Finite Group Databasebeta

← #20284902

Subgroups — #20284902

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811246C211yesmaximal, characteristic, complemented
2#16811097C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810912C211yesmaximal, characteristic, complemented
5#16810902C211yesmaximal, characteristic, complemented
6#16810841C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810773C211yesmaximal, characteristic, complemented
9#16810759C211yesmaximal, characteristic
10#16810734C211yesmaximal, characteristic, complemented
11#16810726C211yesmaximal, characteristic, complemented
12#16810708C211yesmaximal, characteristic, complemented
13#16810705C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21#13133216C2211yescharacteristic, complemented
22#13133212—21no—
23#13133206C2211yescharacteristic, complemented
24#13133203—21no—
25#13133199—21no—
26#13133197—21no—
27#13133191C2211yescharacteristic
28Q8×Dic53C2211yescharacteristic, complemented
29C4×D212—21no—
30C4×D212—21no—
31#13133179C2211yescharacteristic
32#13133179C2211yescharacteristic
33#13133179C2211yescharacteristic
34#13133179C2211yescharacteristic
35#13133179C2211yescharacteristic, complemented
36#13133176C2211yescharacteristic
37C22⋊C4×D53—21no—
38C22⋊C4×D53—21no—
39#13133150C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×D212—21no—
64C2×D212—22no—
65C2×D212—22no—
66C2×D212C2311yescharacteristic
67C2×D212C2311yescharacteristic
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×Dic106—21no—
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C42×C53—21no—
84C22⋊Dic53D411yescharacteristic
85C22⋊Dic53D411yescharacteristic
86#11719924—22no—
87#11719924—22no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53D411yescharacteristic
98C4⋊Dic53D411yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53C2311yescharacteristic
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106D424—22no—
107D424—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—24no—
112C22×D53—21no—
113C22×D53—81no—
114C22×D53—81no—
115Q8×C53—21no—
116Q8×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—81no—
126C2×Dic53—21no—
127C2×Dic53C2×D411yescharacteristic
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131D212—24no—
132D212—21no—
133D212—21no—
134D212—21no—
135D212—21no—
136D212#4112yes—
137D212#4112yes—
138C4×D53—81no—
139C4×D53—81no—
140C4×D53—81no—
141C4×D53—81no—
142C4×D53—81no—
143Dic106—81no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—84no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53C2×#4111yescharacteristic
169Dic53—21no—
170Dic53—21no—
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174#3118—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53#52511yescharacteristic
177C2×C53#57611yescharacteristic
178C2×C53#57711yescharacteristic
179D53—42no—
180D53—42no—
181D53—81no—
182#542—531no—
183#518—531no—
184#485D5311yescharacteristic, complemented
185#482—531no—
186#476—531no—
187#474—531no—
188#463—531no—
189#463—531no—
190#450—531no—
191#447—531no—
192#442—531no—
193#440—531no—
194C4×SD16—531no—
195C4×D8—531no—
196#433—531no—
197C53#311811yescharacteristic, complemented, Sylow
198C22×D4—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×D8—531no—
202C2×D8—531no—
203C4⋊Q8—531no—
204#126—1061no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208C22⋊Q8—1061no—
209C22⋊Q8C2×D5311yescharacteristic
210#121—531no—
211C4×Q8—531no—
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—1061no—
218C4×D4—1061no—
219#117—531no—
220#117C2×D5311yescharacteristic
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223#107—531no—
224#107C2×D5311yescharacteristic, complemented
225#106—531no—
226#106C2×D5311yescharacteristic, complemented
227C4⋊C8—531no—
228C4⋊C8—531no—
229#103C2×D5311yescharacteristic, complemented
230#103C2×D5311yescharacteristic, complemented
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239#97—531no—
240C4×C8—531no—
241C24—1061no—
242C2×Q8—1061no—
243C2×Q8C22×D5311yescharacteristic
244C2×D4—531no—
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4C22×D5311yescharacteristic
257SD16—1062no—
258SD16—1062no—
259D8—1062no—
260D8—1062no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4C22×D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—531no—
277C22⋊C4—531no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—21no—
287C42—531no—
288C42—531no—
289C42—1061no—
290C23—1064no—
291C23—1061no—
292C23—1061no—
293C23—1061no—
294C23—4241no—
295C23—4241no—
296C23D4×D5311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—4241no—
300D4—532no—
301D4—532no—
302D4—1064no—
303D4—1061no—
304D4—1061no—
305D4—1061no—
306D4—1061no—
307D4—4241no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1061no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4D4×D5311yescharacteristic
325C2×C4C23×D5311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1061no—
329C8—1061no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—1062no—
334C22—1062no—
335C22—4244no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—1061no—
350C4—1061no—
351C4—1061no—
352C4—1061no—
353C4—2121no—
354C4#41×D5311yescharacteristic
355C4#41×D5311yescharacteristic
356C2—41no—
357C2—4241no—
358C2—2122no—
359C2—2122no—
360C2#1681113111yescharacteristic
361C2#1681142511yescharacteristic
362C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3118
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #16811246C2semidirect
2. #16811097C2semidirect
3. #16810947C2semidirect
4. #16810912C2semidirect
5. #16810902C2semidirect
6. #16810841C2semidirect
7. #16810841C2semidirect
8. #16810773C2semidirect
10. #16810734C2semidirect
11. #16810726C2semidirect
12. #16810708C2semidirect
13. #16810705C2semidirect
14. #16810681C2semidirect
15. C53×#485C2semidirect
21. #13133216C22semidirect
23. #13133206C22semidirect
28. Q8×Dic53C22semidirect
35. #13133179C22semidirect
39. #13133150C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
184. #485D53semidirect
197. C53#3118semidirect
224. #107C2×D53semidirect
226. #106C2×D53semidirect
229. #103C2×D53semidirect
230. #103C2×D53semidirect