Finite Group Databasebeta

← #20286676

Subgroups — #20286676

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811277C211yesmaximal, characteristic, complemented
2#16810982C211yesmaximal, characteristic, complemented
3#16810951C211yesmaximal, characteristic, complemented
4#16810939C211yesmaximal, characteristic, complemented
5#16810913C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810757C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719924C2211yescharacteristic, complemented
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719917C2211yescharacteristic, complemented
21C22×#88322C2211yescharacteristic
22#13133217C2211yescharacteristic, complemented
23#13133201—21no—
24#13133201—21no—
25#13133197—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic, complemented
28C4×D212C2211yescharacteristic, complemented
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31#13133148C2211yescharacteristic, complemented
32#13133148C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic, complemented
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, 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
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79#11719947—44no—
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×Dic106—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53D411yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53D411yescharacteristic
92C22⋊C4×C53D411yescharacteristic
93C42×C53C2311yescharacteristic
94#11719924—41no—
95#11719922—22no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—81no—
117C23×C53C2×D411yescharacteristic
118C23×C53C2×D411yescharacteristic
119C22×D53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—22no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—42no—
127D4×C53—42no—
128D4×C53—81no—
129D4×C53#4112yescomplemented
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2411yescharacteristic
143C22⋊D53—42no—
144C22⋊D53—42no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150D212—21no—
151D212—21no—
152C4×D53—42no—
153Dic106—42no—
154#88322—21no—
155#88322—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—84no—
170C22×C53AGL(1,ℤ/8)12yes—
171C22×C53C22×D411yescharacteristic
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53C2×#4111yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—41no—
189C2×C53—42no—
190C2×C53—81no—
191C2×C53#54611yescharacteristic
192C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
193C2×C53#57711yescharacteristic
194D53—81no—
195#547—531no—
196#546D5311yescharacteristic, complemented
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465—531no—
203#463—531no—
204#449—531no—
205#447—531no—
206#442—531no—
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—1061no—
212C2×#41C2×D5311yescharacteristic
213C22×D4C2×D5311yescharacteristic
214C2×SD16—531no—
215C2×D8—531no—
216C2×D8—531no—
217C2×M16—531no—
218C22×C8—531no—
219C4⋊Q8—531no—
220#124C2×D5311yescharacteristic
221#123—21no—
222#123—1061no—
223#123—1061no—
224C22⋊Q8—1061no—
225C22⋊Q8C2×D5311yescharacteristic
226#121—21no—
227#121—531no—
228#121—531no—
229#121C2×D5311yescharacteristic
230#120—21no—
231C4×D4—531no—
232C4×D4C2×D5311yescharacteristic
233C4×D4C2×D5311yescharacteristic
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C22⋊C4—21no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8—531no—
241#103—531no—
242#103—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97—531no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—2124no—
258C2×Q8—1061no—
259C2×Q8C22×D5311yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—531no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4C22⋊D5311yescharacteristic
269C2×D4C22×D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278SD16—2121no—
279D8—1062no—
280D8—2121no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—2121no—
287C4⋊C4—21no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4C22⋊D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4C22⋊D5311yescharacteristic
305C22⋊C4C22⋊D5311yescharacteristic
306C42C22×D5311yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—1061no—
313C23C2×C22⋊D5311yescharacteristic
314C23C2×C22⋊D5311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—2122no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—1061no—
326D4—1061no—
327D4—2122no—
328D4—2122no—
329D4#1171994712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4C2×C22⋊D5311yescharacteristic
345C2×C4C2×C22⋊D5311yescharacteristic
346C2×C4C2×C22⋊D5311yescharacteristic
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—1061no—
350C8—1061no—
351C8—1061no—
352C8—1061no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22#1313324212yes—
369C22C22×C22⋊D5311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4C2×#1171994711yescharacteristic
380C4#1313326211yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—4241no—
386C2#1681126711yescharacteristic
387C2C2×#1313324211yescharacteristic
388C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3306
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. #16811277C2semidirect
2. #16810982C2semidirect
3. #16810951C2semidirect
4. #16810939C2semidirect
5. #16810913C2semidirect
6. #16810874C2semidirect
7. #16810854C2semidirect
8. #16810835C2semidirect
9. #16810768C2semidirect
10. #16810757C2semidirect
11. #16810733C2semidirect
12. #16810695C2semidirect
13. #16810624C2semidirect
14. C2×#13133138C2semidirect
15. C53×#546C2semidirect
17. C2×#11719924C22semidirect
20. C2×#11719917C22semidirect
22. #13133217C22semidirect
27. #13133191C22semidirect
28. C4×D212C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. #13133148C22semidirect
32. #13133148C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
44. #13133137C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
129. D4×C53#41semidirect
196. #546D53semidirect
210. C53#3306semidirect
329. D4#11719947semidirect