Finite Group Databasebeta

← #20284969

Subgroups — #20284969

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811236C211yesmaximal, characteristic, complemented
2#16811095C211yesmaximal, characteristic
3#16810967C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810829C211yesmaximal, characteristic, complemented
7#16810828C211yesmaximal, characteristic, complemented
8#16810775C211yesmaximal, characteristic
9#16810775C211yesmaximal, characteristic
10#16810722C211yesmaximal, characteristic, complemented
11#16810720C211yesmaximal, characteristic, complemented
12#16810714C211yesmaximal, characteristic, complemented
13#16810710C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21#13133217C2211yescharacteristic, complemented
22#13133212—21no—
23#13133212—21no—
24#13133208—21no—
25#13133208—21no—
26#13133197C2211yescharacteristic
27#13133197C2211yescharacteristic
28#13133182C2211yescharacteristic, complemented
29#13133182C2211yescharacteristic, complemented
30C4×D212C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32#13133179C2211yescharacteristic
33C4×C22⋊D53—21no—
34#13133177—21no—
35#13133175C2211yescharacteristic
36#13133174C2211yescharacteristic
37C2×#11719921—21no—
38C22⋊C4×D53—21no—
39#13133150C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41C4⋊#88322C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52#125×C53C2211yescharacteristic, complemented
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C22×Dic53—21no—
64D4×D53—22no—
65D4×D53—22no—
66C2×D212C2311yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53C2311yescharacteristic
70C2×Dic106C2311yescharacteristic
71C2×Dic106C2311yescharacteristic
72C2×C8×C53C2311yescharacteristic
73C2×C8×C53C2311yescharacteristic
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53C2311yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C42×C53C2311yescharacteristic
81C22⋊Dic53—21no—
82#11719923—22no—
83#11719923—22no—
84#11719921—21no—
85#11719921—21no—
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921D411yescharacteristic
90#11719921D411yescharacteristic
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919D411yescharacteristic
99#11719919D411yescharacteristic
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719913—22no—
107#11719913—22no—
108C23×C53—21no—
109C22×D53—21no—
110C22×D53—24no—
111C22×D53—81no—
112C22×D53—81no—
113C22×D53C2×D411yescharacteristic
114D4×C53—21no—
115D4×C53#4112yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—24no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—81no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131D212—21no—
132D212#4112yes—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—81no—
136C4×D53—81no—
137Dic106—21no—
138Dic106—21no—
139Dic106—21no—
140Dic106—21no—
141C8×C53—21no—
142C8×C53—21no—
143#88322—21no—
144#88322—21no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53C22×D411yescharacteristic
149C2×D53—21no—
150C2×D53—22no—
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—84no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#4111yescharacteristic
162C4×C53C2×#4111yescharacteristic
163Dic53—21no—
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3123—531nomaximal, Sylow
170C2×C53—42no—
171C2×C53#52511yescharacteristic
172C2×C53#57711yescharacteristic
173C2×C53#57711yescharacteristic
174D53—41no—
175D53—42no—
176#541—531no—
177#518—531no—
178#485—531no—
179#485—531no—
180#478—531no—
181#478D5311yescharacteristic, complemented
182#461—531no—
183#461—531no—
184#450—531no—
185#450—531no—
186#440—531no—
187#440—531no—
188C4×SD16—531no—
189C4×SD16—531no—
190#433—531no—
191C53#312311yescharacteristic, complemented, Sylow
192C22×D4—531no—
193C2×SD16—531no—
194C2×SD16—531no—
195C2×SD16—531no—
196C2×SD16—531no—
197C4⋊Q8—531no—
198#126—1061no—
199#126—1061no—
200#125—1061no—
201#125—1061no—
202#125C2×D5311yescharacteristic, complemented
203C22⋊Q8—531no—
204C22⋊Q8—531no—
205C4×Q8—531no—
206C4×Q8—531no—
207C4×D4—531no—
208C4×D4—531no—
209C4×D4—531no—
210C4×D4—1061no—
211C4×D4—1061no—
212C4×D4C2×D5311yescharacteristic
213#117—531no—
214#117—531no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217#107—531no—
218#107C2×D5311yescharacteristic, complemented
219#106—531no—
220#106C2×D5311yescharacteristic, complemented
221C4⋊C8—531no—
222C4⋊C8C2×D5311yescharacteristic, complemented
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#102—531no—
228#102—531no—
229#102C2×D5311yescharacteristic
230#102C2×D5311yescharacteristic
231C22⋊C8—531no—
232C22⋊C8—531no—
233#97—531no—
234C4×C8—531no—
235C24—1061no—
236C2×Q8—531no—
237C2×Q8—531no—
238C2×D4—531no—
239C2×D4—1061no—
240C2×D4—1062no—
241C2×D4—1062no—
242C2×D4C22×D5311yescharacteristic
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248SD16—1062no—
249SD16—1062no—
250SD16—1062no—
251SD16—1062no—
252C2×C8—531no—
253C2×C8—531no—
254C2×C8C22×D5311yescharacteristic
255C2×C8C22×D5311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—531no—
259C4⋊C4—531no—
260C4⋊C4—531no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4C22×D5311yescharacteristic
268C4⋊C4C22×D5311yescharacteristic
269C4⋊C4C22×D5311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—531no—
272C22⋊C4—531no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C42—531no—
280C42—1061no—
281C42—1061no—
282C42—1061no—
283C42C22×D5311yescharacteristic
284C23—21no—
285C23—531no—
286C23—1064no—
287C23—1061no—
288C23—4241no—
289C23—4241no—
290Q8—1061no—
291Q8—1061no—
292Q8—1061no—
293Q8—1061no—
294D4—21no—
295D4—532no—
296D4—1064no—
297D4—1061no—
298D4#1171994912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—4241no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4D4×D5311yescharacteristic
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—21no—
319C8—1061no—
320C8—1061no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—1061no—
325C22—1062no—
326C22—1061no—
327C22—1061no—
328C22—4244no—
329C22—2121no—
330C22—2121no—
331C22C2×D4×D5311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—1061no—
338C4—1061no—
339C4—1061no—
340C4—2121no—
341C4—2121no—
342C4—2121no—
343C4#41×D5311yescharacteristic
344C4C2×#1171994911yescharacteristic
345C2—42no—
346C2—2121no—
347C2—2122no—
348C2#1681113611yescharacteristic
349C2#1681143011yescharacteristic
350C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811236C2semidirect
3. #16810967C2semidirect
4. #16810966C2semidirect
5. #16810920C2semidirect
6. #16810829C2semidirect
7. #16810828C2semidirect
10. #16810722C2semidirect
11. #16810720C2semidirect
12. #16810714C2semidirect
13. #16810710C2semidirect
14. #16810683C2semidirect
15. C53×#478C2semidirect
21. #13133217C22semidirect
28. #13133182C22semidirect
29. #13133182C22semidirect
39. #13133150C22semidirect
40. #13133148C22semidirect
41. C4⋊#88322C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
45. #13133140C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
52. #125×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. C4⋊C8×C53C22semidirect
181. #478D53semidirect
191. C53#3123semidirect
202. #125C2×D53semidirect
218. #107C2×D53semidirect
220. #106C2×D53semidirect
222. C4⋊C8C2×D53semidirect