Finite Group Databasebeta

← #20285068

Subgroups — #20285068

Order 6784.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811253C211yesmaximal, characteristic, complemented
2#16811191C211yesmaximal, characteristic
3#16810989C211yesmaximal, characteristic, complemented
4#16810986C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810982C211yesmaximal, characteristic, complemented
7#16810979C211yesmaximal, characteristic, complemented
8#16810970C211yesmaximal, characteristic, complemented
9#16810968C211yesmaximal, characteristic, complemented
10#16810626C211yesmaximal, characteristic, complemented
11#16810623C211yesmaximal, characteristic, complemented
12#16810618C211yesmaximal, characteristic, complemented
13#102×D53C211yesmaximal, characteristic, complemented
14#16810577C211yesmaximal, characteristic, complemented
15C53×#488C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133217C2211yescharacteristic, complemented
21#13133216—21no—
22#13133216C2211yescharacteristic, complemented
23#13133214C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133208C2211yescharacteristic, complemented
26#13133207—21no—
27#13133206C2211yescharacteristic
28#13133197—21no—
29#13133195—21no—
30#13133190—21no—
31#13133190—21no—
32#13133187—21no—
33#13133187—21no—
34#13133175C2211yescharacteristic
35#13133174C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37#13133141C2211yescharacteristic, complemented
38#13133141C2211yescharacteristic, complemented
39#13133139C2211yescharacteristic
40#13133139C2211yescharacteristic
41#13133139C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52#125×C53C2211yescharacteristic, complemented
53#124×C53C2211yescharacteristic
54#103×C53C2211yescharacteristic, complemented
55#103×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65#11719949—22no—
66#11719949—22no—
67D4×D53—22no—
68D4×D53—22no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53C2311yescharacteristic
77C2×Dic106—21no—
78C2×Dic106—21no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C42×C53C2311yescharacteristic
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107C8×D53—22no—
108C23×C53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—42no—
112C22×D53—81no—
113C22×D53—81no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—81no—
127C22⋊D53—81no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—21no—
136D212—21no—
137D212—42no—
138D212—81no—
139D212—81no—
140C4×D53—42no—
141C4×D53—81no—
142C4×D53#4112yes—
143C4×D53#4112yes—
144Dic106—81no—
145Dic106—81no—
146C8×C53—21no—
147C8×C53—21no—
148#88322—21no—
149#88322—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—84no—
163C2×D53#12412yes—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53C2×#4111yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53#12412yes—
176#3136—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53C2×#12411yescharacteristic
179C2×C53#57611yescharacteristic
180C2×C53#57711yescharacteristic
181D53—22no—
182D53—81no—
183D53—81no—
184#544—531no—
185#535—531no—
186#490—531no—
187#490—531no—
188#489—531no—
189#489—531no—
190#488—531no—
191#488D5311yescharacteristic, complemented
192#486—531no—
193#486—531no—
194#418—531no—
195#417—531no—
196#416—531no—
197C2×#102—531no—
198#405—531no—
199C53#313611yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C22×D4—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206C4⋊Q8—1061no—
207#127—531no—
208#127—531no—
209#125—531no—
210#125—1061no—
211#125C2×D5311yescharacteristic, complemented
212#124—531no—
213#124C2×D5311yescharacteristic
214C22⋊Q8—1061no—
215C22⋊Q8—1061no—
216#121—1061no—
217#121—1061no—
218#120—1061no—
219#120—1061no—
220#117—531no—
221#117—531no—
222C2×C4⋊C4—531no—
223#103—531no—
224#103—531no—
225#103C2×D5311yescharacteristic, complemented
226#103C2×D5311yescharacteristic, complemented
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102C2×D5311yescharacteristic
238#102C2×D5311yescharacteristic
239#97—531no—
240#97—531no—
241C4×C8—531no—
242C4×C8C2×D5311yescharacteristic, complemented
243C24—1061no—
244#41—1062no—
245#41—1062no—
246C2×Q8—1061no—
247C2×Q8—1061no—
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1062no—
253C2×D4—1062no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263M16—1062no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8—1062no—
267C2×C8C22×D5311yescharacteristic
268C2×C8C22×D5311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—531no—
285C22⋊C4—531no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—531no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—21no—
293C23—531no—
294C23—1061no—
295C23—1061no—
296C23—4241no—
297C23—4241no—
298C23—2122no—
299Q8—21no—
300Q8—21no—
301Q8—4241no—
302Q8—4241no—
303D4—21no—
304D4—21no—
305D4—1061no—
306D4—1061no—
307D4—1061no—
308D4—1061no—
309D4—4241no—
310D4—4241no—
311D4—4241no—
312D4—4241no—
313D4—2122no—
314D4—2122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—531no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—2122no—
327C2×C4D4×D5311yescharacteristic
328C2×C4D4×D5311yescharacteristic
329C2×C4C23×D5311yescharacteristic
330C8—21no—
331C8—21no—
332C8—1061no—
333C8—1061no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—532no—
338C22—1061no—
339C22—1061no—
340C22—4244no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—532no—
354C4—1061no—
355C4—1061no—
356C4—2121no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4#41×D5311yescharacteristic
360C2—81no—
361C2—1062no—
362C2—4241no—
363C2—4241no—
364C2#124×D5311yescharacteristic
365C2#1681142511yescharacteristic
366C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#488C2
socleC22×C53C22×D4
hypercenter#488D53
hypocenterC53#3136
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #16811253C2semidirect
3. #16810989C2semidirect
4. #16810986C2semidirect
5. #16810984C2semidirect
6. #16810982C2semidirect
7. #16810979C2semidirect
8. #16810970C2semidirect
9. #16810968C2semidirect
10. #16810626C2semidirect
11. #16810623C2semidirect
12. #16810618C2semidirect
13. #102×D53C2semidirect
14. #16810577C2semidirect
15. C53×#488C2semidirect
20. #13133217C22semidirect
22. #13133216C22semidirect
25. #13133208C22semidirect
37. #13133141C22semidirect
38. #13133141C22semidirect
41. #13133139C22semidirect
42. #13133139C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. #13133128C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
52. #125×C53C22semidirect
54. #103×C53C22semidirect
55. #103×C53C22semidirect
58. C4×C8×C53C22semidirect
191. #488D53semidirect
199. C53#3136semidirect
211. #125C2×D53semidirect
225. #103C2×D53semidirect
226. #103C2×D53semidirect
242. C4×C8C2×D53semidirect