Finite Group Databasebeta

← #20286863

Subgroups — #20286863

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#16811323C211yesmaximal, characteristic, complemented
2#16811258C211yesmaximal, characteristic, complemented
3#16810987C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810940C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810873C211yesmaximal, characteristic, complemented
8#16810856C211yesmaximal, characteristic, complemented
9#16810755C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×D4×D53—21no—
17C2×D4×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133210—21no—
23#13133209C2211yescharacteristic, complemented
24#13133200—21no—
25#13133190—21no—
26#13133188—21no—
27#13133188C2211yescharacteristic
28#13133188C2211yescharacteristic
29#13133188C2211yescharacteristic, complemented
30#13133188C2211yescharacteristic, complemented
31#13133185—21no—
32#13133185—21no—
33C4×D212C2211yescharacteristic
34C4×D212C2211yescharacteristic
35C4×C22⋊D53—21no—
36#13133175C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic, complemented
38#13133151C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133138C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133128C2211yescharacteristic, complemented
52C2×D8×C53C2211yescharacteristic, complemented
53C2×D8×C53C2211yescharacteristic, complemented
54#127×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic, complemented
56C4⋊C8×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
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73D4×D53—22no—
74D4×D53—22no—
75D4×D53—41no—
76D4×D53—41no—
77D4×D53—44no—
78C2×D212D411yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×C4×D53C2311yescharacteristic
84D8×C53—41no—
85D8×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719922—22no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99#11719921D411yescharacteristic
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53C2×D411yescharacteristic
123C22×D53C2×D411yescharacteristic
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—42no—
131D4×C53—81no—
132C2×C4×C53—21no—
133C2×C4×C53—81no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—42no—
138C22⋊D53—42no—
139C22⋊D53—42no—
140C22⋊D53—42no—
141C22⋊D53—42no—
142C22⋊D53—81no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148D212—22no—
149D212—21no—
150D212#4112yes—
151C4×D53—21no—
152C4×D53—21no—
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—41no—
162C22×C53—41no—
163C22×C53—41no—
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—21no—
174C2×D53—22no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—84no—
180C2×D53—84no—
181C2×D53AGL(1,ℤ/8)12yes—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53C2×#4111yescharacteristic
186C4×C53H(5,𝔽2)11yescharacteristic
187Dic53—21no—
188Dic53—21no—
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192#3318—531nomaximal, Sylow
193C2×C53—81no—
194C2×C53—81no—
195C2×C53—81no—
196C2×C53#54611yescharacteristic
197C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
198C2×C53#57611yescharacteristic
199D53—22no—
200D53—41no—
201D53—42no—
202#553—531no—
203D42—531no—
204#490—531no—
205#482—531no—
206#481—531no—
207#478—531no—
208#469—531no—
209#466—531no—
210#459D5311yescharacteristic, complemented
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×D8C2×D5311yescharacteristic, complemented
223C2×D8C2×D5311yescharacteristic, complemented
224C2×M16—531no—
225C22×C8—531no—
226#127C2×D5311yescharacteristic
227#126—1061no—
228#125—531no—
229#123—1061no—
230#121—531no—
231#121—531no—
232#121—531no—
233#121—531no—
234#121—1061no—
235#121—1061no—
236#120—1061no—
237#120—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—1061no—
241C4×D4C2×D5311yescharacteristic, complemented
242#117—531no—
243C2×C4⋊C4—531no—
244#107—531no—
245#106—531no—
246#106—531no—
247C4⋊C8C2×D5311yescharacteristic, complemented
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—1061no—
268C2×D4—1061no—
269C2×D4—1062no—
270C2×D4—1062no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—2121no—
276C2×D4—2121no—
277C2×D4—2124no—
278C2×D4C22×D5311yescharacteristic
279C2×D4C22×D5311yescharacteristic
280C2×D4C22×D5311yescharacteristic
281C22×C4—21no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—1061no—
285C22×C4—1061no—
286D8—41no—
287D8—41no—
288D8—1062no—
289M16—2121no—
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—531no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—531no—
305C22⋊C4—531no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42—1061no—
313C42C22×D5311yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—531no—
318C23—531no—
319C23—1061no—
320C23—4241no—
321C23—4241no—
322C23—2122no—
323C23—2122no—
324C23—2122no—
325C23—2122no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—81no—
334D4—532no—
335D4—1062no—
336D4—1061no—
337D4—4241no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—4241no—
355C2×C4—2122no—
356C2×C4D4×D5311yescharacteristic
357C2×C4D4×D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—532no—
373C22—1061no—
374C22—1061no—
375C22—1061no—
376C22—1062no—
377C22—1061no—
378C22—1061no—
379C22—4244no—
380C22—4244no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—1061no—
388C4—1061no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C4#41×D5311yescharacteristic
393C4#1313326211yescharacteristic
394C2—81no—
395C2—81no—
396C2—81no—
397C2—1062no—
398C2—2121no—
399C2—2122no—
400C2#1681127011yescharacteristic
401C2AGL(1,ℤ/8)×D5311yescharacteristic
402C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3318
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811323C2semidirect
2. #16811258C2semidirect
3. #16810987C2semidirect
4. #16810945C2semidirect
5. #16810940C2semidirect
6. #16810924C2semidirect
7. #16810873C2semidirect
8. #16810856C2semidirect
9. #16810755C2semidirect
10. #16810755C2semidirect
11. #16810733C2semidirect
12. #16810693C2semidirect
13. #16810626C2semidirect
14. #102×D53C2semidirect
15. C53×#459C2semidirect
20. C2×#11719912C22semidirect
21. C2×C8×D53C22semidirect
23. #13133209C22semidirect
29. #13133188C22semidirect
30. #13133188C22semidirect
36. #13133175C22semidirect
37. C4⋊C4×D53C22semidirect
38. #13133151C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
43. #13133138C22semidirect
44. #13133138C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
49. #13133129C22semidirect
50. #13133129C22semidirect
51. #13133128C22semidirect
52. C2×D8×C53C22semidirect
53. C2×D8×C53C22semidirect
55. C4×D4×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #102×C53C22semidirect
58. #102×C53C22semidirect
210. #459D53semidirect
217. C53#3318semidirect
222. C2×D8C2×D53semidirect
223. C2×D8C2×D53semidirect
241. C4×D4C2×D53semidirect
247. C4⋊C8C2×D53semidirect
256. #102C2×D53semidirect
257. #102C2×D53semidirect