Finite Group Databasebeta

← #20284072

Subgroups — #20284072

Order 6784.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#16811421C212yesmaximal, complemented
2#16811190C211yesmaximal, characteristic, complemented
3#16810801C214yesmaximal, complemented
4#16810791C214yesmaximal, complemented
5#16810640C212yesmaximal, complemented
6C2×#13133134C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C53C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic, complemented
9#13133248—22no—
10#13133247C2214yescomplemented
11#13133243—22no—
12#13133242C2214yescomplemented
13#13133237—24no—
14C2×#11719917C2212yescomplemented
15#13133213C2212yescomplemented
16#13133204C2212yescomplemented
17#13133188—22no—
18#13133186—22no—
19#13133176C2211yescharacteristic, complemented
20#13133144C2218yescomplemented
21#13133134C2214yescomplemented
22H(5,𝔽2)×C53—24no—
23H(5,𝔽2)×C53—22no—
24H(5,𝔽2)×C53C2214yes—
25C2×#41×C53—22no—
26C2×#41×C53C2212yes—
27C22×D4×C53—22no—
28C22×D4×C53—22no—
29C22×D4×C53C2211yescharacteristic
30C24×C53—22no—
31C24×C53D412yes—
32#41×C53—28no—
33#41×C53—28no—
34#41×C53—22no—
35#41×C53—22no—
36#41×C53D414yescomplemented
37#41×C53C2314yes—
38C2×Q8×C53D412yescomplemented
39C2×D4×C53—24no—
40C2×D4×C53—28no—
41C2×D4×C53—28no—
42C2×D4×C53—22no—
43C2×D4×C53—28no—
44C2×D4×C53—28no—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53D412yes—
48C2×D4×C53D412yescomplemented
49C2×D4×C53C2312yes—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53C2311yescharacteristic
53C2×C22⋊D53—21no—
54C2×C22⋊D53—24no—
55#11719945—22no—
56#11719945—22no—
57C2×D212C2311yescharacteristic, complemented
58C2×C4×D53—21no—
59C2×Dic106C2311yescharacteristic, complemented
60C22⋊Dic53—21no—
61C22⋊Dic53—24no—
62#11719925—24no—
63#11719924—24no—
64#11719923—24no—
65#11719922—24no—
66C4⋊Dic53—21no—
67C4×Dic53C2312yescomplemented
68#11719917—22no—
69#11719917C2314yescomplemented
70C2×#88322—22no—
71C23×C53—28no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—82no—
76C23×C53—84no—
77C23×C53C2×D414yes—
78C23×C53C2×D411yescharacteristic
79C22×D53—21no—
80Q8×C53—22no—
81Q8×C53C2×D414yes—
82D4×C53—216no—
83D4×C53—24no—
84D4×C53—24no—
85D4×C53—24no—
86D4×C53—22no—
87D4×C53—24no—
88D4×C53C2×D414yes—
89C2×C4×C53—28no—
90C2×C4×C53—22no—
91C2×C4×C53—24no—
92C2×C4×C53—24no—
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2×D412yes—
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—41no—
98C22⋊D53—41no—
99C22⋊D53—84no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102D212—22no—
103C4×D53—42no—
104Dic106—22no—
105#88322—24no—
106C22×C53—24no—
107C22×C53—24no—
108C22×C53—41no—
109C22×C53—44no—
110C22×C53—44no—
111C22×C53—44no—
112C22×C53—82no—
113C22×C53—88no—
114C22×C53#12011yescharacteristic
115C22×C53#12011yescharacteristic
116C22×C53C22×D411yescharacteristic
117C2×D53—42no—
118C2×D53—41no—
119C4×C53—24no—
120C4×C53#12012yes—
121C4×C53C22×D412yes—
122Dic53—42no—
123Dic53—41no—
124#3019—531nomaximal, Sylow
125C2×C53—21no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—44no—
129C2×C53—44no—
130C2×C53C2×#12011yescharacteristic
131D53—81no—
132C2×H(5,𝔽2)D5311yescharacteristic, complemented
133#575—532no—
134#535—531no—
135#455—534no—
136#453—534no—
137#421—532no—
138C2×#100—531no—
139C53#301911yescharacteristic, complemented, Sylow
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D5314yes—
143C2×#41—22no—
144C2×#41—531no—
145C2×#41C2×D5312yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D5311yescharacteristic
149#137—534no—
150#137—1062no—
151AGL(1,ℤ/8)—534no—
152AGL(1,ℤ/8)—1062no—
153#135—1064no—
154C2×M16—532no—
155#127—532no—
156#124—532no—
157#121—1062no—
158#120—1062no—
159#117—531no—
160#104—538no—
161#100—534no—
162C24—22no—
163C24C22⋊D5312yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—1062no—
169#41—1062no—
170#41C22⋊D5314yescomplemented
171#41C22×D5314yes—
172C2×Q8—531no—
173C2×Q8C22⋊D5312yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—531no—
183C2×D4—1064no—
184C2×D4—1061no—
185C2×D4C22⋊D5312yes—
186C2×D4C22⋊D5312yescomplemented
187C2×D4C22×D5312yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1061no—
191C22×C4C22×D5311yescharacteristic
192Q16—1064no—
193SD16—1064no—
194SD16—1064no—
195D8—1064no—
196M16—534no—
197M16—1062no—
198C2×C8—1062no—
199C4⋊C4—1061no—
200C22⋊C4—1061no—
201C22⋊C4—1064no—
202C42—532no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—1061no—
210C23C2×C22⋊D5314yes—
211C23C2×C22⋊D5311yescharacteristic
212Q8—22no—
213Q8—1062no—
214Q8C2×C22⋊D5314yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—1062no—
222D4—4244no—
223D4—2121no—
224D4—2121no—
225D4C2×C22⋊D5314yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—2122no—
233C2×C4C2×C22⋊D5311yescharacteristic
234C2×C4C2×C22⋊D5312yes—
235C2×C4C2×C22⋊D5312yes—
236C2×C4C23×D5311yescharacteristic
237C8—1064no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—2122no—
247C22—2121no—
248C22#1313318611yescharacteristic
249C22#1313318611yescharacteristic
250C22C22×C22⋊D5311yescharacteristic
251C4—24no—
252C4—2122no—
253C4—2121no—
254C4#1313318612yes—
255C4C22×C22⋊D5312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—4241no—
262C2C2×#1313318611yescharacteristic
subgroupG/H
centerC2C2×#13133186
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×H(5,𝔽2)×C53C2
socleC2×C53C2×#120
hypercenterC2×H(5,𝔽2)D53
hypocenterC53#3019
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #16811421C2semidirect
2. #16811190C2semidirect
3. #16810801C2semidirect
4. #16810791C2semidirect
5. #16810640C2semidirect
6. C2×#13133134C2semidirect
7. C2×H(5,𝔽2)×C53C2semidirect
8. C2×#11719945C22semidirect
10. #13133247C22semidirect
12. #13133242C22semidirect
14. C2×#11719917C22semidirect
15. #13133213C22semidirect
16. #13133204C22semidirect
19. #13133176C22semidirect
20. #13133144C22semidirect
21. #13133134C22semidirect
36. #41×C53D4semidirect
38. C2×Q8×C53D4semidirect
48. C2×D4×C53D4semidirect
57. C2×D212C23semidirect
59. C2×Dic106C23semidirect
67. C4×Dic53C23semidirect
69. #11719917C23semidirect
132. C2×H(5,𝔽2)D53semidirect
139. C53#3019semidirect
170. #41C22⋊D53semidirect
173. C2×Q8C22⋊D53semidirect
186. C2×D4C22⋊D53semidirect