Finite Group Databasebeta

← #20286955

Subgroups — #20286955

Order 6784.

460 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#16811336C211yesmaximal, characteristic, complemented
2#16811257C211yesmaximal, characteristic, complemented
3#16810968C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810912C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810874C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810759C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic, complemented
11#16810705C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810627C211yesmaximal, characteristic, complemented
14#16810626C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic, complemented
21C2×#11719922C2211yescharacteristic, complemented
22C2×#11719917C2211yescharacteristic, complemented
23C2×#11719912C2211yescharacteristic, complemented
24#13133216C2211yescharacteristic, complemented
25#13133213C2211yescharacteristic
26#13133205—21no—
27#13133203—21no—
28#13133192C2211yescharacteristic
29#13133192C2211yescharacteristic, complemented
30#13133191C2211yescharacteristic, complemented
31#13133190—21no—
32#13133190—21no—
33#13133187—21no—
34#13133185—21no—
35#13133179C2211yescharacteristic
36#13133179C2211yescharacteristic, complemented
37C4×C22⋊D53C2211yescharacteristic
38#13133175C2211yescharacteristic, complemented
39#13133174—21no—
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69D4×D53—22no—
70D4×D53—41no—
71D4×D53—44no—
72C2×D212—22no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C42×C53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719922—41no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—22no—
110D424—41no—
111#11719912—41no—
112C23×C53—21no—
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—
123C22×D53C2×D411yescharacteristic
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—22no—
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212—42no—
150D212—81no—
151D212#4112yes—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—42no—
155C4×D53—42no—
156Dic106—81no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—22no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—84no—
181C2×D53—84no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53C2×#4111yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—41no—
191Dic53AGL(1,ℤ/8)12yes—
192#3325—531nomaximal, Sylow
193C2×C53—41no—
194C2×C53—81no—
195C2×C53#54611yescharacteristic
196C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C53#57611yescharacteristic
198D53—41no—
199D53—42no—
200D53—81no—
201D53—81no—
202#555—531no—
203D42—531no—
204#486—531no—
205#482—531no—
206#482D5311yescharacteristic, complemented
207#476—531no—
208#469—531no—
209#469—531no—
210#459—531no—
211#447—531no—
212#447—531no—
213C4×D8—531no—
214#436—531no—
215#418—531no—
216#418—531no—
217C53#332511yescharacteristic, complemented, Sylow
218C22×D4—531no—
219C22×D4—531no—
220C22×D4—1061no—
221C2×D8—531no—
222C2×D8—531no—
223C2×D8—531no—
224C2×M16—531no—
225C2×M16—531no—
226C4⋊Q8—531no—
227#127—531no—
228#124—1061no—
229#124—1061no—
230#121—531no—
231#121—531no—
232#121—531no—
233#121—1061no—
234#121—1061no—
235#121C2×D5311yescharacteristic
236#120—1061no—
237#120—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241#117—531no—
242#117—1061no—
243#117C2×D5311yescharacteristic, complemented
244#107C2×D5311yescharacteristic, complemented
245#106—531no—
246#106C2×D5311yescharacteristic, complemented
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#102C2×D5311yescharacteristic, complemented
257#102C2×D5311yescharacteristic, complemented
258C22⋊C8—531no—
259C22⋊C8C2×D5311yescharacteristic, complemented
260C4×C8—531no—
261C24—1061no—
262C24—1061no—
263C2×Q8—1061no—
264C2×D4—21no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—1062no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1062no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2124no—
281C2×D4C22×D5311yescharacteristic
282C22×C4—531no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4C22×D5311yescharacteristic
286D8—1062no—
287D8—2121no—
288D8—2121no—
289M16—2121no—
290M16—2121no—
291C2×C8—531no—
292C2×C8—531no—
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—1061no—
298C4⋊C4C22×D5311yescharacteristic
299C4⋊C4C22×D5311yescharacteristic
300C4⋊C4C22×D5311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C42—21no—
312C42—531no—
313C42—1061no—
314C23—21no—
315C23—531no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—4241no—
320C23—4241no—
321C23—2122no—
322C23—2122no—
323C23—2122no—
324C23—2122no—
325C23D4×D5311yescharacteristic
326Q8—4241no—
327D4—21no—
328D4—21no—
329D4—42no—
330D4—532no—
331D4—1062no—
332D4—1061no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—4241no—
338D4—4241no—
339D4—4241no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343D4—2122no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4D4×D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—1061no—
370C22—1062no—
371C22—1061no—
372C22—1061no—
373C22—4244no—
374C22—4244no—
375C22—2121no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—532no—
389C4—1061no—
390C4—1061no—
391C4—2121no—
392C4#41×D5311yescharacteristic
393C4#1313326311yescharacteristic
394C2—41no—
395C2—81no—
396C2—4241no—
397C2—4241no—
398C2—2121no—
399C2—2122no—
400C2#1681126511yescharacteristic
401C2AGL(1,ℤ/8)×D5311yescharacteristic
402C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3325
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811336C2semidirect
2. #16811257C2semidirect
3. #16810968C2semidirect
4. #16810950C2semidirect
5. #16810912C2semidirect
6. #16810877C2semidirect
7. #16810874C2semidirect
8. #16810817C2semidirect
9. #16810759C2semidirect
10. #16810758C2semidirect
11. #16810705C2semidirect
12. #16810698C2semidirect
13. #16810627C2semidirect
14. #16810626C2semidirect
15. C53×#482C2semidirect
20. C2×D424C22semidirect
21. C2×#11719922C22semidirect
22. C2×#11719917C22semidirect
23. C2×#11719912C22semidirect
24. #13133216C22semidirect
29. #13133192C22semidirect
30. #13133191C22semidirect
36. #13133179C22semidirect
38. #13133175C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
43. #13133139C22semidirect
44. #13133138C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. #13133137C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
53. #117×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
206. #482D53semidirect
217. C53#3325semidirect
243. #117C2×D53semidirect
244. #107C2×D53semidirect
246. #106C2×D53semidirect
256. #102C2×D53semidirect
257. #102C2×D53semidirect
259. C22⋊C8C2×D53semidirect