Finite Group Databasebeta

← #20285762

Subgroups — #20285762

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#16811224C211yesmaximal, characteristic, complemented
2#16811191C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic, complemented
4#16810998C211yesmaximal, characteristic, complemented
5#16810986C211yesmaximal, characteristic, complemented
6#16810979C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810819C211yesmaximal, characteristic, complemented
9#16810818C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810767C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810755C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic, complemented
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719922C2211yescharacteristic, complemented
24#13133214C2211yescharacteristic
25#13133213C2211yescharacteristic
26#13133213C2211yescharacteristic, complemented
27#13133208C2211yescharacteristic, complemented
28#13133206C2211yescharacteristic
29#13133200—21no—
30#13133199—21no—
31#13133192C2211yescharacteristic
32#13133191—21no—
33#13133190—21no—
34#13133190—21no—
35#13133190—21no—
36#13133188C2211yescharacteristic
37#13133187—21no—
38#13133187—21no—
39C4×D212C2211yescharacteristic, complemented
40#13133174C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70#11719949—41no—
71#11719949—41no—
72D4×D53—41no—
73D4×D53—41no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84SD16×C53—41no—
85D8×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94#11719924—41no—
95#11719922—41no—
96#11719922—41no—
97#11719922—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921D411yescharacteristic
103#11719921D411yescharacteristic
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919D411yescharacteristic
108#11719919D411yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113D424—41no—
114C23×C53—21no—
115C23×C53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—81no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—81no—
137C22⋊D53—81no—
138C22⋊D53—81no—
139C22⋊D53—81no—
140C22⋊D53—81no—
141C2×Dic53—21no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212—21no—
150D212—21no—
151D212—42no—
152D212—81no—
153D212—81no—
154D212—81no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—42no—
158C4×D53—81no—
159C8×C53—21no—
160C8×C53—21no—
161#88322—21no—
162#88322—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53C22×D411yescharacteristic
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
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—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53H(5,𝔽2)11yescharacteristic
188C4×C53H(5,𝔽2)11yescharacteristic
189Dic53—21no—
190Dic53—21no—
191Dic53—21no—
192Dic53—41no—
193#3220—531nomaximal, Sylow
194C2×C53—81no—
195C2×C53—81no—
196C2×C53#53511yescharacteristic
197C2×C53#57611yescharacteristic
198C2×C53#57611yescharacteristic
199D53—41no—
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#463D5311yescharacteristic, complemented
211#459—531no—
212#459—531no—
213#449—531no—
214#449—531no—
215#447—531no—
216#447—531no—
217#433—531no—
218C53#322011yescharacteristic, complemented, Sylow
219C2×#41—531no—
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
229#127—531no—
230#127—531no—
231#127—531no—
232#125—531no—
233#124—531no—
234#124C2×D5311yescharacteristic
235#123—1061no—
236#123—1061no—
237#121—531no—
238#121—531no—
239#121—1061no—
240#121—1061no—
241#121—1061no—
242#121—1061no—
243#120—1061no—
244#120—1061no—
245C4×D4—531no—
246C4×D4C2×D5311yescharacteristic, complemented
247#117—531no—
248C4⋊C8—531no—
249C4⋊C8C2×D5311yescharacteristic, complemented
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
258C22⋊C8—531no—
259C22⋊C8—531no—
260#97—531no—
261C4×C8—531no—
262C24—1061no—
263#41—2121no—
264#41—2121no—
265C2×Q8C22×D5311yescharacteristic
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—1061no—
270C2×D4—1061no—
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—531no—
285C22×C4—1061no—
286C22×C4—1061no—
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—21no—
307C22⋊C4—21no—
308C22⋊C4—531no—
309C22⋊C4—531no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C42—531no—
316C42C22×D5311yescharacteristic
317C23—21no—
318C23—21no—
319C23—531no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326Q8—21no—
327Q8—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—1061no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—1061no—
338D4—4241no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—4241no—
345D4—4241no—
346D4—2122no—
347D4—2122no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—81no—
351C2×C4—531no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—4241no—
358C2×C4—2122no—
359C2×C4D4×D5311yescharacteristic
360C2×C4D4×D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—21no—
363C8—21no—
364C8—1061no—
365C8—1061no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1061no—
373C22—1061no—
374C22—1061no—
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—21no—
388C4—41no—
389C4—41no—
390C4—1061no—
391C4—1061no—
392C4—1061no—
393C4—2121no—
394C4#1313326211yescharacteristic
395C4#1313326311yescharacteristic
396C2—81no—
397C2—81no—
398C2—4241no—
399C2—4241no—
400C2—4241no—
401C2—2121no—
402C2#1681118911yescharacteristic
403C2#1681142511yescharacteristic
404C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3220
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #16811224C2semidirect
2. #16811191C2semidirect
3. #16811009C2semidirect
4. #16810998C2semidirect
5. #16810986C2semidirect
6. #16810979C2semidirect
7. #16810835C2semidirect
8. #16810819C2semidirect
9. #16810818C2semidirect
10. #16810771C2semidirect
11. #16810767C2semidirect
12. #16810758C2semidirect
13. #16810755C2semidirect
14. #16810683C2semidirect
15. C53×#463C2semidirect
18. C2×#11719924C22semidirect
20. C2×D424C22semidirect
23. C2×#11719922C22semidirect
26. #13133213C22semidirect
27. #13133208C22semidirect
39. C4×D212C22semidirect
41. C4⋊#88322C22semidirect
42. #13133141C22semidirect
44. #13133139C22semidirect
47. #13133137C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
52. C2×SD16×C53C22semidirect
55. C4×D4×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
210. #463D53semidirect
218. C53#3220semidirect
222. C2×SD16C2×D53semidirect
246. C4×D4C2×D53semidirect
249. C4⋊C8C2×D53semidirect
251. #103C2×D53semidirect