Finite Group Databasebeta

← #28377842

Subgroups — #28377842

Order 13952.

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#20295862C2112yesmaximal, complemented
2#20295861C2116yesmaximal, complemented
3C2×#16812744C2118yesmaximal, complemented
4H(5,𝔽2)×D109C2116yesmaximal, complemented
5C2×H(5,𝔽2)×C109C211yesmaximal, characteristic, complemented
6#16812747C2218yescomplemented
7#16812746C22148yescomplemented
8#16812745C22148yescomplemented
9#16812744C221144yescomplemented
10#16812744C22172yescomplemented
11C2×#13182029C22136yescomplemented
12C2×#13182028C22172yescomplemented
13#41×D109C22196yescomplemented
14C22×C22⋊D109C22124yescomplemented
15C2×D4×D109C22172yescomplemented
16H(5,𝔽2)×C109C22116yescomplemented
17C2×#41×C109C2216yescomplemented
18C22×D4×C109C2219yescomplemented
19C23×D109C23124yescomplemented
20#13182030C23132yescomplemented
21#13182029C231144yescomplemented
22#13182029C23148yescomplemented
23#13182028C231288yescomplemented
24C2×Dic218C23136yescomplemented
25Q8×D109C23132yescomplemented
26C2×D436C23136yescomplemented
27C2×C22⋊D109C231144yescomplemented
28C2×C22⋊D109C23172yescomplemented
29D4×D109C231288yescomplemented
30C22×Dic109C23124yescomplemented
31C2×C4×D109C23172yescomplemented
32C24×C109C2316yescomplemented
33#41×C109C23148yescomplemented
34C2×Q8×C109C2312yescomplemented
35C2×D4×C109C23118yescomplemented
36C2×D4×C109C23172yescomplemented
37C22×C4×C109C2319yescomplemented
38C23×C109—86no—
39C23×C109C2419yes—
40C23×C109C24124yes—
41C22×D109—824no—
42C22×D109C24172yes—
43Q8×C109C2418yes—
44D4×C109C24172yes—
45C2×C4×C109C2416yes—
46C2×C4×C109C24136yes—
47C22⋊D109C241144yes—
48C22⋊D109C24116yes—
49C2×Dic109C24172yes—
50D436C24148yes—
51C4×D109C24196yes—
52Dic218C24148yes—
53C22×C109—424no—
54C22×C109—49no—
55C22×C109C25118yes—
56C22×C109C2511yescharacteristic
57C2×D109—472no—
58C2×D109C25116yes—
59C4×C109C25112yes—
60Dic109C25116yes—
61C2×C109—218no—
62C2×C109—21no—
63C2×C109C2611yescharacteristic
64D109—216no—
65H(7,𝔽2)—1091nomaximal, Sylow
66C109H(7,𝔽2)11yescharacteristic, complemented, Sylow
67#585—10916no—
68#585—10912no—
69C2×H(5,𝔽2)—10916no—
70C2×H(5,𝔽2)—10918no—
71C2×H(5,𝔽2)D10911yescharacteristic, complemented
722−1+4—10948no—
732−1+4—1098no—
74H(5,𝔽2)—10948no—
75H(5,𝔽2)—109144no—
76H(5,𝔽2)—10972no—
77H(5,𝔽2)C2×D109116yescomplemented
78C2×#41—10996no—
79C2×#41—10972no—
80C2×#41—10936no—
81C2×#41C2×D10916yescomplemented
82C22×D4—10924no—
83C22×D4—10972no—
84C22×D4C2×D10919yescomplemented
85C24—10924no—
86C24C22×D10916yescomplemented
87#41—10932no—
88#41—109288no—
89#41—109144no—
90#41—10948no—
91#41C22×D109148yescomplemented
92C2×Q8—10932no—
93C2×Q8—10936no—
94C2×Q8C22×D10912yescomplemented
95C2×D4—109144no—
96C2×D4—10972no—
97C2×D4—109288no—
98C2×D4—10936no—
99C2×D4C22×D109172yescomplemented
100C2×D4C22×D109118yescomplemented
101C22×C4—10924no—
102C22×C4—10972no—
103C22×C4C22×D10919yescomplemented
104C23—86no—
105C23—87224no—
106C23—10972no—
107C23C23×D109124yes—
108C23C23×D10919yes—
109Q8—10948no—
110Q8C23×D10918yes—
111D4—109144no—
112D4—10916no—
113D4—10948no—
114D4C23×D109172yes—
115C2×C4—10972no—
116C2×C4—10996no—
117C2×C4C23×D109136yes—
118C2×C4C23×D10916yes—
119C22—49no—
120C22—424no—
121C22—10916no—
122C22—43672no—
123C22C24×D109118yes—
124C22C24×D10911yescharacteristic
125C4—10916no—
126C4C24×D109112yes—
127C2—21no—
128C2—218no—
129C2—21816no—
130C2C25×D10911yescharacteristic
subgroupG/H
center, FrattiniC2C25×D109
commutator, socle, cosocleC2×C109C26
FittingC2×H(5,𝔽2)×C109C2
hypercenterC2×H(5,𝔽2)D109
hypocenterC109H(7,𝔽2)
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. #20295861C2semidirect
3. C2×#16812744C2semidirect
4. H(5,𝔽2)×D109C2semidirect
5. C2×H(5,𝔽2)×C109C2semidirect
6. #16812747C22semidirect
7. #16812746C22semidirect
8. #16812745C22semidirect
9. #16812744C22semidirect
10. #16812744C22semidirect
11. C2×#13182029C22semidirect
12. C2×#13182028C22semidirect
13. #41×D109C22semidirect
14. C22×C22⋊D109C22semidirect
15. C2×D4×D109C22semidirect
16. H(5,𝔽2)×C109C22semidirect
17. C2×#41×C109C22semidirect
18. C22×D4×C109C22semidirect
19. C23×D109C23semidirect
20. #13182030C23semidirect
21. #13182029C23semidirect
22. #13182029C23semidirect
23. #13182028C23semidirect
24. C2×Dic218C23semidirect
25. Q8×D109C23semidirect
26. C2×D436C23semidirect
27. C2×C22⋊D109C23semidirect
28. C2×C22⋊D109C23semidirect
29. D4×D109C23semidirect
30. C22×Dic109C23semidirect
31. C2×C4×D109C23semidirect
32. C24×C109C23semidirect
33. #41×C109C23semidirect
34. C2×Q8×C109C23semidirect
35. C2×D4×C109C23semidirect
36. C2×D4×C109C23semidirect
37. C22×C4×C109C23semidirect
66. C109H(7,𝔽2)semidirect
71. C2×H(5,𝔽2)D109semidirect
77. H(5,𝔽2)C2×D109semidirect
81. C2×#41C2×D109semidirect
84. C22×D4C2×D109semidirect
86. C24C22×D109semidirect
91. #41C22×D109semidirect
94. C2×Q8C22×D109semidirect
99. C2×D4C22×D109semidirect
100. C2×D4C22×D109semidirect
103. C22×C4C22×D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
#202958612
C2×#168127442
H(5,𝔽2)×D1092
C2×H(5,𝔽2)×C1092
H(7,𝔽2)109