Finite Group Databasebeta

← #20285763

Subgroups — #20285763

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811226C211yesmaximal, characteristic, complemented
2#16811187C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic, complemented
4#16810998C211yesmaximal, characteristic, complemented
5#16810986C211yesmaximal, characteristic, complemented
6#16810979C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810816C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810759C211yesmaximal, characteristic, complemented
13#16810756C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×#11719924C2211yescharacteristic, complemented
18C2×D424C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic, complemented
23#13133213C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133213C2211yescharacteristic, complemented
26#13133208C2211yescharacteristic, complemented
27#13133206C2211yescharacteristic
28#13133206C2211yescharacteristic
29#13133201—21no—
30#13133199—21no—
31#13133191—21no—
32#13133191—21no—
33#13133191C2211yescharacteristic
34#13133189—21no—
35#13133188—21no—
36#13133185—21no—
37#13133185—21no—
38D4×Dic53C2211yescharacteristic, complemented
39#13133179C2211yescharacteristic, complemented
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic, complemented
54C2×D8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53—21no—
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C22×Dic53—21no—
77C2×D212—21no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×D212C2311yescharacteristic
83C2×C4×D53—21no—
84SD16×C53—41no—
85D8×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53D411yescharacteristic
92C22⋊Dic53D411yescharacteristic
93#11719924—41no—
94#11719922—41no—
95#11719922—41no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53D411yescharacteristic
105C4⋊Dic53D411yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113D424—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C2×D411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—42no—
130D4×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—81no—
139C22⋊D53—81no—
140C22⋊D53—81no—
141C22⋊D53—81no—
142C22⋊D53—81no—
143C22⋊D53—81no—
144C22⋊D53—81no—
145C2×Dic53—21no—
146C2×Dic53—81no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150C2×Dic53C2×D411yescharacteristic
151D212—21no—
152D212—21no—
153D212—21no—
154D212—21no—
155D212—21no—
156D212—21no—
157D212—42no—
158C4×D53—81no—
159C8×C53—21no—
160C8×C53—21no—
161#88322—21no—
162#88322—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53C22×D411yescharacteristic
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—84no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53H(5,𝔽2)11yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—21no—
191Dic53—21no—
192Dic53—41no—
193#3220—531nomaximal, Sylow
194C2×C53—41no—
195C2×C53—81no—
196C2×C53—81no—
197C2×C53#53511yescharacteristic
198C2×C53#57611yescharacteristic
199C2×C53#57611yescharacteristic
200D53—81no—
201D53—81no—
202D53—81no—
203#540—531no—
204#535—531no—
205#496—531no—
206#493—531no—
207#490—531no—
208#488—531no—
209#463—531no—
210#463—531no—
211#459—531no—
212#459—531no—
213#449—531no—
214#449D5311yescharacteristic, complemented
215#447—531no—
216#447—531no—
217#433—531no—
218C53#322011yescharacteristic, complemented, Sylow
219C2×#41C2×D5311yescharacteristic
220C22×D4—531no—
221C2×SD16—531no—
222C2×SD16C2×D5311yescharacteristic, complemented
223C2×D8—531no—
224C2×D8—531no—
225C2×D8—531no—
226C2×D8—531no—
227C2×D8—531no—
228C2×D8C2×D5311yescharacteristic, complemented
229#127—531no—
230#127—531no—
231#127—531no—
232#125—531no—
233#124—531no—
234#124—531no—
235#123—1061no—
236#123—1061no—
237#121—531no—
238#121—1061no—
239#121—1061no—
240#121—1061no—
241#121—1061no—
242#121C2×D5311yescharacteristic
243#120—1061no—
244#120—1061no—
245C4×D4—531no—
246C4×D4—531no—
247#117—531no—
248C4⋊C8—531no—
249C4⋊C8—531no—
250#103—531no—
251#103C2×D5311yescharacteristic, complemented
252#102—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102C2×D5311yescharacteristic, complemented
258C22⋊C8—531no—
259C22⋊C8C2×D5311yescharacteristic
260#97—531no—
261C4×C8—531no—
262C24—1061no—
263#41—41no—
264#41—41no—
265C2×Q8C22×D5311yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—531no—
269C2×D4—531no—
270C2×D4—531no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4C22×D5311yescharacteristic
282C2×D4C22×D5311yescharacteristic
283C22×C4—21no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4C22×D5311yescharacteristic
287SD16—41no—
288SD16—2121no—
289D8—41no—
290D8—2121no—
291D8—2121no—
292D8—2121no—
293D8—2121no—
294D8—2121no—
295C2×C8—531no—
296C2×C8—531no—
297C2×C8C22×D5311yescharacteristic
298C2×C8C22×D5311yescharacteristic
299C4⋊C4—531no—
300C4⋊C4—531no—
301C4⋊C4—531no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4C22×D5311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—531no—
307C22⋊C4—531no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C42—531no—
316C42—531no—
317C23—21no—
318C23—21no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—4241no—
323C23—4241no—
324C23—2122no—
325C23D4×D5311yescharacteristic
326Q8—21no—
327Q8—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—81no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—1061no—
338D4—1061no—
339D4—1061no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—4241no—
345D4—4241no—
346D4—4241no—
347D4—2122no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—531no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—1061no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4D4×D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—21no—
363C8—21no—
364C8—1061no—
365C8—1061no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—4244no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22C2×D4×D5311yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—1061no—
390C4—1061no—
391C4—1061no—
392C4—1061no—
393C4—2121no—
394C4#1313326211yescharacteristic
395C4#1313326211yescharacteristic
396C2—41no—
397C2—81no—
398C2—81no—
399C2—4241no—
400C2—4241no—
401C2—4241no—
402C2#1681118811yescharacteristic
403C2#1681142511yescharacteristic
404C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3220
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #16811226C2semidirect
2. #16811187C2semidirect
3. #16811009C2semidirect
4. #16810998C2semidirect
5. #16810986C2semidirect
6. #16810979C2semidirect
7. #16810841C2semidirect
8. #16810840C2semidirect
9. #16810817C2semidirect
10. #16810816C2semidirect
11. #16810768C2semidirect
12. #16810759C2semidirect
13. #16810756C2semidirect
14. #16810681C2semidirect
15. C53×#449C2semidirect
17. C2×#11719924C22semidirect
22. C2×#11719922C22semidirect
25. #13133213C22semidirect
26. #13133208C22semidirect
38. D4×Dic53C22semidirect
39. #13133179C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133141C22semidirect
46. #13133138C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
53. C2×SD16×C53C22semidirect
54. C2×D8×C53C22semidirect
56. #103×C53C22semidirect
57. #102×C53C22semidirect
214. #449D53semidirect
218. C53#3220semidirect
222. C2×SD16C2×D53semidirect
228. C2×D8C2×D53semidirect
251. #103C2×D53semidirect
257. #102C2×D53semidirect