Finite Group Databasebeta

← #20284852

Subgroups — #20284852

Order 6784.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811152C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic, complemented
3#16810952C211yesmaximal, characteristic, complemented
4#16810935C211yesmaximal, characteristic, complemented
5#16810924C211yesmaximal, characteristic, complemented
6#16810923C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810772C211yesmaximal, characteristic, complemented
9#16810755C211yesmaximal, characteristic
10#16810735C211yesmaximal, characteristic, complemented
11#16810732C211yesmaximal, characteristic, complemented
12#16810726C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810679C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C22×C4×D53—21no—
18C2×#11719924C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133210—21no—
21#13133209C2211yescharacteristic, complemented
22#13133199—21no—
23#13133193C2211yescharacteristic
24#13133188C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic, complemented
26C4×D212C2211yescharacteristic
27D4×Dic53C2211yescharacteristic
28D4×Dic53C2211yescharacteristic, complemented
29#13133179C2211yescharacteristic
30C4×C22⋊D53—21no—
31#13133177—21no—
32#13133177—21no—
33#13133173—21no—
34C4⋊C4×D53C2211yescharacteristic
35C22⋊C4×D53—21no—
36C42×D53C2211yescharacteristic
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41C4⋊#88322C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic, complemented
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68D4×D53—22no—
69D4×D53—22no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—44no—
75C2×C4×D53D411yescharacteristic
76C2×C4×D53D411yescharacteristic
77C2×C4×D53C2311yescharacteristic
78SD16×C53—22no—
79D8×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719924—22no—
91#11719922—22no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C4×Dic53D411yescharacteristic
105C4×Dic53D411yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C23×C53—21no—
110C23×C53—21no—
111C22×D53—24no—
112C22×D53—21no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53#4112yes—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—24no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212#4112yes—
140C4×D53—22no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—41no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—22no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—22no—
161C2×D53—48no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53AGL(1,ℤ/8)12yes—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53C2×#4111yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3114—531nomaximal, Sylow
178C2×C53—42no—
179C2×C53—81no—
180C2×C53#52511yescharacteristic
181C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C53#57511yescharacteristic
183D53—22no—
184D53—42no—
185#529—531no—
186C2×C4×D4—531no—
187#483—531no—
188#480—531no—
189#478—531no—
190#478—531no—
191#463—531no—
192#463D5311yescharacteristic, complemented
193#450—531no—
194#447—531no—
195#442—531no—
196#442—531no—
197#440—531no—
198#440—531no—
199#432—531no—
200C53#311411yescharacteristic, complemented, Sylow
201C22×D4—531no—
202C23×C4—1061no—
203C2×SD16—531no—
204C2×SD16C2×D5311yescharacteristic
205C2×D8—531no—
206C2×D8C2×D5311yescharacteristic
207#126—1061no—
208#125—531no—
209#124C2×D5311yescharacteristic, complemented
210#123—1061no—
211C22⋊Q8—531no—
212#121—531no—
213C4×Q8—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—531no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4—1061no—
221C4×D4C2×D5311yescharacteristic
222#117—1061no—
223C2×C4⋊C4—531no—
224C2×C22⋊C4—1061no—
225C2×C42—531no—
226#107—531no—
227#107—531no—
228#106—531no—
229#106—531no—
230C4⋊C8—531no—
231C4⋊C8C2×D5311yescharacteristic, complemented
232#103—531no—
233#103C2×D5311yescharacteristic, complemented
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102C2×D5311yescharacteristic, complemented
240C22⋊C8—531no—
241C22⋊C8—531no—
242#97—531no—
243#97—531no—
244C24—1061no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1062no—
249C2×D4—1062no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C2×D4C22×D5311yescharacteristic
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—2124no—
263SD16—22no—
264SD16—1062no—
265D8—22no—
266D8—1062no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—531no—
290C42—531no—
291C42—531no—
292C42—1061no—
293C42C22×D5311yescharacteristic
294C23—21no—
295C23—21no—
296C23—531no—
297C23—1064no—
298C23—1061no—
299C23—2122no—
300C23—2122no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—532no—
307D4—1064no—
308D4—1061no—
309D4—2122no—
310D4#1171994712yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—1062no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2121no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4D4×D5311yescharacteristic
330C2×C4D4×D5311yescharacteristic
331C2×C4C23×D5311yescharacteristic
332C8—21no—
333C8—21no—
334C8—1061no—
335C8—1061no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—532no—
343C22—1061no—
344C22—1061no—
345C22—1062no—
346C22—2128no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—1061no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4#41×D5311yescharacteristic
361C4C2×#1171994711yescharacteristic
362C2—42no—
363C2—81no—
364C2—1062no—
365C2—2122no—
366C2#1681113311yescharacteristic
367C2AGL(1,ℤ/8)×D5311yescharacteristic
368C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3114
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. #16811152C2semidirect
2. C4×D4×D53C2semidirect
3. #16810952C2semidirect
4. #16810935C2semidirect
5. #16810924C2semidirect
6. #16810923C2semidirect
7. #16810835C2semidirect
8. #16810772C2semidirect
10. #16810735C2semidirect
11. #16810732C2semidirect
12. #16810726C2semidirect
13. #16810724C2semidirect
14. #16810679C2semidirect
15. C53×#463C2semidirect
21. #13133209C22semidirect
25. Q8×Dic53C22semidirect
28. D4×Dic53C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. C4⋊#88322C22semidirect
42. #13133141C22semidirect
47. #13133138C22semidirect
50. #13133127C22semidirect
51. #13133127C22semidirect
54. #124×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
192. #463D53semidirect
200. C53#3114semidirect
209. #124C2×D53semidirect
231. C4⋊C8C2×D53semidirect
233. #103C2×D53semidirect
239. #102C2×D53semidirect