Finite Group Databasebeta

← #20286858

Subgroups — #20286858

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
2D4×D212C211yesmaximal, characteristic, complemented
3#16810988C211yesmaximal, characteristic, complemented
4#16810948C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810859C211yesmaximal, characteristic, complemented
9#16810819C211yesmaximal, characteristic, complemented
10#16810759C211yesmaximal, characteristic, complemented
11#16810759C211yesmaximal, characteristic, complemented
12#16810736C211yesmaximal, characteristic, complemented
13#16810629C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×D4×D53—21no—
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic, complemented
21C2×D424C2211yescharacteristic, complemented
22#13133214C2211yescharacteristic
23#13133212—21no—
24#13133209C2211yescharacteristic, complemented
25#13133199—21no—
26#13133192—21no—
27#13133191C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic, complemented
30#13133191C2211yescharacteristic, complemented
31#13133189—21no—
32#13133187—21no—
33#13133187—21no—
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic, complemented
36#13133177—21no—
37#13133176C2211yescharacteristic, complemented
38C2×C4⋊Dic53C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic, complemented
52C2×M16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic, complemented
57C22⋊C8×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53D411yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69D4×D53—44no—
70C2×D212—22no—
71C2×D212—22no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84M16×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C2×C8×C53C2311yescharacteristic
90C4⋊C4×C53D411yescharacteristic
91C22⋊C4×C53D411yescharacteristic
92C22⋊C4×C53D411yescharacteristic
93C42×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108C4×Dic53—21no—
109D424—22no—
110D424—41no—
111D424—41no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—81no—
123C22×D53—81no—
124D4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—81no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212—21no—
150D212—42no—
151D212—42no—
152D212—42no—
153D212—81no—
154D212#4112yes—
155C4×D53—42no—
156C4×D53—81no—
157C8×C53—21no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53AGL(1,ℤ/8)12yes—
167C22×C53C22×D411yescharacteristic
168C2×D53—22no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
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—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53C2×#4111yescharacteristic
187C4×C53H(5,𝔽2)11yescharacteristic
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192#3318—531nomaximal, Sylow
193C2×C53—22no—
194C2×C53—41no—
195C2×C53#54611yescharacteristic
196C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C53#57611yescharacteristic
198D53—42no—
199D53—81no—
200D53—81no—
201D53—81no—
202#553—531no—
203D42—531no—
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#435D5311yescharacteristic, complemented
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×M16C2×D5311yescharacteristic, complemented
225C22×C8C2×D5311yescharacteristic, complemented
226#127—531no—
227#126—1061no—
228#125—531no—
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—1061no—
241C4×D4C2×D5311yescharacteristic
242#117—531no—
243C2×C4⋊C4—531no—
244#107—531no—
245#106—531no—
246#106—531no—
247C4⋊C8C2×D5311yescharacteristic, complemented
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⋊C8C2×D5311yescharacteristic, complemented
259C22⋊C8C2×D5311yescharacteristic, complemented
260#97C2×D5311yescharacteristic, complemented
261C24—1061no—
262C24—1061no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—1062no—
268C2×D4—1062no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2124no—
280C2×D4D21211yescharacteristic
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286D8—1062no—
287D8—2121no—
288D8—2121no—
289M16—41no—
290C2×C8—41no—
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—531no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4D21211yescharacteristic
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4D21211yescharacteristic
311C22⋊C4D21211yescharacteristic
312C42—1061no—
313C42C22×D5311yescharacteristic
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—4241no—
319C23—4241no—
320C23—2122no—
321C23—2122no—
322C23—2122no—
323C23—2122no—
324C23C2×D21211yescharacteristic
325C23C2×D21211yescharacteristic
326D4—22no—
327D4—532no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—1061no—
333D4—1061no—
334D4—1061no—
335D4—4241no—
336D4—4241no—
337D4—2122no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—2122no—
354C2×C4C2×D21211yescharacteristic
355C2×C4C2×D21211yescharacteristic
356C2×C4C2×D21211yescharacteristic
357C2×C4C2×D21211yescharacteristic
358C2×C4C23×D5311yescharacteristic
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—1062no—
369C22—4244no—
370C22—4244no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22#1313324512yes—
383C22C22×D21211yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—2121no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C4#41×D5311yescharacteristic
393C4#1313326211yescharacteristic
394C2—22no—
395C2—41no—
396C2—4241no—
397C2—4241no—
398C2—4241no—
399C2—2122no—
400C2#1681126811yescharacteristic
401C2C2×#1313324511yescharacteristic
402C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3318
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. D4×D212C2semidirect
3. #16810988C2semidirect
4. #16810948C2semidirect
5. #16810942C2semidirect
6. #16810924C2semidirect
7. #16810876C2semidirect
8. #16810859C2semidirect
9. #16810819C2semidirect
10. #16810759C2semidirect
11. #16810759C2semidirect
12. #16810736C2semidirect
13. #16810629C2semidirect
14. C2×#13133139C2semidirect
15. C53×#435C2semidirect
20. C2×D424C22semidirect
21. C2×D424C22semidirect
24. #13133209C22semidirect
29. #13133191C22semidirect
30. #13133191C22semidirect
35. C4×D212C22semidirect
37. #13133176C22semidirect
38. C2×C4⋊Dic53C22semidirect
39. #13133151C22semidirect
40. #13133149C22semidirect
41. #13133149C22semidirect
45. #13133139C22semidirect
46. #13133139C22semidirect
47. #13133139C22semidirect
48. #13133139C22semidirect
49. #13133139C22semidirect
50. #13133139C22semidirect
51. #13133139C22semidirect
52. C2×M16×C53C22semidirect
53. C22×C8×C53C22semidirect
55. C4⋊C8×C53C22semidirect
56. C22⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. #97×C53C22semidirect
214. #435D53semidirect
217. C53#3318semidirect
224. C2×M16C2×D53semidirect
225. C22×C8C2×D53semidirect
247. C4⋊C8C2×D53semidirect
258. C22⋊C8C2×D53semidirect
259. C22⋊C8C2×D53semidirect
260. #97C2×D53semidirect