Finite Group Databasebeta

← #20285064

Subgroups — #20285064

Order 6784.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811250C211yesmaximal, characteristic, complemented
2#16811186C211yesmaximal, characteristic
3#16810987C211yesmaximal, characteristic, complemented
4#16810986C211yesmaximal, characteristic, complemented
5#16810985C211yesmaximal, characteristic, complemented
6#16810981C211yesmaximal, characteristic, complemented
7#16810978C211yesmaximal, characteristic, complemented
8#16810976C211yesmaximal, characteristic, complemented
9#16810968C211yesmaximal, characteristic, complemented
10#16810626C211yesmaximal, characteristic, complemented
11#16810621C211yesmaximal, characteristic, complemented
12#16810616C211yesmaximal, characteristic, complemented
13#102×D53C211yesmaximal, characteristic, complemented
14#16810577C211yesmaximal, characteristic, complemented
15C53×#486C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133216C2211yescharacteristic, complemented
21#13133215—21no—
22#13133213C2211yescharacteristic
23#13133209C2211yescharacteristic, complemented
24#13133208C2211yescharacteristic, complemented
25#13133207—21no—
26#13133205C2211yescharacteristic
27#13133204C2211yescharacteristic
28#13133195—21no—
29#13133193—21no—
30#13133189—21no—
31#13133189—21no—
32#13133185—21no—
33#13133185—21no—
34#13133175C2211yescharacteristic
35#13133174C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37#13133142C2211yescharacteristic, complemented
38#13133142C2211yescharacteristic, complemented
39#13133140C2211yescharacteristic, complemented
40#13133140C2211yescharacteristic, complemented
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133128C2211yescharacteristic, complemented
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C4⋊Q8×C53C2211yescharacteristic, complemented
53#127×C53C2211yescharacteristic
54#102×C53C2211yescharacteristic
55#102×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70#11719947—22no—
71#11719947—22no—
72D4×D53—22no—
73D4×D53—22no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53C2311yescharacteristic
77C2×Dic106—21no—
78C2×Dic106C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921D411yescharacteristic
91#11719921D411yescharacteristic
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107C8×D53—22no—
108C23×C53—21no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53—42no—
112C22×D53—81no—
113C22×D53—81no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—81no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—81no—
121D4×C53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—42no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140C4×D53—81no—
141C4×D53#4112yes—
142C4×D53#4112yes—
143Dic106—21no—
144Dic106—21no—
145Dic106—81no—
146C8×C53—21no—
147C8×C53—21no—
148#88322—21no—
149#88322—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53C22×D411yescharacteristic
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—84no—
163C2×D53#12412yes—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53C2×#4111yescharacteristic
170Dic53—21no—
171Dic53—21no—
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175Dic53#12412yes—
176#3136—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53—81no—
179C2×C53C2×#12411yescharacteristic
180C2×C53#57611yescharacteristic
181C2×C53#57711yescharacteristic
182D53—22no—
183D53—81no—
184#544—531no—
185#535—531no—
186#490—531no—
187#490—531no—
188#489—531no—
189#489—531no—
190#488—531no—
191#488—531no—
192#486—531no—
193#486D5311yescharacteristic, complemented
194#418—531no—
195#417—531no—
196#416—531no—
197C2×#102—531no—
198#405—531no—
199C53#313611yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C22×D4—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205C4⋊Q8—1061no—
206C4⋊Q8C2×D5311yescharacteristic, complemented
207#127—531no—
208#127C2×D5311yescharacteristic
209#125—531no—
210#125—531no—
211#125—1061no—
212#124—531no—
213#124—531no—
214C22⋊Q8—1061no—
215C22⋊Q8—1061no—
216#121—1061no—
217#121—1061no—
218#120—1061no—
219#120—1061no—
220#117—531no—
221#117—531no—
222C2×C4⋊C4—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102C2×D5311yescharacteristic
236#102C2×D5311yescharacteristic
237#102C2×D5311yescharacteristic, complemented
238#102C2×D5311yescharacteristic, complemented
239#97—531no—
240#97—531no—
241C4×C8—531no—
242C4×C8C2×D5311yescharacteristic, complemented
243C24—1061no—
244#41—1062no—
245#41—1062no—
246C2×Q8—21no—
247C2×Q8—531no—
248C2×Q8—1061no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—531no—
252C2×D4—1062no—
253C2×D4—1062no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4C22×D5311yescharacteristic
259C2×D4C22×D5311yescharacteristic
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263M16—1062no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8—1062no—
267C2×C8C22×D5311yescharacteristic
268C2×C8C22×D5311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—531no—
283C22⋊C4—531no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—531no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—21no—
293C23—21no—
294C23—531no—
295C23—1061no—
296C23—4241no—
297C23—4241no—
298C23—2122no—
299Q8—81no—
300Q8—1061no—
301Q8—1061no—
302Q8—4241no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—1061no—
310D4—1061no—
311D4—4241no—
312D4—4241no—
313D4—2122no—
314D4—2122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—531no—
318C2×C4—532no—
319C2×C4—532no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—2122no—
327C2×C4D4×D5311yescharacteristic
328C2×C4D4×D5311yescharacteristic
329C2×C4C23×D5311yescharacteristic
330C8—21no—
331C8—21no—
332C8—1061no—
333C8—1061no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—532no—
341C22—1061no—
342C22—1061no—
343C22—4244no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—532no—
353C4—1061no—
354C4—1061no—
355C4—2121no—
356C4—2121no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4#41×D5311yescharacteristic
360C2—81no—
361C2—81no—
362C2—1062no—
363C2—4241no—
364C2#124×D5311yescharacteristic
365C2#1681142411yescharacteristic
366C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3136
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811250C2semidirect
3. #16810987C2semidirect
4. #16810986C2semidirect
5. #16810985C2semidirect
6. #16810981C2semidirect
7. #16810978C2semidirect
8. #16810976C2semidirect
9. #16810968C2semidirect
10. #16810626C2semidirect
11. #16810621C2semidirect
12. #16810616C2semidirect
13. #102×D53C2semidirect
14. #16810577C2semidirect
15. C53×#486C2semidirect
20. #13133216C22semidirect
23. #13133209C22semidirect
24. #13133208C22semidirect
37. #13133142C22semidirect
38. #13133142C22semidirect
39. #13133140C22semidirect
40. #13133140C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
49. #13133128C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
52. C4⋊Q8×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C4×C8×C53C22semidirect
193. #486D53semidirect
199. C53#3136semidirect
206. C4⋊Q8C2×D53semidirect
237. #102C2×D53semidirect
238. #102C2×D53semidirect
242. C4×C8C2×D53semidirect