Finite Group Databasebeta

← #20285813

Subgroups — #20285813

Order 6784.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811246C211yesmaximal, characteristic, complemented
2#16811190C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic, complemented
4#16811004C211yesmaximal, characteristic, complemented
5#16810981C211yesmaximal, characteristic, complemented
6#16810968C211yesmaximal, characteristic, complemented
7#16810843C211yesmaximal, characteristic, complemented
8#16810837C211yesmaximal, characteristic, complemented
9#16810821C211yesmaximal, characteristic, complemented
10#16810817C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic, complemented
13#16810757C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719925C2211yescharacteristic, complemented
18C2×#11719913C2211yescharacteristic, complemented
19C2×#11719924C2211yescharacteristic, complemented
20C2×D424C2211yescharacteristic, complemented
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133216C2211yescharacteristic, complemented
24#13133213C2211yescharacteristic
25#13133213C2211yescharacteristic
26#13133212—21no—
27#13133206C2211yescharacteristic, complemented
28#13133204C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133203—21no—
31#13133199—21no—
32#13133197—21no—
33#13133191C2211yescharacteristic
34#13133188—21no—
35#13133188—21no—
36#13133186—21no—
37#13133186—21no—
38Q8×Dic53C2211yescharacteristic, complemented
39#13133179C2211yescharacteristic, complemented
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C24×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73#11719945—41no—
74#11719945—41no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×Dic106—21no—
80C2×Dic106C2311yescharacteristic
81SD16×C53—41no—
82SD16×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53D411yescharacteristic
93C22⋊Dic53D411yescharacteristic
94#11719925—41no—
95#11719924—41no—
96#11719922—41no—
97#11719922—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53D411yescharacteristic
104C4⋊Dic53D411yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113#11719913—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—42no—
117C23×C53—81no—
118C23×C53—81no—
119C23×C53C2×D411yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122Q8×C53—21no—
123Q8×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
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×Dic53—21no—
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147D212—21no—
148D212—21no—
149D212—21no—
150D212—21no—
151C4×D53—42no—
152C4×D53—81no—
153Dic106—21no—
154Dic106—21no—
155Dic106—81no—
156C8×C53—21no—
157C8×C53—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—84no—
170C22×C53C22×D411yescharacteristic
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53H(5,𝔽2)11yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—21no—
183Dic53—21no—
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—81no—
191C2×C53—81no—
192C2×C53#53511yescharacteristic
193C2×C53#57611yescharacteristic
194C2×C53#57711yescharacteristic
195D53—81no—
196D53—81no—
197#542—531no—
198#535—531no—
199#496—531no—
200#495—531no—
201#489—531no—
202#486—531no—
203#464—531no—
204#463—531no—
205#460—531no—
206#459—531no—
207#452—531no—
208#450D5311yescharacteristic, complemented
209#449—531no—
210#447—531no—
211#433—531no—
212C53#322411yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C22×D4C2×D5311yescharacteristic
215C2×Q16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16C2×D5311yescharacteristic
219C2×SD16C2×D5311yescharacteristic
220C2×D8—531no—
221C2×D8—531no—
222C2×D8—531no—
223C4⋊Q8—531no—
224#127—531no—
225#127—531no—
226#126—1061no—
227#124—531no—
228#124—531no—
229#124—531no—
230#124—1061no—
231#123—1061no—
232C22⋊Q8—1061no—
233C22⋊Q8C2×D5311yescharacteristic
234#121—531no—
235#121—1061no—
236#121—1061no—
237#120—1061no—
238#120—1061no—
239C4×Q8—531no—
240C4×D4—531no—
241#117—531no—
242C4⋊C8—531no—
243C4⋊C8—531no—
244#103—531no—
245#103—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102C2×D5311yescharacteristic
251#102C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8C2×D5311yescharacteristic
254#97—531no—
255C4×C8—531no—
256C24—21no—
257#41—2121no—
258#41—2121no—
259C2×Q8—531no—
260C2×Q8—1061no—
261C2×Q8C22×D5311yescharacteristic
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4C22×D5311yescharacteristic
274C2×D4C22×D5311yescharacteristic
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278Q16—2121no—
279SD16—41no—
280SD16—41no—
281SD16—2121no—
282SD16—2121no—
283D8—2121no—
284D8—2121no—
285D8—2121no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8C22×D5311yescharacteristic
289C2×C8C22×D5311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—531no—
292C4⋊C4—531no—
293C4⋊C4—531no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—531no—
299C22⋊C4—531no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—531no—
309C42—531no—
310C42—1061no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—1061no—
317C23—1061no—
318C23D4×D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—1061no—
322Q8—1061no—
323Q8—4241no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—1061no—
333D4—4241no—
334D4—4241no—
335D4—4241no—
336D4—4241no—
337D4—4241no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4D4×D5311yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1061no—
356C8—1061no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—1061no—
378C4—1061no—
379C4—1061no—
380C4—1061no—
381C4—2121no—
382C4—2121no—
383C4#1313326211yescharacteristic
384C4#1313326211yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—4241no—
389C2—4241no—
390C2#1681118811yescharacteristic
391C2#1681142711yescharacteristic
392C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811246C2semidirect
2. #16811190C2semidirect
3. #16811009C2semidirect
4. #16811004C2semidirect
5. #16810981C2semidirect
6. #16810968C2semidirect
7. #16810843C2semidirect
8. #16810837C2semidirect
9. #16810821C2semidirect
10. #16810817C2semidirect
11. #16810783C2semidirect
12. #16810770C2semidirect
13. #16810757C2semidirect
14. #16810681C2semidirect
15. C53×#450C2semidirect
17. C2×#11719925C22semidirect
18. C2×#11719913C22semidirect
19. C2×#11719924C22semidirect
20. C2×D424C22semidirect
23. #13133216C22semidirect
27. #13133206C22semidirect
38. Q8×Dic53C22semidirect
39. #13133179C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
45. #13133139C22semidirect
48. #13133137C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
208. #450D53semidirect
212. C53#3224semidirect