Finite Group Databasebeta

← #20286683

Subgroups — #20286683

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#16811278C211yesmaximal, characteristic, complemented
2#16811266C211yesmaximal, characteristic, complemented
3#16810982C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810940C211yesmaximal, characteristic, complemented
6#16810914C211yesmaximal, characteristic, complemented
7#16810873C211yesmaximal, characteristic, complemented
8#16810849C211yesmaximal, characteristic, complemented
9#16810767C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic
22#13133217C2211yescharacteristic, complemented
23#13133199—21no—
24#13133199—21no—
25#13133199—21no—
26#13133196—21no—
27#13133193C2211yescharacteristic
28#13133190—21no—
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic
31#13133188C2211yescharacteristic, complemented
32#13133187—21no—
33C4×D212C2211yescharacteristic
34C4×D212C2211yescharacteristic
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37C22⋊C4×D53—21no—
38#13133149C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133128C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68#11719949—22no—
69#11719949—41no—
70D4×D53—22no—
71D4×D53—41no—
72#11719945—44no—
73C2×D212—21no—
74C2×D212D411yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83SD16×C53—41no—
84D8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95#11719922—22no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99#11719921D411yescharacteristic
100C4⋊Dic53—21no—
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118C22×D53C2×D411yescharacteristic
119C22×D53C2×D411yescharacteristic
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—42no—
133C22⋊D53—42no—
134C22⋊D53—42no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—22no—
142D212—21no—
143D212—42no—
144D212#4112yescomplemented
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—42no—
150C4×D53—42no—
151C4×D53—42no—
152C4×D53—81no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—22no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—84no—
174C2×D53AGL(1,ℤ/8)12yes—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C2×#4111yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—21no—
182Dic53—21no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—81no—
188C2×C53—81no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—22no—
193D53—41no—
194D53—42no—
195#547—531no—
196#546—531no—
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465—531no—
203#463D5311yescharacteristic, complemented
204#449—531no—
205#447—531no—
206#442—531no—
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—1061no—
213C22×D4—531no—
214C2×SD16C2×D5311yescharacteristic, complemented
215C2×D8—531no—
216C2×D8C2×D5311yescharacteristic
217C2×M16—531no—
218C22×C8—531no—
219C4⋊Q8—531no—
220#124C2×D5311yescharacteristic
221#123—1061no—
222#123—1061no—
223#123—1061no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226#121—531no—
227#121—531no—
228#121—531no—
229#121—1061no—
230#120—1061no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4C2×D5311yescharacteristic, complemented
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C22⋊C4—1061no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8C2×D5311yescharacteristic, complemented
241#103—531no—
242#103—531no—
243#103C2×D5311yescharacteristic, complemented
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102C2×D5311yescharacteristic
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97—531no—
254C24—1061no—
255#41—1062no—
256#41—2124no—
257#41—2121no—
258C2×Q8—1061no—
259C2×Q8C22×D5311yescharacteristic
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1062no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C2×D4C22×D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C22×C4—21no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278SD16—41no—
279D8—41no—
280D8—1062no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—2121no—
285C2×C8C22×D5311yescharacteristic
286C2×C8C22×D5311yescharacteristic
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C42C22×D5311yescharacteristic
307C23—21no—
308C23—21no—
309C23—531no—
310C23—531no—
311C23—1061no—
312C23—4241no—
313C23—2122no—
314C23—2122no—
315Q8—21no—
316Q8—21no—
317Q8—2122no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—532no—
323D4—1062no—
324D4—1061no—
325D4—4241no—
326D4—2122no—
327D4—2122no—
328D4—2122no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4D4×D5311yescharacteristic
347C2×C4D4×D5311yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—21no—
350C8—21no—
351C8—1061no—
352C8—1061no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—532no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—1062no—
364C22—1061no—
365C22—1061no—
366C22—4244no—
367C22—2121no—
368C22—2121no—
369C22C2×D4×D5311yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—1061no—
375C4—1061no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4#41×D5311yescharacteristic
380C4#1313326211yescharacteristic
381C2—81no—
382C2—81no—
383C2—1062no—
384C2—2121no—
385C2—2122no—
386C2#1681127011yescharacteristic
387C2AGL(1,ℤ/8)×D5311yescharacteristic
388C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #16811278C2semidirect
2. #16811266C2semidirect
3. #16810982C2semidirect
4. #16810952C2semidirect
5. #16810940C2semidirect
6. #16810914C2semidirect
7. #16810873C2semidirect
8. #16810849C2semidirect
9. #16810767C2semidirect
10. #16810755C2semidirect
11. #16810733C2semidirect
12. #16810693C2semidirect
13. #16810623C2semidirect
14. #102×D53C2semidirect
15. C53×#463C2semidirect
20. C2×#11719912C22semidirect
22. #13133217C22semidirect
31. #13133188C22semidirect
36. C4⋊C4×D53C22semidirect
38. #13133149C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
47. #13133138C22semidirect
50. #13133129C22semidirect
51. #13133128C22semidirect
52. C2×SD16×C53C22semidirect
55. C4×D4×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
144. D212#41semidirect
203. #463D53semidirect
210. C53#3306semidirect
214. C2×SD16C2×D53semidirect
233. C4×D4C2×D53semidirect
240. C4⋊C8C2×D53semidirect
243. #103C2×D53semidirect