Finite Group Databasebeta

← #20286550

Subgroups — #20286550

Order 6784.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811313C211yesmaximal, characteristic, complemented
2#16810977C211yesmaximal, characteristic, complemented
3#16810951C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810862C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810842C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11C4×#11719923C211yesmaximal, characteristic, complemented
12D4×#88322C211yesmaximal, characteristic, complemented
13#16810619C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic
16C2×#11719925C2211yescharacteristic, complemented
17C2×#11719923C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C22×#88322C2211yescharacteristic
20C22×#88322C2211yescharacteristic, complemented
21#13133210—21no—
22#13133209C2211yescharacteristic, complemented
23#13133207—21no—
24#13133198—21no—
25#13133196C2211yescharacteristic
26#13133196C2211yescharacteristic, complemented
27C4×Dic106C2211yescharacteristic, complemented
28#13133176C2211yescharacteristic, complemented
29#13133173—21no—
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic, complemented
32#13133148C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133140C2211yescharacteristic, complemented
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48C2×#41×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50#124×C53C2211yescharacteristic
51#123×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53—21no—
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56C4×D4×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58C2×C22⋊C4×C53—21no—
59C24×C53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—22no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×Dic53—21no—
76C2×Dic106C2311yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53D411yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53D411yescharacteristic
86C22⋊C4×C53D411yescharacteristic
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719925—41no—
92#11719923—22no—
93#11719923—41no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C23×C53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120D4×C53—22no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—81no—
127D4×C53#4112yes—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53—81no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—42no—
146Dic106—21no—
147Dic106—21no—
148#88322—21no—
149#88322—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—22no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—84no—
164C22×C53#13512yes—
165C22×C53C22×D411yescharacteristic
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3296—531nomaximal, Sylow
179C2×C53—22no—
180C2×C53—41no—
181C2×C53—42no—
182C2×C53—81no—
183C2×C53#54611yescharacteristic
184C2×C53C2×#13511yescharacteristic
185C2×C53#57711yescharacteristic
186#552—531no—
187#546D5311yescharacteristic
188#488—531no—
189#483—531no—
190#482—531no—
191#478—531no—
192#467—531no—
193#465—531no—
194#464—531no—
195#452—531no—
196#450—531no—
197C4×SD16—531no—
198C8×D4—531no—
199#416—531no—
200C2×#102—531no—
201C53#329611yescharacteristic, complemented, Sylow
202C2×#41C2×D5311yescharacteristic
203C22×D4C2×D5311yescharacteristic
204C2×Q16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C22×C8—531no—
208C22×C8—531no—
209#126—1061no—
210#125—531no—
211#125—1061no—
212#124C2×D5311yescharacteristic
213#123—21no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8C2×D5311yescharacteristic
218#121—21no—
219#121C2×D5311yescharacteristic
220#120—21no—
221C4×Q8—531no—
222C4×D4C2×D5311yescharacteristic
223C4×D4C2×D5311yescharacteristic
224#117—531no—
225#117—1061no—
226C2×C4⋊C4—531no—
227C2×C22⋊C4—21no—
228#107—531no—
229#106—531no—
230#106—531no—
231C4⋊C8—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242C22⋊C8—531no—
243C22⋊C8—531no—
244C4×C8—531no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—531no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D5311yescharacteristic
256C2×D4C22×D5311yescharacteristic
257C2×D4C22×D5311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1061no—
261C22×C4C22×D5311yescharacteristic
262C22×C4C22×D5311yescharacteristic
263Q16—2121no—
264SD16—1062no—
265SD16—2121no—
266C2×C8—531no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4C22⋊D5311yescharacteristic
283C4⋊C4C22×D5311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4C22⋊D5311yescharacteristic
293C22⋊C4C22⋊D5311yescharacteristic
294C42—1061no—
295C42—1061no—
296C42C22×D5311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D5311yescharacteristic
303C23C2×C22⋊D5311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1061no—
307Q8—1061no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1171994712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—2122no—
329C2×C4C2×C22⋊D5311yescharacteristic
330C2×C4C2×C22⋊D5311yescharacteristic
331C2×C4C2×C22⋊D5311yescharacteristic
332C2×C4C2×C22⋊D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—1061no—
335C8—1061no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1313323712yes—
351C22C22×C22⋊D5311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—2121no—
359C4—2121no—
360C4—2121no—
361C4—2121no—
362C4C2×#1171994711yescharacteristic
363C4#1313326211yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#1681126711yescharacteristic
369C2C2×#1313323711yescharacteristic
370C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3296
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #16811313C2semidirect
2. #16810977C2semidirect
3. #16810951C2semidirect
4. #16810944C2semidirect
5. #16810923C2semidirect
6. #16810862C2semidirect
7. #16810850C2semidirect
8. #16810842C2semidirect
9. #16810785C2semidirect
10. #16810774C2semidirect
11. C4×#11719923C2semidirect
12. D4×#88322C2semidirect
13. #16810619C2semidirect
14. C2×#13133138C2semidirect
16. C2×#11719925C22semidirect
20. C22×#88322C22semidirect
22. #13133209C22semidirect
26. #13133196C22semidirect
27. C4×Dic106C22semidirect
28. #13133176C22semidirect
31. #13133150C22semidirect
32. #13133148C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
38. #13133140C22semidirect
46. C22⋊#88322C22semidirect
47. C4×#88322C22semidirect
201. C53#3296semidirect