Finite Group Databasebeta

← #20286730

Subgroups — #20286730

Order 6784.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D212C211yesmaximal, characteristic
2#16810989C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810823C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810773C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810627C211yesmaximal, characteristic, complemented
14C2×#13133137C211yesmaximal, characteristic, complemented
15C53×#552C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×D4×D53—21no—
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic, complemented
21C2×#11719924C2211yescharacteristic, complemented
22C2×#11719917C2211yescharacteristic, complemented
23C22×#88322C2211yescharacteristic, complemented
24#13133214C2211yescharacteristic
25#13133192—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133187—21no—
29#13133187—21no—
30C4×D212C2211yescharacteristic
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133137C2211yescharacteristic
37#13133137C2211yescharacteristic
38#13133137C2211yescharacteristic
39#13133137C2211yescharacteristic, complemented
40#13133137C2211yescharacteristic, complemented
41#13133137C2211yescharacteristic, complemented
42#13133137C2211yescharacteristic, complemented
43#13133137C2211yescharacteristic, complemented
44#13133137C2211yescharacteristic, complemented
45C22⋊#88322C2211yescharacteristic, complemented
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48#126×C53—21no—
49#125×C53—21no—
50#125×C53C2211yescharacteristic, complemented
51#123×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic, complemented
53C22⋊Q8×C53C2211yescharacteristic, complemented
54C4×Q8×C53C2211yescharacteristic, complemented
55C4×D4×C53C2211yescharacteristic
56#117×C53—21no—
57#117×C53C2211yescharacteristic, complemented
58C2×C4⋊C4×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53D411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68D4×D53—44no—
69C2×D212—22no—
70C2×D212—22no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53D411yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53D411yescharacteristic
95C22⋊C4×C53D411yescharacteristic
96C42×C53—21no—
97C42×C53—21no—
98C42×C53C2311yescharacteristic
99#11719924—22no—
100#11719924—41no—
101#11719924—41no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
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×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—42no—
141C22⋊D53—42no—
142C2×Dic53—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212—42no—
150D212—42no—
151D212—81no—
152D212#4112yes—
153C4×D53—42no—
154#88322—21no—
155#88322—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53AGL(1,ℤ/8)12yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—22no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—84no—
175C2×D53—84no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53C2×#4111yescharacteristic
185C4×C53H(5,𝔽2)11yescharacteristic
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—22no—
189C2×C53—41no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—42no—
194D53—81no—
195D53—81no—
196#552D5311yescharacteristic, complemented
197D42—531no—
198#490—531no—
199#482—531no—
200#480—531no—
201#478—531no—
202#468—531no—
203#465—531no—
204#460—531no—
205#450—531no—
206#450—531no—
207#440—531no—
208#435—531no—
209#418—531no—
210C2×#102—531no—
211C53#331011yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C22×D4—531no—
214C22×D4—1061no—
215C2×SD16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×M16—531no—
219C22×C8—531no—
220#127—531no—
221#126—21no—
222#125—21no—
223#125C2×D5311yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D5311yescharacteristic, complemented
226C22⋊Q8C2×D5311yescharacteristic, complemented
227#121—531no—
228#121—531no—
229#121—1061no—
230#120—1061no—
231#120—1061no—
232C4×Q8C2×D5311yescharacteristic, complemented
233C4×D4—531no—
234C4×D4C2×D5311yescharacteristic
235#117—21no—
236#117C2×D5311yescharacteristic, complemented
237C2×C4⋊C4C2×D5311yescharacteristic, complemented
238#107—531no—
239#107—531no—
240#106—531no—
241C4⋊C8—531no—
242#103—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C24—1061no—
256C24—1061no—
257C2×Q8C22×D5311yescharacteristic
258C2×D4—531no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—1062no—
262C2×D4—1062no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2124no—
272C2×D4C22⋊D5311yescharacteristic
273C22×C4—21no—
274C22×C4—1061no—
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277SD16—1062no—
278SD16—2121no—
279SD16—2121no—
280M16—2121no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—2121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—531no—
293C4⋊C4C22⋊D5311yescharacteristic
294C4⋊C4C22×D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4C22⋊D5311yescharacteristic
304C22⋊C4C22⋊D5311yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D5311yescharacteristic
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23—4241no—
312C23—4241no—
313C23—2122no—
314C23—2122no—
315C23—2122no—
316C23—2122no—
317C23C2×C22⋊D5311yescharacteristic
318C23C2×C22⋊D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—532no—
323D4—1061no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—1061no—
328D4—4241no—
329D4—2122no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1061no—
344C2×C4—2122no—
345C2×C4C2×C22⋊D5311yescharacteristic
346C2×C4C2×C22⋊D5311yescharacteristic
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C2×C22⋊D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—1061no—
351C8—1061no—
352C8—1061no—
353C8—1061no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—1062no—
360C22—4244no—
361C22—4244no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22#1313324212yes—
371C22C22×C22⋊D5311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2121no—
381C4#41×D5311yescharacteristic
382C4#1313326311yescharacteristic
383C2—22no—
384C2—41no—
385C2—4241no—
386C2—4241no—
387C2—2122no—
388C2#1681126111yescharacteristic
389C2C2×#1313324211yescharacteristic
390C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#552C2
socleC22×C53C22×D4
hypercenter#552D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #16810989C2semidirect
3. #16810947C2semidirect
4. #16810936C2semidirect
5. #16810920C2semidirect
6. #16810871C2semidirect
7. #16810854C2semidirect
8. #16810823C2semidirect
9. #16810773C2semidirect
10. #16810773C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810627C2semidirect
14. C2×#13133137C2semidirect
15. C53×#552C2semidirect
20. C2×#11719924C22semidirect
21. C2×#11719924C22semidirect
22. C2×#11719917C22semidirect
23. C22×#88322C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
39. #13133137C22semidirect
40. #13133137C22semidirect
41. #13133137C22semidirect
42. #13133137C22semidirect
43. #13133137C22semidirect
44. #13133137C22semidirect
45. C22⋊#88322C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
50. #125×C53C22semidirect
52. C22⋊Q8×C53C22semidirect
53. C22⋊Q8×C53C22semidirect
54. C4×Q8×C53C22semidirect
57. #117×C53C22semidirect
58. C2×C4⋊C4×C53C22semidirect
196. #552D53semidirect
211. C53#3310semidirect
223. #125C2×D53semidirect
225. C22⋊Q8C2×D53semidirect
226. C22⋊Q8C2×D53semidirect
232. C4×Q8C2×D53semidirect
236. #117C2×D53semidirect
237. C2×C4⋊C4C2×D53semidirect