Finite Group Databasebeta

← #20288493

Subgroups — #20288493

Order 6784.

1748 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 557 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
1C2×#13133263C212yesmaximal, complemented
2#16811295C214yesmaximal, complemented
3D4×D212C2112yesmaximal, complemented
4#16811192C216yesmaximal, complemented
5#16811173C213yesmaximal, complemented
6C2×#13133214C213yesmaximal, complemented
7C4×#41×C53C211yesmaximal, characteristic, complemented
8#13133263—216no—
9C2×#11719949C2214yescomplemented
10C2×#11719945C2216yescomplemented
11C22×D212C2216yescomplemented
12C22×D212C22112yescomplemented
13C2×D4×D53C22112yescomplemented
14#13133217C2211yescharacteristic, complemented
15#13133214C22112yescomplemented
16#13133214C2213yescomplemented
17#13133214C2216yescomplemented
18#13133205C2216yescomplemented
19#13133192C22124yescomplemented
20#13133191C22112yescomplemented
21#13133187C22124yescomplemented
22C4×D212C22112yescomplemented
23C2×#41×C53C2211yescharacteristic
24C4×Q8×C53C2212yescomplemented
25C4×D4×C53C2216yescomplemented
26#117×C53C2213yescomplemented
27C2×C42×C53C2213yescomplemented
28C23×D53C23112yes—
29#41×C53D418yescomplemented
30C2×Q8×C53C2311yescharacteristic
31C2×D4×C53C2313yes—
32C22×C4×C53C2316yes—
33C22×C4×C53C2313yes—
34C2×C22⋊D53C23112yes—
35#11719949—216no—
36D4×D53—248no—
37#11719945—224no—
38C2×D212—224no—
39C2×D212—424no—
40C2×D212C23112yes—
41C2×D212C23112yes—
42C2×D212C2316yes—
43C2×D212C23124yes—
44C2×C4×D53C23112yes—
45C2×Dic106C2312yes—
46C4⋊C4×C53C2316yes—
47C22⋊C4×C53C2316yes—
48C42×C53C2311yescharacteristic
49C42×C53C2313yes—
50C42×C53C2316yes—
51#11719921C23124yes—
52C4⋊Dic53C2314yes—
53C23×C53C2413yes—
54C22×D53—224no—
55C22×D53—424no—
56C22×D53C24112yes—
57Q8×C53C2×D414yescomplemented
58D4×C53C2×D4112yescomplemented
59C2×C4×C53—212no—
60C2×C4×C53C2×D4112yescomplemented
61C2×C4×C53C2416yes—
62C2×C4×C53C2411yescharacteristic
63C2×C4×C53C2413yes—
64C2×C4×C53C2412yes—
65C22⋊D53—224no—
66C2×Dic53C2414yes—
67D212—212no—
68D212—224no—
69D212—224no—
70D212—424no—
71C4×D53—224no—
72Dic106—24no—
73C22×C53—26no—
74C22×C53C22×D416yes—
75C22×C53C2511yescharacteristic
76C2×D53—212no—
77C2×D53—224no—
78C2×D53—824no—
79C4×C53—26no—
80C4×C53C22×D412yes—
81C4×C53C22×D416yes—
82C4×C53H(5,𝔽2)14yescomplemented
83Dic53—24no—
84#3498—531nomaximal, Sylow
85C2×C53—26no—
86C2×C53C23×D411yescharacteristic
87C2×C53C2×H(5,𝔽2)12yes—
88D53—412no—
89C2×H(5,𝔽2)—532no—
90#550—534no—
91D42—5312no—
92#535—536no—
93#532—533no—
94C2×#127—533no—
95C4×#41D5311yescharacteristic, complemented
96C53#349811yescharacteristic, complemented, Sylow
97H(5,𝔽2)—10616no—
98C2×#41—534no—
99C2×#41—536no—
100C2×#41C2×D5311yescharacteristic
101C22×D4—5312no—
102C22×D4—536no—
103C22×D4—5312no—
104C4⋊Q8—531no—
105#127—533no—
106#127—536no—
107#127—5312no—
108#124—536no—
109#121—5312no—
110#121—5324no—
111#120—5324no—
112C4×Q8C2×D5312yescomplemented
113C4×D4—5312no—
114C4×D4C2×D5316yescomplemented
115#117C2×D5313yescomplemented
116C2×C42C2×D5313yescomplemented
117C24—5312no—
118#41—10616no—
119#41—10624no—
120#41D21218yescomplemented
121C2×Q8—532no—
122C2×Q8C22×D5311yescharacteristic
123C2×D4—5312no—
124C2×D4—5312no—
125C2×D4—5312no—
126C2×D4—536no—
127C2×D4—5324no—
128C2×D4—10648no—
129C2×D4—10624no—
130C2×D4—21224no—
131C2×D4C22×D5313yes—
132C22×C4—5312no—
133C22×C4C22×D5316yes—
134C22×C4C22×D5313yes—
135C4⋊C4—534no—
136C4⋊C4C22×D5316yes—
137C22⋊C4—5324no—
138C22⋊C4C22×D5316yes—
139C42C22×D5311yescharacteristic
140C42C22×D5313yes—
141C42C22×D5316yes—
142C23—5312no—
143C23—10624no—
144C23—21224no—
145C23C23×D5313yes—
146Q8—1064no—
147Q8C2×D21214yescomplemented
148D4—10624no—
149D4—10624no—
150D4—10612no—
151D4—10624no—
152D4—21224no—
153D4C2×D212112yescomplemented
154C2×C4—212no—
155C2×C4—534no—
156C2×C4—10624no—
157C2×C4C2×D212112yescomplemented
158C2×C4C23×D5316yes—
159C2×C4C23×D5311yescharacteristic
160C2×C4C23×D5313yes—
161C2×C4C23×D5312yes—
162C22—26no—
163C22—10612no—
164C22—10624no—
165C22—42424no—
166C22C22×D21216yes—
167C22C24×D5311yescharacteristic
168C4—26no—
169C4—1064no—
170C4C22×D21216yes—
171C4C22×D21212yes—
172C4#1313326314yescomplemented
173C2—26no—
174C2—21212no—
175C2C23×D21211yescharacteristic
176C2C2×#1313326312yes—
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC4×#41×C53C2
hypercenterC4×#41D53
hypocenterC53#3498
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. #16811295C2semidirect
3. D4×D212C2semidirect
4. #16811192C2semidirect
5. #16811173C2semidirect
6. C2×#13133214C2semidirect
7. C4×#41×C53C2semidirect
9. C2×#11719949C22semidirect
10. C2×#11719945C22semidirect
11. C22×D212C22semidirect
12. C22×D212C22semidirect
13. C2×D4×D53C22semidirect
14. #13133217C22semidirect
15. #13133214C22semidirect
16. #13133214C22semidirect
17. #13133214C22semidirect
18. #13133205C22semidirect
19. #13133192C22semidirect
20. #13133191C22semidirect
21. #13133187C22semidirect
22. C4×D212C22semidirect
24. C4×Q8×C53C22semidirect
25. C4×D4×C53C22semidirect
26. #117×C53C22semidirect
27. C2×C42×C53C22semidirect
29. #41×C53D4semidirect
57. Q8×C53C2×D4semidirect
58. D4×C53C2×D4semidirect
60. C2×C4×C53C2×D4semidirect
82. C4×C53H(5,𝔽2)semidirect
95. C4×#41D53semidirect
96. C53#3498semidirect
112. C4×Q8C2×D53semidirect
114. C4×D4C2×D53semidirect
115. #117C2×D53semidirect
116. C2×C42C2×D53semidirect
120. #41D212semidirect
147. Q8C2×D212semidirect
153. D4C2×D212semidirect
157. C2×C4C2×D212semidirect
172. C4#13133263semidirect