Finite Group Databasebeta

← #20285809

Subgroups — #20285809

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#16811244C211yesmaximal, characteristic, complemented
2#16811010C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16810982C211yesmaximal, characteristic, complemented
5#16810969C211yesmaximal, characteristic, complemented
6#16810842C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810820C211yesmaximal, characteristic, complemented
9#16810815C211yesmaximal, characteristic, complemented
10#16810785C211yesmaximal, characteristic, complemented
11#16810774C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810757C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#535C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic, complemented
17C2×#11719923C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic, complemented
20C2×#11719924C2211yescharacteristic, complemented
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719922C2211yescharacteristic, complemented
24#13133217C2211yescharacteristic, complemented
25#13133212—21no—
26#13133205C2211yescharacteristic, complemented
27#13133203—21no—
28#13133199—21no—
29#13133196C2211yescharacteristic
30#13133195—21no—
31#13133191C2211yescharacteristic
32C4×Dic106C2211yescharacteristic, complemented
33C4×D212C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35C4⋊#88322C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic, complemented
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48C2×#41×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50#127×C53C2211yescharacteristic
51#127×C53C2211yescharacteristic
52#124×C53C2211yescharacteristic
53#124×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53—21no—
56#120×C53—21no—
57#120×C53—21no—
58#117×C53C2211yescharacteristic
59C24×C53—21no—
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C22×C4×C53—21no—
75C22×C4×C53C2311yescharacteristic
76C2×C22⋊D53—21no—
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×Dic106—21no—
80C2×Dic106C2311yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53D411yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53C2311yescharacteristic
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92#11719925—41no—
93#11719924—41no—
94#11719923—41no—
95#11719923—41no—
96#11719923—41no—
97#11719922—41no—
98#11719922—41no—
99#11719922—41no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53C2×D411yescharacteristic
121C22×D53—21no—
122Q8×C53—21no—
123Q8×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—42no—
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—42no—
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—81no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150D212—21no—
151D212—21no—
152C4×D53—81no—
153Dic106—21no—
154Dic106—21no—
155Dic106—81no—
156#88322—21no—
157#88322—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—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—84no—
173C22×C53C22×D411yescharacteristic
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53H(5,𝔽2)11yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—81no—
191C2×C53—81no—
192C2×C53—81no—
193C2×C53#53511yescharacteristic
194C2×C53#57611yescharacteristic
195C2×C53#57711yescharacteristic
196D53—81no—
197#542—531no—
198#535D5311yescharacteristic, complemented
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#450—531no—
209#449—531no—
210#447—531no—
211#433—531no—
212C53#322411yescharacteristic, complemented, Sylow
213C2×#41C2×D5311yescharacteristic
214C22×D4C2×D5311yescharacteristic
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⋊Q8—531no—
224#127C2×D5311yescharacteristic
225#127C2×D5311yescharacteristic
226#126—1061no—
227#124—531no—
228#124—1061no—
229#124C2×D5311yescharacteristic
230#124C2×D5311yescharacteristic
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234#121—21no—
235#121—21no—
236#121—531no—
237#120—21no—
238#120—21no—
239C4×Q8—531no—
240C4×D4—531no—
241#117C2×D5311yescharacteristic
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#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C4×C8—531no—
256C24—21no—
257#41—41no—
258#41—41no—
259C2×Q8—531no—
260C2×Q8—1061no—
261C2×Q8C22×D5311yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—531no—
271C2×D4—1061no—
272C2×D4C22×D5311yescharacteristic
273C2×D4C22×D5311yescharacteristic
274C2×D4C22×D5311yescharacteristic
275C22×C4—21no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278Q16—2121no—
279SD16—2121no—
280SD16—2121no—
281SD16—2121no—
282SD16—2121no—
283D8—2121no—
284D8—2121no—
285D8—2121no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—531no—
290C4⋊C4—531no—
291C4⋊C4—531no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22⋊D5311yescharacteristic
296C4⋊C4C22⋊D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4C22⋊D5311yescharacteristic
307C22⋊C4C22⋊D5311yescharacteristic
308C42—1061no—
309C42C22×D5311yescharacteristic
310C42C22×D5311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—1061no—
318C23C2×C22⋊D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—1061no—
322Q8—1061no—
323Q8—4241no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—1061no—
337D4—1061no—
338D4—4241no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—4241no—
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C2×C22⋊D5311yescharacteristic
349C2×C4C2×C22⋊D5311yescharacteristic
350C2×C4C2×C22⋊D5311yescharacteristic
351C2×C4C2×C22⋊D5311yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—1061no—
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—84no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C22×C22⋊D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4#1313326211yescharacteristic
384C4#1313326211yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—4241no—
390C2#1681119011yescharacteristic
391C2#1681142611yescharacteristic
392C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC22×C53C22×D4
hypercenter#535D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16811010C2semidirect
3. #16811005C2semidirect
4. #16810982C2semidirect
5. #16810969C2semidirect
6. #16810842C2semidirect
7. #16810835C2semidirect
8. #16810820C2semidirect
9. #16810815C2semidirect
10. #16810785C2semidirect
11. #16810774C2semidirect
12. #16810768C2semidirect
13. #16810757C2semidirect
14. #16810684C2semidirect
15. C53×#535C2semidirect
16. C2×#11719925C22semidirect
19. C2×#11719923C22semidirect
20. C2×#11719924C22semidirect
23. C2×#11719922C22semidirect
24. #13133217C22semidirect
26. #13133205C22semidirect
32. C4×Dic106C22semidirect
33. C4×D212C22semidirect
34. C4⋊#88322C22semidirect
35. C4⋊#88322C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
42. #13133138C22semidirect
43. #13133138C22semidirect
46. #13133128C22semidirect
47. C4×#88322C22semidirect
198. #535D53semidirect
212. C53#3224semidirect