Finite Group Databasebeta

← #20286851

Subgroups — #20286851

Order 6784.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811320C211yesmaximal, characteristic, complemented
2#16811257C211yesmaximal, characteristic, complemented
3#16810986C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810943C211yesmaximal, characteristic, complemented
6#16810922C211yesmaximal, characteristic, complemented
7#16810858C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810756C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11#16810732C211yesmaximal, characteristic, complemented
12#16810692C211yesmaximal, characteristic, complemented
13#16810629C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#469C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133213C2211yescharacteristic
23#13133210—21no—
24#13133208C2211yescharacteristic, complemented
25#13133198—21no—
26#13133192C2211yescharacteristic
27#13133190—21no—
28#13133188—21no—
29#13133188C2211yescharacteristic, complemented
30#13133187—21no—
31#13133185—21no—
32D4×Dic53C2211yescharacteristic, complemented
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36#13133176C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×D8×C53C2211yescharacteristic, complemented
53C2×M16×C53C2211yescharacteristic, complemented
54#121×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic, complemented
56#106×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71C22×Dic53—21no—
72D4×D53—22no—
73D4×D53—41no—
74D4×D53—44no—
75C2×D212—22no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—41no—
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84D8×C53—41no—
85M16×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53D411yescharacteristic
96#11719922—22no—
97#11719922—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53C2×D411yescharacteristic
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2411yescharacteristic
136C22⋊D53—22no—
137C22⋊D53—42no—
138C22⋊D53—42no—
139C22⋊D53—42no—
140C22⋊D53—81no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—81no—
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147D212—21no—
148D212—21no—
149D212—21no—
150D212—42no—
151D212—42no—
152D212#4112yes—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—42no—
156C4×D53—81no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—22no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—84no—
180C2×D53—84no—
181C2×D53AGL(1,ℤ/8)12yes—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53C2×#4111yescharacteristic
186C4×C53H(5,𝔽2)11yescharacteristic
187Dic53—21no—
188Dic53—21no—
189Dic53—21no—
190Dic53—41no—
191Dic53—41no—
192#3318—531nomaximal, Sylow
193C2×C53—41no—
194C2×C53—81no—
195C2×C53—81no—
196C2×C53#54611yescharacteristic
197C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
198C2×C53#57611yescharacteristic
199D53—22no—
200D53—42no—
201D53—81no—
202#553—531no—
203D42—531no—
204#490—531no—
205#482—531no—
206#481—531no—
207#478—531no—
208#469D5311yescharacteristic, complemented
209#466—531no—
210#459—531no—
211#447—531no—
212#447—531no—
213#442—531no—
214#435—531no—
215#418—531no—
216C2×#102—531no—
217C53#331811yescharacteristic, complemented, Sylow
218C22×D4—531no—
219C22×D4—531no—
220C22×D4—1061no—
221C2×D8—531no—
222C2×D8—531no—
223C2×D8C2×D5311yescharacteristic, complemented
224C2×M16C2×D5311yescharacteristic, complemented
225C22×C8—531no—
226#127—531no—
227#126—1061no—
228#125—531no—
229#123—1061no—
230#121—531no—
231#121—531no—
232#121—1061no—
233#121—1061no—
234#121C2×D5311yescharacteristic
235#121C2×D5311yescharacteristic, complemented
236#120—1061no—
237#120—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—1061no—
242#117—531no—
243C2×C4⋊C4—531no—
244#107—531no—
245#106—531no—
246#106C2×D5311yescharacteristic, complemented
247C4⋊C8—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102C2×D5311yescharacteristic, complemented
257#102C2×D5311yescharacteristic, complemented
258C22⋊C8—531no—
259C22⋊C8—531no—
260#97—531no—
261C24—1061no—
262C24—1061no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—1062no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1062no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2124no—
279C2×D4C22×D5311yescharacteristic
280C2×D4C22×D5311yescharacteristic
281C22×C4—531no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4C22×D5311yescharacteristic
286D8—41no—
287D8—1062no—
288D8—2121no—
289M16—41no—
290C2×C8—531no—
291C2×C8—531no—
292C2×C8—2121no—
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4C22×D5311yescharacteristic
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—531no—
306C22⋊C4—531no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42—531no—
313C42—1061no—
314C23—21no—
315C23—21no—
316C23—531no—
317C23—1061no—
318C23—1061no—
319C23—4241no—
320C23—4241no—
321C23—2122no—
322C23—2122no—
323C23—2122no—
324C23—2122no—
325C23D4×D5311yescharacteristic
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—81no—
332D4—532no—
333D4—1062no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—4241no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—531no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—2122no—
357C2×C4D4×D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—532no—
373C22—1061no—
374C22—1061no—
375C22—1062no—
376C22—4244no—
377C22—4244no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—1061no—
388C4—1061no—
389C4—1061no—
390C4—2121no—
391C4—2121no—
392C4#41×D5311yescharacteristic
393C4#1313326211yescharacteristic
394C2—41no—
395C2—81no—
396C2—81no—
397C2—1062no—
398C2—4241no—
399C2—2122no—
400C2#1681126211yescharacteristic
401C2AGL(1,ℤ/8)×D5311yescharacteristic
402C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3318
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16811320C2semidirect
2. #16811257C2semidirect
3. #16810986C2semidirect
4. #16810944C2semidirect
5. #16810943C2semidirect
6. #16810922C2semidirect
7. #16810858C2semidirect
8. #16810816C2semidirect
9. #16810756C2semidirect
10. #16810755C2semidirect
11. #16810732C2semidirect
12. #16810692C2semidirect
13. #16810629C2semidirect
14. #102×D53C2semidirect
15. C53×#469C2semidirect
20. C2×#11719922C22semidirect
21. C2×C8×D53C22semidirect
24. #13133208C22semidirect
29. #13133188C22semidirect
32. D4×Dic53C22semidirect
36. #13133176C22semidirect
37. C4⋊C4×D53C22semidirect
38. #13133150C22semidirect
39. #13133148C22semidirect
40. #13133146C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
48. #13133137C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
52. C2×D8×C53C22semidirect
53. C2×M16×C53C22semidirect
55. #121×C53C22semidirect
56. #106×C53C22semidirect
57. #102×C53C22semidirect
58. #102×C53C22semidirect
208. #469D53semidirect
217. C53#3318semidirect
223. C2×D8C2×D53semidirect
224. C2×M16C2×D53semidirect
235. #121C2×D53semidirect
246. #106C2×D53semidirect
256. #102C2×D53semidirect
257. #102C2×D53semidirect