Finite Group Databasebeta

← #20285567

Subgroups — #20285567

Order 6784.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811201C211yesmaximal, characteristic, complemented
2#16810964C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810939C211yesmaximal, characteristic, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810857C211yesmaximal, characteristic, complemented
9#16810850C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11#16810766C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810757C211yesmaximal, characteristic, complemented
14#16810590C211yesmaximal, characteristic, complemented
15C53×#535C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C2×#11719923C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic, complemented
22C22×#88322C2211yescharacteristic, complemented
23#13133196C2211yescharacteristic
24#13133196C2211yescharacteristic, complemented
25#13133193—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic, complemented
28#13133189—21no—
29#13133189—21no—
30#13133188—21no—
31C2×C4⋊Dic53C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133150C2211yescharacteristic, complemented
34#13133148C2211yescharacteristic, complemented
35#13133148C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic, complemented
43#13133137C2211yescharacteristic, complemented
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47C22⋊#88322C2211yescharacteristic, complemented
48C2×#41×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50#127×C53—21no—
51#124×C53—21no—
52#121×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#120×C53—21no—
57#120×C53—21no—
58#117×C53C2211yescharacteristic
59C24×C53—21no—
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53D411yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53D411yescharacteristic
72C2×D4×C53D411yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×C4×C53C2311yescharacteristic
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C22×Dic53—21no—
82#11719945—41no—
83#11719945—41no—
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×Dic106C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C42×C53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719923—41no—
98#11719923—41no—
99#11719922—41no—
100#11719922—41no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108#11719917—41no—
109C2×#88322—41no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—42no—
117C23×C53—81no—
118C23×C53—81no—
119C23×C53C2×D411yescharacteristic
120C23×C53C2×D411yescharacteristic
121C23×C53C2×D411yescharacteristic
122C22×D53—21no—
123Q8×C53—41no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—41no—
129D4×C53—41no—
130D4×C53—41no—
131D4×C53—42no—
132D4×C53—81no—
133D4×C53—81no—
134D4×C53—81no—
135D4×C53—81no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2×D411yescharacteristic
147C2×C4×C53C2411yescharacteristic
148C22⋊D53—42no—
149C22⋊D53—81no—
150C22⋊D53—81no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—81no—
155D212—21no—
156D212—21no—
157C4×D53—42no—
158Dic106—21no—
159Dic106—21no—
160#88322—21no—
161#88322—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—84no—
180C22×C53C22×D411yescharacteristic
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53H(5,𝔽2)11yescharacteristic
190C4×C53H(5,𝔽2)11yescharacteristic
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3198—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—41no—
197C2×C53—41no—
198C2×C53—81no—
199C2×C53—81no—
200C2×C53#53411yescharacteristic
201C2×C53#57611yescharacteristic
202C2×C53#57711yescharacteristic
203D53—81no—
204#537—531no—
205#535D5311yescharacteristic, complemented
206#485—531no—
207#482—531no—
208#481—531no—
209#480—531no—
210#469—531no—
211#468—531no—
212#466—531no—
213#465—531no—
214#450—531no—
215#449—531no—
216#449—531no—
217#447—531no—
218#408—531no—
219C53#319811yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41C2×D5311yescharacteristic
222C22×D4C2×D5311yescharacteristic
223C2×SD16—531no—
224C2×SD16—531no—
225C2×D8—531no—
226C2×D8—531no—
227C2×M16—531no—
228C22×C8—531no—
229#127—21no—
230#124—21no—
231C22⋊Q8—531no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234#121—531no—
235#121—531no—
236#121—1061no—
237#121—1061no—
238#121—1061no—
239#121C2×D5311yescharacteristic
240#121C2×D5311yescharacteristic
241#121C2×D5311yescharacteristic
242#121C2×D5311yescharacteristic
243#120—21no—
244#120—21no—
245#117C2×D5311yescharacteristic
246C2×C4⋊C4—531no—
247#107—531no—
248#107—531no—
249#106—531no—
250#106—531no—
251#103—531no—
252#103—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259C22⋊C8—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262C22⋊C8—531no—
263C24—21no—
264#41—41no—
265#41—41no—
266#41—2121no—
267#41—2121no—
268C2×Q8—531no—
269C2×Q8C22⋊D5311yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—531no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4C22⋊D5311yescharacteristic
282C2×D4C22⋊D5311yescharacteristic
283C2×D4C22⋊D5311yescharacteristic
284C2×D4C22×D5311yescharacteristic
285C2×D4C22×D5311yescharacteristic
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290C22×C4C22×D5311yescharacteristic
291SD16—2121no—
292SD16—2121no—
293D8—2121no—
294D8—2121no—
295M16—2121no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—531no—
300C2×C8—2121no—
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4C22×D5311yescharacteristic
307C4⋊C4C22×D5311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—21no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—1061no—
324C23C2×C22⋊D5311yescharacteristic
325C23C2×C22⋊D5311yescharacteristic
326C23C2×C22⋊D5311yescharacteristic
327Q8—41no—
328Q8—1061no—
329Q8—1061no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—42no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—1061no—
343D4—1061no—
344D4—4241no—
345D4—4241no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—4241no—
359C2×C4—2122no—
360C2×C4C2×C22⋊D5311yescharacteristic
361C2×C4C2×C22⋊D5311yescharacteristic
362C2×C4C2×C22⋊D5311yescharacteristic
363C2×C4C23×D5311yescharacteristic
364C8—1061no—
365C8—1061no—
366C8—1061no—
367C8—1061no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—84no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C22×C22⋊D5311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—2121no—
394C4—2121no—
395C4—2121no—
396C4#1313326211yescharacteristic
397C4#1313326211yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—81no—
403C2—4241no—
404C2#1681118211yescharacteristic
405C2#1681142611yescharacteristic
406C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC22×C53C22×D4
hypercenter#535D53
hypocenterC53#3198
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16811201C2semidirect
2. #16810964C2semidirect
3. #16810947C2semidirect
4. #16810939C2semidirect
5. #16810934C2semidirect
6. #16810874C2semidirect
7. #16810867C2semidirect
8. #16810857C2semidirect
9. #16810850C2semidirect
10. #16810774C2semidirect
11. #16810766C2semidirect
12. #16810766C2semidirect
13. #16810757C2semidirect
14. #16810590C2semidirect
15. C53×#535C2semidirect
16. C2×#11719945C22semidirect
21. C2×#11719917C22semidirect
22. C22×#88322C22semidirect
24. #13133196C22semidirect
27. #13133191C22semidirect
31. C2×C4⋊Dic53C22semidirect
32. #13133150C22semidirect
33. #13133150C22semidirect
34. #13133148C22semidirect
35. #13133148C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
42. #13133137C22semidirect
43. #13133137C22semidirect
46. C22⋊#88322C22semidirect
47. C22⋊#88322C22semidirect
205. #535D53semidirect
219. C53#3198semidirect