Finite Group Databasebeta

← #20284859

Subgroups — #20284859

Order 6784.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811168C211yesmaximal, characteristic, complemented
2#41×Dic53C211yesmaximal, characteristic, complemented
3#16810948C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810906C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810821C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810735C211yesmaximal, characteristic, complemented
11#16810732C211yesmaximal, characteristic, complemented
12#16810726C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810677C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719924C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19#13133216C2211yescharacteristic
20#13133213C2211yescharacteristic
21#13133203—21no—
22#13133191C2211yescharacteristic
23#13133189—21no—
24Q8×Dic53C2211yescharacteristic
25Q8×Dic53C2211yescharacteristic
26D4×Dic53—21no—
27D4×Dic53—21no—
28D4×Dic53C2211yescharacteristic
29D4×Dic53C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31#13133179C2211yescharacteristic
32#13133177—21no—
33#13133176—21no—
34#13133176C2211yescharacteristic
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51#13133127C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—22no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53D411yescharacteristic
75C22×Dic53D411yescharacteristic
76#11719947—44no—
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×Dic106—21no—
80SD16×C53—22no—
81D8×C53—22no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719924—22no—
91#11719922—22no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C23×C53—21no—
110C23×C53—21no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113Q8×C53—21no—
114Q8×C53#4112yescomplemented
115D4×C53—24no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—42no—
120D4×C53#4112yescomplemented
121C2×C4×C53—24no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—42no—
130C22⋊D53—42no—
131C2×Dic53—22no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—41no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—21no—
144D212—21no—
145C4×D53—42no—
146Dic106—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C22×D411yescharacteristic
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53C2×#4111yescharacteristic
169Dic53—21no—
170Dic53—21no—
171Dic53—21no—
172Dic53—41no—
173Dic53—41no—
174Dic53—41no—
175Dic53AGL(1,ℤ/8)12yes—
176#3115—531nomaximal, Sylow
177C2×C53—41no—
178C2×C53—42no—
179C2×C53—81no—
180C2×C53#52511yescharacteristic
181C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C53#57511yescharacteristic
183D53—81no—
184#532—531no—
185C4×#41—531no—
186#482—531no—
187#482—531no—
188#475—531no—
189#474—531no—
190#460—531no—
191#459—531no—
192#449—531no—
193#449D5311yescharacteristic, complemented
194#442—531no—
195#442—531no—
196#440—531no—
197#440—531no—
198#432—531no—
199C53#311511yescharacteristic, complemented, Sylow
200C2×#41—1061no—
201C2×#41C2×D5311yescharacteristic
202C2×SD16—531no—
203C2×SD16C2×D5311yescharacteristic
204C2×D8—531no—
205C2×D8C2×D5311yescharacteristic
206C4⋊Q8—531no—
207#127—531no—
208#124—1061no—
209#121—531no—
210#121—1061no—
211#121C2×D5311yescharacteristic
212C4×Q8—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—
221#117—531no—
222#117—1061no—
223C2×C42—531no—
224C2×C42—1061no—
225#107—531no—
226#107—531no—
227#106—531no—
228#106—531no—
229C4⋊C8—531no—
230C4⋊C8—531no—
231#103—531no—
232#103C2×D5311yescharacteristic
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238#102C2×D5311yescharacteristic
239C22⋊C8—531no—
240C22⋊C8C2×D5311yescharacteristic
241#97—531no—
242#97—531no—
243#41—22no—
244#41—22no—
245#41—2124no—
246C2×Q8—1061no—
247C2×Q8C22×D5311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4C22×D5311yescharacteristic
255C2×D4C22×D5311yescharacteristic
256C22×C4—21no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4C22×D5311yescharacteristic
264SD16—22no—
265SD16—1062no—
266D8—22no—
267D8—1062no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C42—531no—
288C42—531no—
289C42—531no—
290C42—1061no—
291C42—1061no—
292C42—1061no—
293C23—21no—
294C23—21no—
295C23—1061no—
296C23D4×D5311yescharacteristic
297Q8—21no—
298Q8—2122no—
299Q8#1171994712yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—1061no—
306D4—1061no—
307D4—2122no—
308D4—2122no—
309D4—2122no—
310D4#1171994712yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2121no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—21no—
332C8—21no—
333C8—1061no—
334C8—1061no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—532no—
352C4—1061no—
353C4—1061no—
354C4—1061no—
355C4—2121no—
356C4—2121no—
357C4—2121no—
358C4C2×#1171994711yescharacteristic
359C4C2×#1171994711yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—4241no—
364C2#1681112911yescharacteristic
365C2AGL(1,ℤ/8)×D5311yescharacteristic
366C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3115
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #16811168C2semidirect
2. #41×Dic53C2semidirect
3. #16810948C2semidirect
4. #16810944C2semidirect
5. #16810906C2semidirect
6. #16810901C2semidirect
7. #16810821C2semidirect
8. #16810816C2semidirect
9. #16810768C2semidirect
10. #16810735C2semidirect
11. #16810732C2semidirect
12. #16810726C2semidirect
13. #16810724C2semidirect
14. #16810677C2semidirect
15. C53×#449C2semidirect
114. Q8×C53#41semidirect
120. D4×C53#41semidirect
193. #449D53semidirect
199. C53#3115semidirect
299. Q8#11719947semidirect
310. D4#11719947semidirect