Finite Group Databasebeta

← #20286752

Subgroups — #20286752

Order 6784.

402 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#16811341C211yesmaximal, characteristic, complemented
2#16811267C211yesmaximal, characteristic
3#16810981C211yesmaximal, characteristic, complemented
4#16810964C211yesmaximal, characteristic, complemented
5#16810946C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810881C211yesmaximal, characteristic, complemented
8#16810851C211yesmaximal, characteristic, complemented
9#16810844C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810774C211yesmaximal, characteristic, complemented
12#16810720C211yesmaximal, characteristic, complemented
13#16810692C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic, complemented
15C53×#418C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Dic212C2211yescharacteristic, complemented
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic, complemented
21#13133216C2211yescharacteristic, complemented
22#13133211—21no—
23#13133210—21no—
24#13133204C2211yescharacteristic
25#13133203—21no—
26#13133198—21no—
27#13133197C2211yescharacteristic, complemented
28#13133196C2211yescharacteristic
29#13133196C2211yescharacteristic
30#13133189—21no—
31#13133188C2211yescharacteristic
32#13133186—21no—
33C4×Dic106—21no—
34#13133182C2211yescharacteristic, complemented
35D4×Dic53C2211yescharacteristic
36C4×C22⋊D53C2211yescharacteristic
37#13133175C2211yescharacteristic, complemented
38C2×C22⋊Dic53—21no—
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133140C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54#117×C53C2211yescharacteristic
55#102×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71#11719947—22no—
72#11719947—41no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77M16×C53—41no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719923—22no—
92#11719923—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109Dic212—41no—
110C8×D53—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C22×D53C2×D411yescharacteristic
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—42no—
122D4×C53#4112yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—41no—
130C22⋊D53—42no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137C2×Dic53—81no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C4×D53—21no—
142C4×D53—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—21no—
147Dic106—42no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—84no—
162C22×C53C22×D411yescharacteristic
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3311—531nomaximal, Sylow
179C2×C53—41no—
180C2×C53—42no—
181C2×C53—81no—
182C2×C53#54611yescharacteristic
183C2×C53#57511yescharacteristic
184C2×C53#57711yescharacteristic
185D53—41no—
186#556—531no—
187#546—531no—
188#489—531no—
189#485—531no—
190#482—531no—
191#476—531no—
192#470—531no—
193#465—531no—
194#464—531no—
195#452—531no—
196#450—531no—
197#440—531no—
198#435—531no—
199#418D5311yescharacteristic, complemented
200#416—531no—
201C53#331111yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C22×D4C2×D5311yescharacteristic
204C2×Q16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C2×M16C2×D5311yescharacteristic
208C22×C8—531no—
209C4⋊Q8—531no—
210#126—1061no—
211#126—1061no—
212#124—531no—
213#124—1061no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—531no—
218#121—531no—
219#121—1061no—
220#120—1061no—
221C4×Q8—531no—
222C4×Q8—1061no—
223C4×D4—531no—
224C4×D4—531no—
225#117—531no—
226#117C2×D5311yescharacteristic
227C2×C22⋊C4—1061no—
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#102C2×D5311yescharacteristic
239#102C2×D5311yescharacteristic
240#102C2×D5311yescharacteristic
241#102C2×D5311yescharacteristic
242C22⋊C8—531no—
243C22⋊C8—531no—
244#97—531no—
245C24—21no—
246#41—1062no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—531no—
250C2×Q8—1061no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4C22×D5311yescharacteristic
258C2×D4C22×D5311yescharacteristic
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263Q16—2121no—
264SD16—1062no—
265SD16—2121no—
266M16—41no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—2121no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—531no—
284C22⋊C4—531no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—21no—
294C42—531no—
295C42—1061no—
296C42—1061no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—531no—
303C23D4×D5311yescharacteristic
304Q8—1061no—
305Q8—1061no—
306Q8—1061no—
307Q8—1061no—
308Q8—2122no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—4241no—
314D4—2121no—
315D4—2122no—
316D4#1171994512yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—4241no—
331C2×C4—2122no—
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—1061no—
349C22—1061no—
350C22—1061no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—1061no—
356C4—1061no—
357C4—1061no—
358C4—2121no—
359C4—2121no—
360C4—2121no—
361C4—2121no—
362C4C2×#1171994511yescharacteristic
363C4#1313326211yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—2121no—
368C2#1681126311yescharacteristic
369C2#1681141911yescharacteristic
370C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#418C2
socleC22×C53C22×D4
hypercenter#418D53
hypocenterC53#3311
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #16811341C2semidirect
3. #16810981C2semidirect
4. #16810964C2semidirect
5. #16810946C2semidirect
6. #16810911C2semidirect
7. #16810881C2semidirect
8. #16810851C2semidirect
9. #16810844C2semidirect
10. #16810786C2semidirect
11. #16810774C2semidirect
12. #16810720C2semidirect
13. #16810692C2semidirect
14. #16810616C2semidirect
15. C53×#418C2semidirect
17. C2×Dic212C22semidirect
20. C2×C8×D53C22semidirect
21. #13133216C22semidirect
27. #13133197C22semidirect
34. #13133182C22semidirect
37. #13133175C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133140C22semidirect
46. #13133140C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
199. #418D53semidirect
201. C53#3311semidirect