Finite Group Databasebeta

← #20285783

Subgroups — #20285783

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811233C211yesmaximal, characteristic, complemented
2#16811190C211yesmaximal, characteristic
3#16811014C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810986C211yesmaximal, characteristic, complemented
6#16810976C211yesmaximal, characteristic, complemented
7#16810844C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810787C211yesmaximal, characteristic, complemented
12#16810783C211yesmaximal, characteristic, complemented
13#16810774C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Dic212C2211yescharacteristic, complemented
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic, complemented
20C2×#11719923C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719924C2211yescharacteristic, complemented
23#13133215C2211yescharacteristic, complemented
24#13133213C2211yescharacteristic
25#13133213C2211yescharacteristic
26#13133210—21no—
27#13133210—21no—
28#13133208—21no—
29#13133208C2211yescharacteristic, complemented
30#13133207—21no—
31#13133204C2211yescharacteristic
32#13133204C2211yescharacteristic
33#13133196C2211yescharacteristic
34#13133188—21no—
35#13133188—21no—
36#13133186—21no—
37#13133186—21no—
38Q8×Dic53C2211yescharacteristic, complemented
39#13133182C2211yescharacteristic, complemented
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C24×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72#11719945—41no—
73#11719945—41no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78SD16×C53—41no—
79SD16×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91C22⋊Dic53D411yescharacteristic
92#11719925—41no—
93#11719924—41no—
94#11719923—41no—
95#11719923—41no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53D411yescharacteristic
99C4⋊Dic53D411yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112Dic212—41no—
113#11719913—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—42no—
117C23×C53—81no—
118C23×C53—81no—
119C23×C53C2×D411yescharacteristic
120C22×D53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—81no—
136C22⋊D53—81no—
137C22⋊D53—81no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145C2×Dic53C2×D411yescharacteristic
146D212—21no—
147D212—21no—
148C4×D53—42no—
149Dic106—21no—
150Dic106—21no—
151Dic106—21no—
152Dic106—21no—
153C8×C53—21no—
154C8×C53—21no—
155#88322—21no—
156#88322—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—84no—
167C22×C53C22×D411yescharacteristic
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—21no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53H(5,𝔽2)11yescharacteristic
175C4×C53H(5,𝔽2)11yescharacteristic
176Dic53—21no—
177Dic53—21no—
178Dic53—21no—
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3222—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—81no—
186C2×C53—81no—
187C2×C53#53511yescharacteristic
188C2×C53#57711yescharacteristic
189C2×C53#57711yescharacteristic
190D53—81no—
191#541—531no—
192#535—531no—
193#497—531no—
194#492—531no—
195#490—531no—
196#488—531no—
197#464—531no—
198#464—531no—
199#460—531no—
200#460—531no—
201#452—531no—
202#452—531no—
203#450—531no—
204#450D5311yescharacteristic, complemented
205#433—531no—
206C53#322211yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C22×D4C2×D5311yescharacteristic
209C2×Q16—531no—
210C2×Q16—531no—
211C2×SD16—531no—
212C2×SD16—531no—
213C2×SD16—531no—
214C2×SD16—531no—
215C2×SD16C2×D5311yescharacteristic
216C2×SD16C2×D5311yescharacteristic
217C4⋊Q8—531no—
218#127—531no—
219#127—531no—
220#126—1061no—
221#126—1061no—
222#125—531no—
223#125—1061no—
224#125—1061no—
225#124—531no—
226#124—531no—
227C22⋊Q8—531no—
228C22⋊Q8C2×D5311yescharacteristic
229#121—1061no—
230#121—1061no—
231#120—1061no—
232#120—1061no—
233C4×Q8—531no—
234C4×Q8—531no—
235#117—531no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238#103—531no—
239#103—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102C2×D5311yescharacteristic
245#102C2×D5311yescharacteristic
246C22⋊C8—531no—
247C22⋊C8C2×D5311yescharacteristic
248#97—531no—
249C4×C8—531no—
250C24—21no—
251#41—2121no—
252#41—2121no—
253C2×Q8—531no—
254C2×Q8—531no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—531no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4C22×D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C22×C4—1061no—
268C22×C4C22×D5311yescharacteristic
269Q16—2121no—
270Q16—2121no—
271SD16—41no—
272SD16—41no—
273SD16—2121no—
274SD16—2121no—
275SD16—2121no—
276SD16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4C22×D5311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—531no—
295C22⋊C4—531no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C42—531no—
302C42—531no—
303C42—1061no—
304C42—1061no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—1061no—
311C23D4×D5311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—1061no—
315Q8—1061no—
316Q8—1061no—
317Q8—1061no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—1061no—
324D4—1061no—
325D4—4241no—
326D4—4241no—
327D4—4241no—
328D4—4241no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—2122no—
342C2×C4D4×D5311yescharacteristic
343C2×C4C23×D5311yescharacteristic
344C8—21no—
345C8—21no—
346C8—1061no—
347C8—1061no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—84no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C2×D4×D5311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—1061no—
366C4—1061no—
367C4—1061no—
368C4—1061no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4#1313326211yescharacteristic
373C4#1313326211yescharacteristic
374C2—41no—
375C2—81no—
376C2—81no—
377C2—4241no—
378C2#1681118811yescharacteristic
379C2#1681142911yescharacteristic
380C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3222
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811233C2semidirect
3. #16811014C2semidirect
4. #16810994C2semidirect
5. #16810986C2semidirect
6. #16810976C2semidirect
7. #16810844C2semidirect
8. #16810843C2semidirect
9. #16810822C2semidirect
10. #16810821C2semidirect
11. #16810787C2semidirect
12. #16810783C2semidirect
13. #16810774C2semidirect
14. #16810681C2semidirect
15. C53×#450C2semidirect
17. C2×Dic212C22semidirect
18. C2×#11719925C22semidirect
19. C2×#11719913C22semidirect
22. C2×#11719924C22semidirect
23. #13133215C22semidirect
29. #13133208C22semidirect
38. Q8×Dic53C22semidirect
39. #13133182C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
45. #13133139C22semidirect
48. #13133137C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
204. #450D53semidirect
206. C53#3222semidirect