Finite Group Databasebeta

← #20283981

Subgroups — #20283981

Order 6784.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133263C211yesmaximal, characteristic, complemented
2C2×#13133243C211yesmaximal, characteristic, complemented
3C22×#11719922C211yesmaximal, characteristic, complemented
4#16810766C214yesmaximal, complemented
5#16810756C214yesmaximal, complemented
6#16810627C211yesmaximal, characteristic, complemented
7#16810617C211yesmaximal, characteristic, complemented
8#16810590C211yesmaximal, characteristic, complemented
9C53×#535C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949—21no—
13C2×#11719945C2211yescharacteristic
14C2×#11719945C2212yescomplemented
15C22×D212C2211yescharacteristic
16C22×D212C2212yescomplemented
17C2×D4×D53—21no—
18C2×D4×D53—22no—
19#13133243—24no—
20C2×#11719923C2212yescomplemented
21C2×#11719922—24no—
22C2×#11719922C2212yescomplemented
23C2×#11719922C2214yescomplemented
24C2×#11719917C2211yescharacteristic, complemented
25C22×#88322C2211yescharacteristic, complemented
26#13133140C2212yescomplemented
27#13133137C2212yescomplemented
28#13133137C2214yescomplemented
29C22⋊#88322C2214yescomplemented
30C2×#41×C53C2211yescharacteristic
31C22×D4×C53C2211yescharacteristic, complemented
32#127×C53—21no—
33#124×C53—21no—
34#121×C53C2214yescomplemented
35#120×C53—22no—
36#117×C53C2211yescharacteristic, complemented
37C24×C53—21no—
38C23×D53—21no—
39C23×D53—22no—
40#41×C53—22no—
41#41×C53—22no—
42C2×Q8×C53D411yescharacteristic
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—21no—
47C2×D4×C53D412yes—
48C2×D4×C53D411yescharacteristic
49C2×D4×C53C2312yes—
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—21no—
53C2×C22⋊D53—22no—
54#11719949—28no—
55D4×D53—216no—
56D4×D53—28no—
57#11719945—22no—
58#11719945—24no—
59#11719945—24no—
60#11719945D414yescomplemented
61C2×D212—21no—
62C2×D212—22no—
63C2×D212—22no—
64C2×D212—24no—
65C2×D212—24no—
66C2×D212D414yescomplemented
67C2×D212C2312yes—
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—22no—
71C2×Dic106C2311yescharacteristic
72C4⋊C4×C53C2312yes—
73C22⋊C4×C53—21no—
74C22⋊C4×C53—24no—
75C42×C53—21no—
76#11719923—24no—
77#11719922—28no—
78#11719922—24no—
79#11719917—22no—
80C2×#88322—22no—
81C2×#88322C2312yes—
82C2×#88322C2312yes—
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—82no—
86C23×C53C2×D412yes—
87C23×C53C2×D411yescharacteristic
88C22×D53—21no—
89C22×D53—22no—
90C22×D53—28no—
91C22×D53—24no—
92C22×D53—82no—
93C22×D53—84no—
94Q8×C53—22no—
95D4×C53—24no—
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—22no—
99D4×C53—42no—
100D4×C53—84no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D412yes—
106C2×C4×C53C2×D412yes—
107C2×C4×C53C2×D411yescharacteristic
108C2×C4×C53C2411yescharacteristic
109C22⋊D53—28no—
110C22⋊D53—24no—
111C2×Dic53—21no—
112D212—28no—
113D212—24no—
114D212—21no—
115D212—22no—
116D212C2×D412yes—
117D212C2×D414yes—
118C4×D53—28no—
119C4×D53—24no—
120Dic106—21no—
121Dic106C2×D412yes—
122#88322—22no—
123#88322—22no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—24no—
127C22×C53—42no—
128C22×C53—44no—
129C22×C53—84no—
130C22×C53#12012yes—
131C22×C53C22×D411yescharacteristic
132C2×D53—24no—
133C2×D53—22no—
134C2×D53—44no—
135C2×D53—42no—
136C2×D53—84no—
137C2×D53—88no—
138C4×C53—22no—
139C4×C53—42no—
140C4×C53#12012yes—
141C4×C53C22×D411yescharacteristic
142C4×C53C22×D411yescharacteristic
143Dic53—22no—
144#3008—531nomaximal, Sylow
145C2×C53—22no—
146C2×C53—42no—
147C2×C53—82no—
148C2×C53C2×#12011yescharacteristic
149C2×C53#57612yes—
150D53—44no—
151D53—42no—
152C2×H(5,𝔽2)—531no—
153C2×AGL(1,ℤ/8)—531no—
154C22×D8—531no—
155#535D5311yescharacteristic, complemented
156#449—534no—
157#447—534no—
158#418—531no—
159#416—531no—
160#408—531no—
161C53#300811yescharacteristic, complemented, Sylow
162H(5,𝔽2)—1064no—
163H(5,𝔽2)—1064no—
164C2×#41—531no—
165C2×#41—532no—
166C2×#41—1061no—
167C2×#41C2×D5311yescharacteristic
168C22×D4—531no—
169C22×D4—532no—
170C22×D4—1062no—
171C22×D4—1061no—
172C22×D4C2×D5311yescharacteristic, complemented
173AGL(1,ℤ/8)—1064no—
174C2×SD16—532no—
175C2×D8—534no—
176C2×D8—532no—
177C2×D8—1064no—
178C2×M16—531no—
179C22×C8—531no—
180#127—21no—
181#124—21no—
182#121C2×D5314yescomplemented
183#120—22no—
184#117C2×D5311yescharacteristic, complemented
185#103—532no—
186#102—532no—
187#102—534no—
188C22⋊C8—534no—
189C24—21no—
190C24—1062no—
191C24—1061no—
192#41—22no—
193#41—22no—
194#41—534no—
195#41—1064no—
196#41—1064no—
197#41—1068no—
198#41—1062no—
199C2×Q8—531no—
200C2×Q8C22⋊D5311yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—534no—
206C2×D4—532no—
207C2×D4—531no—
208C2×D4—1064no—
209C2×D4—1064no—
210C2×D4—10616no—
211C2×D4—1068no—
212C2×D4—1062no—
213C2×D4—1061no—
214C2×D4—1062no—
215C2×D4—1062no—
216C2×D4—1061no—
217C2×D4C22⋊D5312yes—
218C2×D4C22⋊D5311yescharacteristic
219C2×D4C22×D5312yes—
220C22×C4—1062no—
221C22×C4—1061no—
222C22×C4C22×D5312yes—
223C22×C4C22×D5311yescharacteristic
224SD16—1064no—
225D8—1068no—
226D8—1064no—
227M16—1062no—
228C2×C8—532no—
229C2×C8—532no—
230C2×C8—1062no—
231C4⋊C4C22×D5312yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—1068no—
239C23—1064no—
240C23—1062no—
241C23—1061no—
242C23—4242no—
243C23—4244no—
244C23C2×C22⋊D5312yes—
245C23C2×C22⋊D5311yescharacteristic
246Q8—22no—
247Q8—532no—
248Q8—1061no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—534no—
256D4—532no—
257D4—1068no—
258D4—1068no—
259D4—1064no—
260D4—1064no—
261D4—1061no—
262D4—1062no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1068no—
267C2×C4—1061no—
268C2×C4—1064no—
269C2×C4D4×D5312yes—
270C2×C4D4×D5312yes—
271C2×C4C2×C22⋊D5312yes—
272C2×C4C2×C22⋊D5311yescharacteristic
273C2×C4C23×D5311yescharacteristic
274C8—1062no—
275C8—1062no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—1064no—
283C22—1062no—
284C22—4244no—
285C22—4248no—
286C22—2124no—
287C22—2122no—
288C22#1313318512yes—
289C22C22×C22⋊D5311yescharacteristic
290C4—22no—
291C4—42no—
292C4—1062no—
293C4#1313318512yes—
294C4C2×D4×D5311yescharacteristic
295C4C2×D4×D5311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—2124no—
300C2—2122no—
301C2C2×#1313318511yescharacteristic
302C2#1681142612yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC22×C53C22×D4
hypercenter#535D53
hypocenterC53#3008
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C2×#13133243C2semidirect
3. C22×#11719922C2semidirect
4. #16810766C2semidirect
5. #16810756C2semidirect
6. #16810627C2semidirect
7. #16810617C2semidirect
8. #16810590C2semidirect
9. C53×#535C2semidirect
14. C2×#11719945C22semidirect
16. C22×D212C22semidirect
20. C2×#11719923C22semidirect
22. C2×#11719922C22semidirect
23. C2×#11719922C22semidirect
24. C2×#11719917C22semidirect
25. C22×#88322C22semidirect
26. #13133140C22semidirect
27. #13133137C22semidirect
28. #13133137C22semidirect
29. C22⋊#88322C22semidirect
31. C22×D4×C53C22semidirect
34. #121×C53C22semidirect
36. #117×C53C22semidirect
60. #11719945D4semidirect
66. C2×D212D4semidirect
155. #535D53semidirect
161. C53#3008semidirect
172. C22×D4C2×D53semidirect
182. #121C2×D53semidirect
184. #117C2×D53semidirect