Finite Group Databasebeta

← #20284894

Subgroups — #20284894

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#16811240C211yesmaximal, characteristic, complemented
2#16811097C211yesmaximal, characteristic, complemented
3#16810964C211yesmaximal, characteristic, complemented
4#16810912C211yesmaximal, characteristic, complemented
5#16810902C211yesmaximal, characteristic, complemented
6#16810837C211yesmaximal, characteristic, complemented
7#16810837C211yesmaximal, characteristic, complemented
8#16810776C211yesmaximal, characteristic, complemented
9#16810756C211yesmaximal, characteristic
10#16810732C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810710C211yesmaximal, characteristic, complemented
13#16810703C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21#13133216C2211yescharacteristic, complemented
22#13133210—21no—
23#13133204—21no—
24#13133204C2211yescharacteristic, complemented
25#13133198—21no—
26#13133196C2211yescharacteristic
27#13133194—21no—
28#13133182C2211yescharacteristic, complemented
29C4×D212—21no—
30C4×D212—21no—
31D4×Dic53C2211yescharacteristic, complemented
32#13133179C2211yescharacteristic
33#13133179C2211yescharacteristic
34#13133179C2211yescharacteristic
35#13133179C2211yescharacteristic
36#13133176C2211yescharacteristic
37C22⋊C4×D53—21no—
38C22⋊C4×D53—21no—
39#13133150C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53—21no—
64C2×D212—21no—
65C2×D212—22no—
66C2×D212—22no—
67C2×D212C2311yescharacteristic
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×Dic106—21no—
74C2×Dic106C2311yescharacteristic
75C2×C8×C53C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88C22⋊Dic53D411yescharacteristic
89#11719922—22no—
90#11719922—22no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53D411yescharacteristic
96C4⋊Dic53D411yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53C2311yescharacteristic
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719913—22no—
107#11719913—22no—
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—24no—
113C22×D53—81no—
114C22×D53—81no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—81no—
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
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—81no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—24no—
133D212—21no—
134D212—21no—
135D212#4112yes—
136D212#4112yes—
137C4×D53—81no—
138C4×D53—81no—
139C4×D53—81no—
140C4×D53—81no—
141Dic106—21no—
142Dic106—21no—
143Dic106—81no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—22no—
156C2×D53—22no—
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×C53C2×#4111yescharacteristic
167C4×C53C2×#4111yescharacteristic
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—21no—
172Dic53—41no—
173Dic53—41no—
174#3118—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—81no—
177C2×C53#52511yescharacteristic
178C2×C53#57611yescharacteristic
179C2×C53#57711yescharacteristic
180D53—42no—
181D53—42no—
182#542—531no—
183#518—531no—
184#485—531no—
185#482D5311yescharacteristic, complemented
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—531no—
209C22⋊Q8—1061no—
210#121C2×D5311yescharacteristic
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#103—531no—
230#103—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102C2×D5311yescharacteristic, complemented
236#102C2×D5311yescharacteristic, complemented
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239#97—531no—
240C4×C8—531no—
241C24—1061no—
242C2×Q8—531no—
243C2×Q8—1061no—
244C2×D4—21no—
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—1061no—
250C2×D4C22×D5311yescharacteristic
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—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
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—21no—
291C23—1064no—
292C23—1061no—
293C23—1061no—
294C23—4241no—
295C23—4241no—
296C23D4×D5311yescharacteristic
297Q8—1061no—
298Q8—1061no—
299Q8—4241no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—532no—
304D4—532no—
305D4—1064no—
306D4—1061no—
307D4—1061no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—1061no—
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—41no—
334C22—41no—
335C22—41no—
336C22—1062no—
337C22—1062no—
338C22—4244no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1061no—
349C4—1061no—
350C4—1061no—
351C4—1061no—
352C4—2121no—
353C4—2121no—
354C4#41×D5311yescharacteristic
355C4#41×D5311yescharacteristic
356C2—41no—
357C2—81no—
358C2—2122no—
359C2—2122no—
360C2#1681113111yescharacteristic
361C2#1681142711yescharacteristic
362C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3118
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811240C2semidirect
2. #16811097C2semidirect
3. #16810964C2semidirect
4. #16810912C2semidirect
5. #16810902C2semidirect
6. #16810837C2semidirect
7. #16810837C2semidirect
8. #16810776C2semidirect
10. #16810732C2semidirect
11. #16810722C2semidirect
12. #16810710C2semidirect
13. #16810703C2semidirect
14. #16810681C2semidirect
15. C53×#482C2semidirect
21. #13133216C22semidirect
24. #13133204C22semidirect
28. #13133182C22semidirect
31. D4×Dic53C22semidirect
39. #13133150C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133140C22semidirect
44. #13133140C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
185. #482D53semidirect
197. C53#3118semidirect
224. #107C2×D53semidirect
226. #106C2×D53semidirect
235. #102C2×D53semidirect
236. #102C2×D53semidirect