H(5,𝔽2)—13948no—
75H(5,𝔽2)—139144no—
76H(5,𝔽2)—13972no—
77H(5,𝔽2)C2×D139116yescomplemented
78C2×#41—13996no—
79C2×#41—13972no—
80C2×#41—13936no—
81C2×#41C2×D13916yescomplemented
82C22×D4—13924no—
83C22×D4—13972no—
84C22×D4C2×D13919yescomplemented
85C24—13924no—
86C24C22×D13916yescomplemented
87#41—13932no—
88#41—139288no—
89#41—139144no—
90#41—13948no—
91#41C22×D139148yescomplemented
92C2×Q8—13932no—
93C2×Q8—13936no—
94C2×Q8C22×D13912yescomplemented
95C2×D4—139144no—
96C2×D4—13972no—
97C2×D4—139288no—
98C2×D4—13936no—
99C2×D4C22×D139172yescomplemented
100C2×D4C22×D139118yescomplemented
101C22×C4—13924no—
102C22×C4—13972no—
103C22×C4C22×D13919yescomplemented
104C23—86no—
105C23—111224no—
106C23—13972no—
107C23C23×D139124yes—
108C23C23×D13919yes—
109Q8—13948no—
110Q8C23×D13918yes—
111D4—139144no—
112D4—13916no—
113D4—13948no—
114D4C23×D139172yes—
115C2×C4—13972no—
116C2×C4—13996no—
117C2×C4C23×D139136yes—
118C2×C4C23×D13916yes—
119C22—49no—
120C22—424no—
121C22—55672no—
122C22—13916no—
123C22C24×D139118yes—
124C22C24×D13911yescharacteristic
125C4—13916no—
126C4C24×D139112yes—
127C2—21no—
128C2—218no—
129C2—27816no—
130C2C25×D13911yescharacteristic
subgroupG/H
center, FrattiniC2C25×D139
commutator, socle, cosocleC2×C139C26
FittingC2×H(5,𝔽2)×C139C2
hypercenterC2×H(5,𝔽2)D139
hypocenterC139H(7,𝔽2)
LCSG ▷ C2×C139 ▷ C139
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22581789C2semidirect
2. #22581788C2semidirect
3. C2×#17984887C2semidirect
4. H(5,𝔽2)×D139C2semidirect
5. C2×H(5,𝔽2)×C139C2semidirect
6. #17984890C22semidirect
7. #17984889C22semidirect
8. #17984888C22semidirect
9. #17984887C22semidirect
10. #17984887C22semidirect
11. C2×#14555785C22semidirect
12. C2×#14555784C22semidirect
13. #41×D139C22semidirect
14. C22×C22⋊D139C22semidirect
15. C2×D4×D139C22semidirect
16. H(5,𝔽2)×C139C22semidirect
17. C2×#41×C139C22semidirect
18. C22×D4×C139C22semidirect
19. C23×D139C23semidirect
20. #14555786C23semidirect
21. #14555785C23semidirect
22. #14555785C23semidirect
23. #14555784C23semidirect
24. C2×Dic278C23semidirect
25. Q8×D139C23semidirect
26. C2×D556C23semidirect
27. C2×C22⋊D139C23semidirect
28. C2×C22⋊D139C23semidirect
29. D4×D139C23semidirect
30. C22×Dic139C23semidirect
31. C2×C4×D139C23semidirect
32. C24×C139C23semidirect
33. #41×C139C23semidirect
34. C2×Q8×C139C23semidirect
35. C2×D4×C139C23semidirect
36. C2×D4×C139C23semidirect
37. C22×C4×C139C23semidirect
65. C139H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D139semidirect
77. H(5,𝔽2)C2×D139semidirect
81. C2×#41C2×D139semidirect
84. C22×D4C2×D139semidirect
86. C24C22×D139semidirect
91. #41C22×D139semidirect
94. C2×Q8C22×D139semidirect
99. C2×D4C22×D139semidirect
100. C2×D4C22×D139semidirect
103. C22×C4C22×D139semidirect

Maximal subgroups

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

subgroupindex
#225817892
#225817882
C2×#179848872
H(5,𝔽2)×D1392
C2×H(5,𝔽2)×C1392
H(7,𝔽2)139