Finite Group Databasebeta

← #20286855

Subgroups — #20286855

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#16811326C211yesmaximal, characteristic, complemented
2#16811258C211yesmaximal, characteristic, complemented
3#16810946C211yesmaximal, characteristic, complemented
4#16810940C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810875C211yesmaximal, characteristic, complemented
7#16810858C211yesmaximal, characteristic, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810758C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic, complemented
11#16810736C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#490C211yesmaximal, characteristic, complemented
16C2×D4×D53—21no—
17C2×D4×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic, complemented
21C2×#11719922C2211yescharacteristic, complemented
22C2×#11719912C2211yescharacteristic, complemented
23C2×C8×D53C2211yescharacteristic, complemented
24#13133212—21no—
25#13133202—21no—
26#13133192—21no—
27#13133192C2211yescharacteristic, complemented
28#13133192C2211yescharacteristic, complemented
29#13133188—21no—
30#13133188C2211yescharacteristic
31#13133188C2211yescharacteristic
32#13133185—21no—
33#13133185—21no—
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic, complemented
37#13133179—21no—
38#13133175C2211yescharacteristic, complemented
39C4⋊C4×D53C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43C4⋊#88322C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#127×C53C2211yescharacteristic
53#125×C53C2211yescharacteristic, complemented
54#102×C53C2211yescharacteristic, complemented
55#102×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69D4×D53—22no—
70D4×D53—22no—
71D4×D53—41no—
72D4×D53—41no—
73D4×D53—44no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212D411yescharacteristic
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×C4×D53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719922—41no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101#11719921D411yescharacteristic
102C4⋊Dic53D411yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—22no—
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53C2×D411yescharacteristic
123C22×D53C2×D411yescharacteristic
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—42no—
138C22⋊D53—42no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—22no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—42no—
148D212—81no—
149D212#4112yes—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—42no—
155C4×D53—81no—
156C4×D53—81no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—22no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—84no—
180C2×D53—84no—
181C2×D53AGL(1,ℤ/8)12yes—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53C2×#4111yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—41no—
191Dic53—41no—
192#3318—531nomaximal, Sylow
193C2×C53—81no—
194C2×C53—81no—
195C2×C53#54611yescharacteristic
196C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C53#57611yescharacteristic
198D53—22no—
199D53—41no—
200D53—42no—
201D53—81no—
202#553—531no—
203D42—531no—
204#490D5311yescharacteristic, complemented
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—531no—
219C22×D4—531no—
220C22×D4—1061no—
221C2×D8—531no—
222C2×D8—531no—
223C2×D8—531no—
224C2×M16—531no—
225C22×C8—531no—
226#127C2×D5311yescharacteristic
227#126—1061no—
228#125C2×D5311yescharacteristic, complemented
229#123—1061no—
230#121—531no—
231#121—531no—
232#121—531no—
233#121—531no—
234#121—1061no—
235#121—1061no—
236#120—1061no—
237#120—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—1061no—
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#102C2×D5311yescharacteristic, complemented
255#102C2×D5311yescharacteristic, complemented
256#102C2×D5311yescharacteristic, complemented
257#102C2×D5311yescharacteristic, complemented
258C22⋊C8—531no—
259C22⋊C8—531no—
260#97C2×D5311yescharacteristic, complemented
261C24—1061no—
262C24—1061no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2124no—
279C2×D4C22×D5311yescharacteristic
280C2×D4C22×D5311yescharacteristic
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286D8—1062no—
287D8—2121no—
288D8—2121no—
289M16—2121no—
290C2×C8—531no—
291C2×C8—531no—
292C2×C8—2121no—
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—531no—
298C4⋊C4—531no—
299C4⋊C4—531no—
300C4⋊C4—1061no—
301C4⋊C4C22×D5311yescharacteristic
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42—1061no—
313C42C22×D5311yescharacteristic
314C23—21no—
315C23—21no—
316C23—531no—
317C23—531no—
318C23—1061no—
319C23—1061no—
320C23—4241no—
321C23—4241no—
322C23—2122no—
323C23—2122no—
324C23—2122no—
325C23—2122no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—81no—
332D4—532no—
333D4—1062no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—4241no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—2122no—
356C2×C4D4×D5311yescharacteristic
357C2×C4D4×D5311yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—1061no—
362C8—1061no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—532no—
370C22—1061no—
371C22—1061no—
372C22—1061no—
373C22—1062no—
374C22—1061no—
375C22—1061no—
376C22—4244no—
377C22—4244no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22C2×D4×D5311yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—1061no—
389C4—1061no—
390C4—2121no—
391C4—2121no—
392C4#41×D5311yescharacteristic
393C4#1313326311yescharacteristic
394C2—81no—
395C2—81no—
396C2—1062no—
397C2—4241no—
398C2—2121no—
399C2—2122no—
400C2#1681126611yescharacteristic
401C2AGL(1,ℤ/8)×D5311yescharacteristic
402C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3318
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811326C2semidirect
2. #16811258C2semidirect
3. #16810946C2semidirect
4. #16810940C2semidirect
5. #16810920C2semidirect
6. #16810875C2semidirect
7. #16810858C2semidirect
8. #16810815C2semidirect
9. #16810758C2semidirect
10. #16810758C2semidirect
11. #16810736C2semidirect
12. #16810696C2semidirect
13. #16810626C2semidirect
14. #102×D53C2semidirect
15. C53×#490C2semidirect
20. C2×#11719922C22semidirect
21. C2×#11719922C22semidirect
22. C2×#11719912C22semidirect
23. C2×C8×D53C22semidirect
27. #13133192C22semidirect
28. #13133192C22semidirect
36. C4×D212C22semidirect
38. #13133175C22semidirect
39. C4⋊C4×D53C22semidirect
40. #13133150C22semidirect
41. #13133149C22semidirect
42. #13133148C22semidirect
43. C4⋊#88322C22semidirect
45. #13133139C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
53. #125×C53C22semidirect
54. #102×C53C22semidirect
55. #102×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. #97×C53C22semidirect
204. #490D53semidirect
217. C53#3318semidirect
228. #125C2×D53semidirect
254. #102C2×D53semidirect
255. #102C2×D53semidirect
256. #102C2×D53semidirect
257. #102C2×D53semidirect
260. #97C2×D53semidirect