Finite Group Databasebeta

← #20286742

Subgroups — #20286742

Order 6784.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811315C211yesmaximal, characteristic, complemented
2#16811257C211yesmaximal, characteristic
3#16810986C211yesmaximal, characteristic, complemented
4#16810949C211yesmaximal, characteristic, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810922C211yesmaximal, characteristic, complemented
7#16810872C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810726C211yesmaximal, characteristic, complemented
12#16810694C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic, complemented
21C2×#11719912C2211yescharacteristic, complemented
22#13133213C2211yescharacteristic
23#13133212—21no—
24#13133208—21no—
25#13133208C2211yescharacteristic, complemented
26#13133201—21no—
27#13133193C2211yescharacteristic, complemented
28#13133192C2211yescharacteristic
29#13133190—21no—
30#13133187—21no—
31#13133185—21no—
32Q8×Dic53C2211yescharacteristic, complemented
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133175C2211yescharacteristic, complemented
36#13133173—21no—
37C2×C4⋊Dic53C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70D4×D53—22no—
71D4×D53—41no—
72D4×D53—44no—
73C2×D212—22no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81SD16×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719924—22no—
92#11719924—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111#11719912—41no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
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
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—22no—
136C22⋊D53—42no—
137C22⋊D53—42no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—42no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212—42no—
150D212#4112yes—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53AGL(1,ℤ/8)12yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—22no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—84no—
175C2×D53—84no—
176C4×C53—21no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53C2×#4111yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—21no—
183Dic53—21no—
184Dic53—21no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—22no—
189C2×C53—81no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—41no—
194D53—42no—
195D53—81no—
196#552—531no—
197D42—531no—
198#490—531no—
199#482—531no—
200#480—531no—
201#478—531no—
202#468—531no—
203#465D5311yescharacteristic, complemented
204#460—531no—
205#450—531no—
206#450—531no—
207#440—531no—
208#435—531no—
209#418—531no—
210C2×#102—531no—
211C53#331011yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C22×D4—531no—
214C22×D4—1061no—
215C2×SD16—531no—
216C2×SD16—531no—
217C2×SD16C2×D5311yescharacteristic, complemented
218C2×M16—531no—
219C22×C8C2×D5311yescharacteristic, complemented
220#127—531no—
221#126—1061no—
222#125—531no—
223#125—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8C2×D5311yescharacteristic, complemented
227#121—531no—
228#121—1061no—
229#121C2×D5311yescharacteristic
230#120—1061no—
231#120—1061no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4—531no—
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—531no—
238#107—531no—
239#107—531no—
240#106C2×D5311yescharacteristic, complemented
241C4⋊C8—531no—
242#103C2×D5311yescharacteristic, complemented
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102C2×D5311yescharacteristic, complemented
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C24—1061no—
256C24—1061no—
257C2×Q8C22×D5311yescharacteristic
258C2×D4—21no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—1062no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1062no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2124no—
272C2×D4C22×D5311yescharacteristic
273C22×C4—531no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4C22×D5311yescharacteristic
277SD16—41no—
278SD16—1062no—
279SD16—2121no—
280M16—2121no—
281C2×C8—41no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C42—531no—
306C42—1061no—
307C42—1061no—
308C23—21no—
309C23—531no—
310C23—1061no—
311C23—1061no—
312C23—4241no—
313C23—4241no—
314C23—2122no—
315C23—2122no—
316C23—2122no—
317C23—2122no—
318C23D4×D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—532no—
325D4—1062no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—4241no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—531no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—1061no—
360C22—1061no—
361C22—1061no—
362C22—1062no—
363C22—4244no—
364C22—4244no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22#1313324412yes—
371C22C2×D4×D5311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—1061no—
377C4—1061no—
378C4—1061no—
379C4—2121no—
380C4—2121no—
381C4#41×D5311yescharacteristic
382C4#1313326211yescharacteristic
383C2—22no—
384C2—81no—
385C2—4241no—
386C2—2121no—
387C2—2122no—
388C2#1681126211yescharacteristic
389C2C2×#1313324411yescharacteristic
390C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811315C2semidirect
3. #16810986C2semidirect
4. #16810949C2semidirect
5. #16810934C2semidirect
6. #16810922C2semidirect
7. #16810872C2semidirect
8. #16810821C2semidirect
9. #16810773C2semidirect
10. #16810772C2semidirect
11. #16810726C2semidirect
12. #16810694C2semidirect
13. #16810626C2semidirect
14. C2×#13133139C2semidirect
15. C53×#465C2semidirect
20. C2×#11719924C22semidirect
21. C2×#11719912C22semidirect
25. #13133208C22semidirect
27. #13133193C22semidirect
32. Q8×Dic53C22semidirect
35. #13133175C22semidirect
37. C2×C4⋊Dic53C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133146C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
48. #13133137C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
52. C2×SD16×C53C22semidirect
53. C22×C8×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #106×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
203. #465D53semidirect
211. C53#3310semidirect
217. C2×SD16C2×D53semidirect
219. C22×C8C2×D53semidirect
226. C22⋊Q8C2×D53semidirect
240. #106C2×D53semidirect
242. #103C2×D53semidirect
251. #102C2×D53semidirect