Finite Group Databasebeta

← #20285552

Subgroups — #20285552

Order 6784.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811134C211yesmaximal, characteristic, complemented
2#16810964C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810944C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810874C211yesmaximal, characteristic, complemented
8#16810871C211yesmaximal, characteristic, complemented
9#16810867C211yesmaximal, characteristic, complemented
10#16810785C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810766C211yesmaximal, characteristic, complemented
14#16810587C211yesmaximal, characteristic, complemented
15C53×#525C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719925C2211yescharacteristic, complemented
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719924C2211yescharacteristic, complemented
20C2×#11719922C2211yescharacteristic, complemented
21C2×#11719917C2211yescharacteristic
22C2×#11719917C2211yescharacteristic
23#13133200—21no—
24#13133196C2211yescharacteristic
25#13133196C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133186—21no—
29C4×C22⋊D53—21no—
30C4×C22⋊D53—21no—
31#13133176C2211yescharacteristic
32#13133150C2211yescharacteristic, complemented
33#13133150C2211yescharacteristic, complemented
34#13133148C2211yescharacteristic, complemented
35#13133148C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133140C2211yescharacteristic, complemented
39#13133140C2211yescharacteristic, complemented
40#13133138C2211yescharacteristic, complemented
41#13133138C2211yescharacteristic, complemented
42#13133137C2211yescharacteristic, complemented
43#13133137C2211yescharacteristic, complemented
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C23×C4×C53C2211yescharacteristic
50#123×C53—21no—
51C22⋊Q8×C53C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58#117×C53C2211yescharacteristic
59C24×C53D411yescharacteristic
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×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53D411yescharacteristic
71C22×C4×C53D411yescharacteristic
72C22×C4×C53D411yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79#11719945—41no—
80#11719945—41no—
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×Dic106C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C42×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719925—41no—
98#11719924—41no—
99#11719923—41no—
100#11719922—41no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108#11719917—41no—
109#11719917—41no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53C2×D411yescharacteristic
119C23×C53C2×D411yescharacteristic
120C23×C53C2×D411yescharacteristic
121C22×D53—21no—
122Q8×C53—21no—
123Q8×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—42no—
127D4×C53—42no—
128D4×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—42no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152D212—21no—
153D212—21no—
154C4×D53—42no—
155Dic106—21no—
156Dic106—21no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—44no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53C22×D411yescharacteristic
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53H(5,𝔽2)11yescharacteristic
185C4×C53H(5,𝔽2)11yescharacteristic
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189#3197—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53—41no—
192C2×C53—41no—
193C2×C53—81no—
194C2×C53#53411yescharacteristic
195C2×C53#57511yescharacteristic
196C2×C53#57511yescharacteristic
197D53—81no—
198#525—531no—
199#525D5311yescharacteristic, complemented
200#485—531no—
201#485—531no—
202#482—531no—
203#482—531no—
204#470—531no—
205#469—531no—
206#468—531no—
207#468—531no—
208#452—531no—
209#452—531no—
210#449—531no—
211#449—531no—
212#407—531no—
213C53#319711yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×#41C2×D5311yescharacteristic
216C23×C4C2×D5311yescharacteristic
217C2×Q16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×D8—531no—
221C2×M16—531no—
222C2×M16—531no—
223#123—21no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227C22⋊Q8C2×D5311yescharacteristic
228C22⋊Q8C2×D5311yescharacteristic
229#121—531no—
230#121—531no—
231#121C2×D5311yescharacteristic
232#121C2×D5311yescharacteristic
233#120—21no—
234#120—1061no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—1061no—
238C4×D4—1061no—
239#117—531no—
240#117C2×D5311yescharacteristic
241#107—531no—
242#107—531no—
243#106—531no—
244#106—531no—
245#103—531no—
246#103—531no—
247#103—531no—
248#103—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8—531no—
256C22⋊C8—531no—
257C24C22⋊D5311yescharacteristic
258#41—41no—
259#41—41no—
260#41—2121no—
261#41—2121no—
262C2×Q8—531no—
263C2×Q8C22×D5311yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—531no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4C22×D5311yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—1061no—
276C22×C4C22⋊D5311yescharacteristic
277C22×C4C22⋊D5311yescharacteristic
278C22×C4C22⋊D5311yescharacteristic
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282Q16—2121no—
283SD16—2121no—
284SD16—2121no—
285D8—2121no—
286M16—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—531no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—531no—
295C4⋊C4—531no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4C22×D5311yescharacteristic
299C4⋊C4C22×D5311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C42—21no—
311C42—1061no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—1061no—
317C23C2×C22⋊D5311yescharacteristic
318C23C2×C22⋊D5311yescharacteristic
319C23C2×C22⋊D5311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1061no—
323Q8—1061no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—1061no—
330D4—1061no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—2122no—
351C2×C4C2×C22⋊D5311yescharacteristic
352C2×C4C2×C22⋊D5311yescharacteristic
353C2×C4C2×C22⋊D5311yescharacteristic
354C2×C4C23×D5311yescharacteristic
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22C22×C22⋊D5311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2121no—
383C4—2121no—
384C4—2121no—
385C4#1313326211yescharacteristic
386C4#1313326211yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—4241no—
392C2#1681118211yescharacteristic
393C2#1681142111yescharacteristic
394C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#525C2
socleC22×C53C22×D4
hypercenter#525D53
hypocenterC53#3197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #16811134C2semidirect
2. #16810964C2semidirect
3. #16810961C2semidirect
4. #16810947C2semidirect
5. #16810944C2semidirect
6. #16810879C2semidirect
7. #16810874C2semidirect
8. #16810871C2semidirect
9. #16810867C2semidirect
10. #16810785C2semidirect
11. #16810783C2semidirect
12. #16810768C2semidirect
13. #16810766C2semidirect
14. #16810587C2semidirect
15. C53×#525C2semidirect
17. C2×#11719925C22semidirect
18. C2×#11719923C22semidirect
19. C2×#11719924C22semidirect
20. C2×#11719922C22semidirect
32. #13133150C22semidirect
33. #13133150C22semidirect
34. #13133148C22semidirect
35. #13133148C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
38. #13133140C22semidirect
39. #13133140C22semidirect
40. #13133138C22semidirect
41. #13133138C22semidirect
42. #13133137C22semidirect
43. #13133137C22semidirect
199. #525D53semidirect
213. C53#3197semidirect