Finite Group Databasebeta

← #20286481

Subgroups — #20286481

Order 6784.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811429C212yesmaximal, complemented
2#16811424C212yesmaximal, complemented
3#16810890C212yesmaximal, complemented
4#16810883C212yesmaximal, complemented
5#16810800C211yesmaximal, characteristic, complemented
6#16810795C211yesmaximal, characteristic, complemented
7#16810790C212yesmaximal, complemented
8#16810746C212yesmaximal, complemented
9C53×#496C211yesmaximal, characteristic, complemented
10#13133265C2211yescharacteristic, complemented
11#13133262C2211yescharacteristic
12#13133262C2212yescomplemented
13#13133248C2212yescomplemented
14#13133246—22no—
15#13133244—22no—
16#13133243C2212yes—
17#13133243C2212yescomplemented
18#13133243C2212yescomplemented
19#13133241—22no—
20#13133241—22no—
21#13133239C2212yes—
22#13133238—22no—
23#13133236—22no—
24SD16×D53—22no—
25D8×D53—22no—
26#13133223C2212yescomplemented
27#13133223C2212yescomplemented
28#13133153C2212yescomplemented
29#13133145C2212yescomplemented
30#13133145C2212yescomplemented
31#13133136C2211yescharacteristic, complemented
32#13133134C2212yescomplemented
33#13133134C2211yescharacteristic, complemented
34C2×SD16×C53C2212yescomplemented
35C2×D8×C53C2212yescomplemented
36#127×C53C2211yescharacteristic, complemented
37#124×C53C2211yescharacteristic, complemented
38#97×C53C2211yescharacteristic, complemented
39C2×Q8×C53C2311yescharacteristic
40C2×D4×C53—41no—
41C2×D4×C53C2311yescharacteristic
42C2×D4×C53C2312yes—
43C2×C22⋊D53—41no—
44C2×C22⋊D53—42no—
45#11719949—22no—
46Q8×D53—22no—
47#11719947—22no—
48#11719947—22no—
49#11719947—22no—
50D4×D53—22no—
51D4×D53—22no—
52D4×D53—22no—
53#11719945—41no—
54#11719945C2312yes—
55#11719945C2312yes—
56SD16×C53—22no—
57SD16×C53—22no—
58D8×C53—22no—
59D8×C53—22no—
60C2×C8×C53C2312yes—
61C22⋊C4×C53—41no—
62C42×C53C2311yescharacteristic
63#11719925—22no—
64#11719924—22no—
65#11719923—22no—
66#11719923—22no—
67#11719923—22no—
68#11719922—22no—
69#11719922—22no—
70#11719922—22no—
71#11719917C2312yes—
72#11719917C2312yes—
73Dic212D412yescomplemented
74D424D412yescomplemented
75#11719913D412yescomplemented
76#11719913D412yescomplemented
77#11719912—22no—
78#11719912—22no—
79C8×D53—22no—
80C8×D53—22no—
81C23×C53—22no—
82C23×C53—21no—
83C22×D53—42no—
84C22×D53—41no—
85Q8×C53—22no—
86D4×C53—22no—
87D4×C53—22no—
88D4×C53—22no—
89D4×C53—82no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—22no—
94C22⋊D53—22no—
95C22⋊D53—42no—
96C22⋊D53—42no—
97C22⋊D53—42no—
98C2×Dic53—42no—
99C2×Dic53—41no—
100D212—41no—
101D212C2×D412yes—
102D212C2×D412yes—
103C4×D53—22no—
104C4×D53—22no—
105C4×D53—42no—
106Dic106—41no—
107Dic106C2×D412yes—
108Dic106C2×D412yes—
109C8×C53C2×D412yes—
110C8×C53C2×D412yes—
111#88322—22no—
112#88322—22no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—82no—
116C22×C53—81no—
117C22×C53H(5,𝔽2)11yescharacteristic
118C2×D53—22no—
119C2×D53—22no—
120C2×D53—162no—
121C2×D53—161no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53C22×D412yes—
125Dic53—22no—
126Dic53—22no—
127#3288—531nomaximal, Sylow
128C2×C53—21no—
129C2×C53—82no—
130C2×C53—81no—
131C2×C53D4211yescharacteristic
132D53—42no—
133D53—42no—
134#577—532no—
135#576—532no—
136#496D5311yescharacteristic, complemented
137#472—532no—
138#471—532no—
139#455—531no—
140#454—531no—
141#453—532no—
142#444—532no—
143C53#328811yescharacteristic, complemented, Sylow
1442−1+4—531no—
145H(5,𝔽2)—531no—
146H(5,𝔽2)—532no—
147#137—532no—
148#137—1062no—
149AGL(1,ℤ/8)—532no—
150AGL(1,ℤ/8)—532no—
151AGL(1,ℤ/8)—532no—
152AGL(1,ℤ/8)—1062no—
153AGL(1,ℤ/8)—1062no—
154AGL(1,ℤ/8)—1062no—
155#135—532no—
156#135—1062no—
157#135—1062no—
158C2×SD16—1062no—
159C2×SD16C2×D5312yescomplemented
160C2×D8—1062no—
161C2×D8C2×D5312yescomplemented
162#131—532no—
163#131—532no—
164#127C2×D5311yescharacteristic, complemented
165#124C2×D5311yescharacteristic, complemented
166#108—532no—
167#104—532no—
168#104—532no—
169#101—531no—
170#100—532no—
171#100—531no—
172#97C2×D5311yescharacteristic, complemented
173#41—532no—
174#41—532no—
175#41—1062no—
176#41—1062no—
177#41—1062no—
178#41—1062no—
179#41—2121no—
180C2×Q8—1062no—
181C2×Q8C22×D5311yescharacteristic
182C2×D4—41no—
183C2×D4—1062no—
184C2×D4—1062no—
185C2×D4—1062no—
186C2×D4—2121no—
187C2×D4—2122no—
188C2×D4C22×D5311yescharacteristic
189C2×D4C22×D5312yes—
190Q16—532no—
191Q16—1062no—
192SD16—22no—
193SD16—22no—
194SD16—532no—
195SD16—532no—
196SD16—1062no—
197SD16—1062no—
198SD16—1062no—
199SD16—1062no—
200D8—22no—
201D8—22no—
202D8—532no—
203D8—1062no—
204D8—1062no—
205D8—1062no—
206M16—532no—
207M16—532no—
208M16—1062no—
209M16—1062no—
210C2×C8—1062no—
211C2×C8—1062no—
212C2×C8C22×D5312yes—
213C22⋊C4—41no—
214C42C22×D5311yescharacteristic
215C23—21no—
216C23—22no—
217C23—2122no—
218C23—2121no—
219Q8—22no—
220Q8—532no—
221Q8—532no—
222Q8—2121no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—532no—
228D4—532no—
229D4—1062no—
230D4—1062no—
231D4—2121no—
232D4—2122no—
233D4—2122no—
234D4—2122no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4—2122no—
242C2×C4C23×D5311yescharacteristic
243C8—1062no—
244C8—1062no—
245C8D4×D5312yes—
246C8D4×D5312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—8481no—
252C22—8482no—
253C22—1062no—
254C22—1062no—
255C22#1313326211yescharacteristic
256C4—41no—
257C4—41no—
258C4—1062no—
259C4—1062no—
260C4C2×D4×D5312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—2122no—
265C2—2122no—
266C2#1681125811yescharacteristic
subgroupG/H
centerC2#16811258
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC2×C53D42
hypercenter#496D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16811429C2semidirect
2. #16811424C2semidirect
3. #16810890C2semidirect
4. #16810883C2semidirect
5. #16810800C2semidirect
6. #16810795C2semidirect
7. #16810790C2semidirect
8. #16810746C2semidirect
9. C53×#496C2semidirect
10. #13133265C22semidirect
12. #13133262C22semidirect
13. #13133248C22semidirect
17. #13133243C22semidirect
18. #13133243C22semidirect
26. #13133223C22semidirect
27. #13133223C22semidirect
28. #13133153C22semidirect
29. #13133145C22semidirect
30. #13133145C22semidirect
31. #13133136C22semidirect
32. #13133134C22semidirect
33. #13133134C22semidirect
34. C2×SD16×C53C22semidirect
35. C2×D8×C53C22semidirect
36. #127×C53C22semidirect
37. #124×C53C22semidirect
38. #97×C53C22semidirect
73. Dic212D4semidirect
74. D424D4semidirect
75. #11719913D4semidirect
76. #11719913D4semidirect
136. #496D53semidirect
143. C53#3288semidirect
159. C2×SD16C2×D53semidirect
161. C2×D8C2×D53semidirect
164. #127C2×D53semidirect
165. #124C2×D53semidirect
172. #97C2×D53semidirect