Finite Group Databasebeta

← #20285810

Subgroups — #20285810

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#16811247C211yesmaximal, characteristic, complemented
2#16811191C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16811005C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810848C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810822C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810775C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic, complemented
13#16810758C211yesmaximal, characteristic, complemented
14#16810685C211yesmaximal, characteristic, complemented
15C53×#489C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic
22C2×#11719924C2211yescharacteristic, complemented
23C2×D424C2211yescharacteristic
24C2×D424C2211yescharacteristic, complemented
25C2×#11719922C2211yescharacteristic
26#13133214C2211yescharacteristic
27#13133213C2211yescharacteristic
28#13133212—21no—
29#13133206—21no—
30#13133206C2211yescharacteristic
31#13133205C2211yescharacteristic, complemented
32#13133202—21no—
33#13133197C2211yescharacteristic
34#13133193—21no—
35#13133192C2211yescharacteristic
36#13133190—21no—
37#13133190—21no—
38#13133187—21no—
39#13133187—21no—
40#13133182C2211yescharacteristic, complemented
41#13133179C2211yescharacteristic, complemented
42#13133174C2211yescharacteristic
43#13133146C2211yescharacteristic, complemented
44#13133146C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic, complemented
52C4⋊Q8×C53C2211yescharacteristic, complemented
53#124×C53C2211yescharacteristic
54#103×C53C2211yescharacteristic, complemented
55#103×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66#11719949—41no—
67#11719949—41no—
68D4×D53—41no—
69D4×D53—41no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53C2311yescharacteristic
89#11719925—41no—
90#11719924—41no—
91#11719923—41no—
92#11719922—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101#11719921D411yescharacteristic
102C4⋊Dic53—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111D424—41no—
112#11719913—41no—
113#11719913—41no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—81no—
125D4×C53—21no—
126D4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—42no—
134C22⋊D53—81no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140D212—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—42no—
147D212—81no—
148D212—81no—
149D212—81no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—42no—
153C4×D53—81no—
154Dic106—21no—
155Dic106—21no—
156C8×C53—21no—
157C8×C53—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—84no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53H(5,𝔽2)11yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—81no—
190C2×C53#53511yescharacteristic
191C2×C53#57611yescharacteristic
192C2×C53#57711yescharacteristic
193D53—41no—
194D53—81no—
195D53—81no—
196D53—81no—
197#542—531no—
198#535—531no—
199#496—531no—
200#495—531no—
201#489D5311yescharacteristic, complemented
202#486—531no—
203#464—531no—
204#463—531no—
205#460—531no—
206#459—531no—
207#452—531no—
208#450—531no—
209#449—531no—
210#447—531no—
211#433—531no—
212C53#322411yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C22×D4—531no—
215C2×Q16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×D8—531no—
221C2×D8—531no—
222C2×D8—531no—
223C4⋊Q8C2×D5311yescharacteristic, complemented
224#127—531no—
225#127—531no—
226#126—1061no—
227#124—531no—
228#124—531no—
229#124—1061no—
230#124C2×D5311yescharacteristic
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
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#103C2×D5311yescharacteristic, complemented
245#103C2×D5311yescharacteristic, complemented
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102C2×D5311yescharacteristic
251#102C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97C2×D5311yescharacteristic, complemented
255C4×C8—531no—
256C24—1061no—
257#41—2121no—
258#41—2121no—
259C2×Q8—21no—
260C2×Q8—531no—
261C2×Q8C22×D5311yescharacteristic
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4C22×D5311yescharacteristic
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278Q16—2121no—
279SD16—2121no—
280SD16—2121no—
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—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—531no—
309C42—1061no—
310C42C22×D5311yescharacteristic
311C23—21no—
312C23—531no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—4241no—
317C23—4241no—
318C23—2122no—
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—1061no—
323Q8—1061no—
324D4—21no—
325D4—21no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—4241no—
333D4—4241no—
334D4—4241no—
335D4—4241no—
336D4—4241no—
337D4—2122no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—4241no—
349C2×C4—2122no—
350C2×C4D4×D5311yescharacteristic
351C2×C4D4×D5311yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1061no—
356C8—1061no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—4244no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—2121no—
383C4#1313326311yescharacteristic
384C4#1313326311yescharacteristic
385C2—81no—
386C2—4241no—
387C2—4241no—
388C2—4241no—
389C2—2121no—
390C2#1681119111yescharacteristic
391C2#1681142711yescharacteristic
392C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16811247C2semidirect
2. #16811191C2semidirect
3. #16811012C2semidirect
4. #16811005C2semidirect
5. #16810971C2semidirect
6. #16810848C2semidirect
7. #16810841C2semidirect
8. #16810822C2semidirect
9. #16810817C2semidirect
10. #16810788C2semidirect
11. #16810775C2semidirect
12. #16810771C2semidirect
13. #16810758C2semidirect
14. #16810685C2semidirect
15. C53×#489C2semidirect
18. C2×#11719925C22semidirect
20. C2×#11719913C22semidirect
22. C2×#11719924C22semidirect
24. C2×D424C22semidirect
31. #13133205C22semidirect
40. #13133182C22semidirect
41. #13133179C22semidirect
43. #13133146C22semidirect
44. #13133146C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
51. C4×#88322C22semidirect
52. C4⋊Q8×C53C22semidirect
54. #103×C53C22semidirect
55. #103×C53C22semidirect
58. #97×C53C22semidirect
201. #489D53semidirect
212. C53#3224semidirect
223. C4⋊Q8C2×D53semidirect
244. #103C2×D53semidirect
245. #103C2×D53semidirect
254. #97C2×D53semidirect