Finite Group Databasebeta

← #20286856

Subgroups — #20286856

Order 6784.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811319C211yesmaximal, characteristic, complemented
2#16810987C211yesmaximal, characteristic, complemented
3#16810944C211yesmaximal, characteristic, complemented
4#16810939C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810757C211yesmaximal, characteristic, complemented
10#16810757C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810628C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15D42×C53C211yesmaximal, characteristic, complemented
16C2×#11719922C2211yescharacteristic
17C2×#11719922C2211yescharacteristic, complemented
18C2×#11719922C2211yescharacteristic, complemented
19C2×#11719917C2211yescharacteristic, complemented
20C22×#88322C2211yescharacteristic, complemented
21#13133212—21no—
22#13133209C2211yescharacteristic, complemented
23#13133199—21no—
24#13133191C2211yescharacteristic, complemented
25#13133191C2211yescharacteristic, complemented
26#13133189—21no—
27C4×D212C2211yescharacteristic, complemented
28#13133177—21no—
29#13133176C2211yescharacteristic, complemented
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133148C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133138C2211yescharacteristic
36#13133138C2211yescharacteristic
37#13133138C2211yescharacteristic
38#13133138C2211yescharacteristic, complemented
39#13133138C2211yescharacteristic, complemented
40#13133138C2211yescharacteristic, complemented
41#13133138C2211yescharacteristic, complemented
42#13133138C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic, complemented
44#13133137C2211yescharacteristic, complemented
45C22⋊#88322C2211yescharacteristic, complemented
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48C22×D4×C53—21no—
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51#127×C53C2211yescharacteristic
52#121×C53—21no—
53#121×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56#120×C53—21no—
57C4×D4×C53C2211yescharacteristic
58C4×D4×C53C2211yescharacteristic
59C24×C53—21no—
60C24×C53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—44no—
72C2×D4×C53D411yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C2×D4×C53C2311yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C22×Dic53—21no—
82C2×D212C2311yescharacteristic
83C2×C4×D53—21no—
84C4⋊C4×C53D411yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53D411yescharacteristic
90C22⋊C4×C53D411yescharacteristic
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93#11719922—22no—
94#11719922—41no—
95#11719922—41no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53C2×D411yescharacteristic
122C23×C53C2×D411yescharacteristic
123C22×D53—21no—
124D4×C53—22no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—81no—
136D4×C53#4112yes—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2×D411yescharacteristic
147C2×C4×C53C2411yescharacteristic
148C22⋊D53—42no—
149C22⋊D53—81no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—81no—
154D212—21no—
155D212—21no—
156C4×D53—81no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—22no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—84no—
176C22×C53—84no—
177C22×C53AGL(1,ℤ/8)12yes—
178C22×C53C22×D411yescharacteristic
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53C2×#4111yescharacteristic
188C4×C53H(5,𝔽2)11yescharacteristic
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192#3318—531nomaximal, Sylow
193C2×C53—22no—
194C2×C53—41no—
195C2×C53—42no—
196C2×C53—81no—
197C2×C53—81no—
198C2×C53#54611yescharacteristic
199C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C53#57611yescharacteristic
201D53—81no—
202#553—531no—
203D42D5311yescharacteristic, complemented
204#490—531no—
205#482—531no—
206#481—531no—
207#478—531no—
208#469—531no—
209#466—531no—
210#459—531no—
211#447—531no—
212#447—531no—
213#442—531no—
214#435—531no—
215#418—531no—
216C2×#102—531no—
217C53#331811yescharacteristic, complemented, Sylow
218C22×D4—21no—
219C22×D4C2×D5311yescharacteristic
220C22×D4C2×D5311yescharacteristic
221C2×D8—531no—
222C2×D8—531no—
223C2×D8—531no—
224C2×M16—531no—
225C22×C8—531no—
226#127C2×D5311yescharacteristic
227#126—1061no—
228#125—531no—
229#123—1061no—
230#121—21no—
231#121—531no—
232#121—531no—
233#121—1061no—
234#121C2×D5311yescharacteristic
235#121C2×D5311yescharacteristic
236#120—21no—
237#120—21no—
238C4×D4—531no—
239C4×D4—1061no—
240C4×D4C2×D5311yescharacteristic
241C4×D4C2×D5311yescharacteristic
242#117—531no—
243C2×C4⋊C4—531no—
244#107—531no—
245#106—531no—
246#106—531no—
247C4⋊C8—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258C22⋊C8—531no—
259C22⋊C8—531no—
260#97—531no—
261C24—21no—
262C24—21no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—531no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4C22⋊D5311yescharacteristic
278C2×D4C22×D5311yescharacteristic
279C2×D4C22×D5311yescharacteristic
280C2×D4C22×D5311yescharacteristic
281C22×C4—21no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286D8—1062no—
287D8—2121no—
288D8—2121no—
289M16—2121no—
290C2×C8—531no—
291C2×C8—531no—
292C2×C8—531no—
293C2×C8—531no—
294C2×C8—2121no—
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4C22⋊D5311yescharacteristic
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4C22⋊D5311yescharacteristic
311C22⋊C4C22⋊D5311yescharacteristic
312C42—1061no—
313C42C22×D5311yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—1061no—
324C23C2×C22⋊D5311yescharacteristic
325C23C2×C22⋊D5311yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—81no—
338D4—1061no—
339D4—1061no—
340D4—4241no—
341D4—2122no—
342D4#1171994712yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4C2×C22⋊D5311yescharacteristic
355C2×C4C2×C22⋊D5311yescharacteristic
356C2×C4C2×C22⋊D5311yescharacteristic
357C2×C4C2×C22⋊D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—1061no—
360C8—1061no—
361C8—1061no—
362C8—1061no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—84no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22#1313324312yes—
383C22C22×C22⋊D5311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C4C2×#1171994711yescharacteristic
393C4#1313326211yescharacteristic
394C2—22no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—81no—
399C2—4241no—
400C2#1681126711yescharacteristic
401C2C2×#1313324311yescharacteristic
402C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD42×C53C2
socleC22×C53C22×D4
hypercenterD42D53
hypocenterC53#3318
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16810987C2semidirect
3. #16810944C2semidirect
4. #16810939C2semidirect
5. #16810923C2semidirect
6. #16810874C2semidirect
7. #16810857C2semidirect
8. #16810815C2semidirect
9. #16810757C2semidirect
10. #16810757C2semidirect
11. #16810733C2semidirect
12. #16810695C2semidirect
13. #16810628C2semidirect
14. C2×#13133138C2semidirect
15. D42×C53C2semidirect
17. C2×#11719922C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719917C22semidirect
20. C22×#88322C22semidirect
22. #13133209C22semidirect
24. #13133191C22semidirect
25. #13133191C22semidirect
27. C4×D212C22semidirect
29. #13133176C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. #13133150C22semidirect
32. #13133148C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
38. #13133138C22semidirect
39. #13133138C22semidirect
40. #13133138C22semidirect
41. #13133138C22semidirect
42. #13133138C22semidirect
43. #13133138C22semidirect
44. #13133137C22semidirect
45. C22⋊#88322C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
203. D42D53semidirect
217. C53#3318semidirect