Finite Group Databasebeta

← #20287736

Subgroups — #20287736

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811345C211yesmaximal, characteristic, complemented
2#16811294C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16810994C211yesmaximal, characteristic, complemented
6#16810841C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810827C211yesmaximal, characteristic, complemented
9#16810823C211yesmaximal, characteristic, complemented
10#16810822C211yesmaximal, characteristic, complemented
11#16810817C211yesmaximal, characteristic, complemented
12#16810751C211yesmaximal, characteristic, complemented
13#16810722C211yesmaximal, characteristic, complemented
14#16810705C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719913C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×D424C2211yescharacteristic
23C2×D424C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133214C2211yescharacteristic
27#13133213C2211yescharacteristic
28#13133213C2211yescharacteristic
29#13133212—21no—
30#13133211—21no—
31#13133206C2211yescharacteristic
32#13133205C2211yescharacteristic
33#13133195—21no—
34#13133191—21no—
35#13133190—21no—
36#13133190—21no—
37C4×Dic106C2211yescharacteristic
38#13133182C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133179C2211yescharacteristic
41#13133179C2211yescharacteristic
42#13133174—21no—
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66#11719945—44no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×Dic106D411yescharacteristic
76C2×Dic106C2311yescharacteristic
77SD16×C53—22no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53—21no—
85C42×C53C2311yescharacteristic
86#11719924—41no—
87#11719924—41no—
88#11719923—41no—
89#11719922—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53D411yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—22no—
109D424—41no—
110#11719913—22no—
111#11719913—41no—
112C23×C53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—42no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136D212—21no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—42no—
143D212—81no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—42no—
147Dic106—22no—
148Dic106—21no—
149Dic106—21no—
150C8×C53—21no—
151C8×C53#4112yescomplemented
152#88322—21no—
153#88322—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—21no—
175Dic53—21no—
176Dic53—21no—
177Dic53—41no—
178Dic53AGL(1,ℤ/8)12yes—
179#3397—531nomaximal, Sylow
180C2×C53—81no—
181C2×C53#55011yescharacteristic
182C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C53#57711yescharacteristic
184D53—81no—
185D53—81no—
186D53—81no—
187#556—531no—
188#550—531no—
189#496—531no—
190#496—531no—
191#492—531no—
192#463—531no—
193#463—531no—
194#461—531no—
195#460—531no—
196#460—531no—
197#459—531no—
198#446—531no—
199#440—531no—
200#440D5311yescharacteristic, complemented
201C4×D8—531no—
202C53#339711yescharacteristic, complemented, Sylow
203C2×#41—1061no—
204C2×SD16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×SD16C2×D5311yescharacteristic
210C2×D8—531no—
211C2×D8—531no—
212C2×D8—531no—
213C4⋊Q8—531no—
214#127—531no—
215#127—531no—
216#127—531no—
217#126—1061no—
218#126—1061no—
219#124—531no—
220#124—531no—
221C22⋊Q8—1061no—
222#121—1061no—
223#121—1061no—
224#121—1061no—
225C4×Q8—531no—
226C4×Q8—531no—
227C4×Q8C2×D5311yescharacteristic
228C4×D4—531no—
229C4×D4—531no—
230C4×D4—531no—
231C4×D4C2×D5311yescharacteristic
232#117—1061no—
233#107C2×D5311yescharacteristic
234C4⋊C8—531no—
235C4⋊C8—531no—
236C4⋊C8—531no—
237#103—531no—
238#103C2×D5311yescharacteristic
239#102—531no—
240#102—531no—
241#102—531no—
242#102C2×D5311yescharacteristic
243#97—531no—
244#97C2×D5311yescharacteristic
245C4×C8—531no—
246#41—2124no—
247C2×Q8—531no—
248C2×Q8—531no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—531no—
251C2×D4—531no—
252C2×D4—531no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—21no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—22no—
265SD16—1062no—
266SD16—2121no—
267SD16—2121no—
268SD16—2121no—
269SD16—2121no—
270D8—1062no—
271D8—2121no—
272D8—2121no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4C22×D5311yescharacteristic
285C4⋊C4C22×D5311yescharacteristic
286C22⋊C4—21no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—21no—
295C42—531no—
296C42—531no—
297C42—1061no—
298C42C22×D5311yescharacteristic
299C23—21no—
300C23—1061no—
301C23—1061no—
302C23—1061no—
303Q8—21no—
304Q8—21no—
305Q8—1062no—
306Q8—1061no—
307Q8—1061no—
308D4—21no—
309D4—21no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—2122no—
320D4—2122no—
321D4—2122no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—1061no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4D4×D5311yescharacteristic
335C2×C4D4×D5311yescharacteristic
336C2×C4C23×D5311yescharacteristic
337C8—21no—
338C8—1061no—
339C8—1061no—
340C8#1171994512yescomplemented
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22C2×D4×D5311yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—532no—
360C4—1061no—
361C4—1061no—
362C4—1061no—
363C4—2121no—
364C4C2×#1171994511yescharacteristic
365C4#1313326311yescharacteristic
366C2—81no—
367C2—4241no—
368C2—4241no—
369C2—4241no—
370C2#1681129411yescharacteristic
371C2AGL(1,ℤ/8)×D5311yescharacteristic
372C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3397
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #16811345C2semidirect
2. #16811294C2semidirect
3. #16811012C2semidirect
4. #16811009C2semidirect
5. #16810994C2semidirect
6. #16810841C2semidirect
7. #16810838C2semidirect
8. #16810827C2semidirect
9. #16810823C2semidirect
10. #16810822C2semidirect
11. #16810817C2semidirect
12. #16810751C2semidirect
13. #16810722C2semidirect
14. #16810705C2semidirect
15. C53×#440C2semidirect
151. C8×C53#41semidirect
200. #440D53semidirect
202. C53#3397semidirect
340. C8#11719945semidirect