Finite Group Databasebeta

← #20284633

Subgroups — #20284633

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811136C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810957C211yesmaximal, characteristic, complemented
5#16810952C211yesmaximal, characteristic, complemented
6#16810948C211yesmaximal, characteristic, complemented
7#16810947C211yesmaximal, characteristic, complemented
8#16810943C211yesmaximal, characteristic, complemented
9#16810938C211yesmaximal, characteristic, complemented
10#16810660C211yesmaximal, characteristic, complemented
11#16810658C211yesmaximal, characteristic, complemented
12#16810618C211yesmaximal, characteristic, complemented
13#16810618C211yesmaximal, characteristic, complemented
14C22⋊C8×D53C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×C8×D53C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133202—21no—
21#13133193C2211yescharacteristic
22#13133192C2211yescharacteristic
23#13133191C2211yescharacteristic
24#13133187—21no—
25#13133179—21no—
26#13133177—21no—
27#13133176C2211yescharacteristic
28#13133175C2211yescharacteristic
29#13133175C2211yescharacteristic
30#13133175C2211yescharacteristic
31#13133174—21no—
32#13133173—21no—
33C4⋊C4×D53C2211yescharacteristic
34C4⋊C4×D53C2211yescharacteristic
35C22⋊C4×D53—21no—
36C42×D53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53D411yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C22×Dic53D411yescharacteristic
64#11719949—22no—
65#11719949—22no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53D411yescharacteristic
75C2×C4×D53D411yescharacteristic
76C2×C4×D53C2311yescharacteristic
77C2×C4×D53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C8×D53—22no—
107C8×D53—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—21no—
112C22×D53—41no—
113C22×D53C2×D411yescharacteristic
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C2×Dic53—22no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—21no—
133D212—21no—
134D212—42no—
135D212—42no—
136C4×D53—22no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—22no—
140C4×D53—42no—
141C4×D53—42no—
142C4×D53—42no—
143C4×D53#4112yes—
144C4×D53#4112yes—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—44no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53#12312yes—
162C2×D53#13512yes—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53C2×#4111yescharacteristic
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174Dic53#12312yes—
175#3091—531nomaximal, Sylow
176C2×C53—41no—
177C2×C53C2×#12311yescharacteristic
178C2×C53C2×#13511yescharacteristic
179C2×C53#57511yescharacteristic
180D53—22no—
181D53—22no—
182D53—81no—
183#525—531no—
184C2×#117—531no—
185#485—531no—
186#485D5311yescharacteristic, complemented
187#484—531no—
188#483—531no—
189#482—531no—
190#482—531no—
191#481—531no—
192#480—531no—
193#427—531no—
194#427—531no—
195#416—531no—
196#416—531no—
197C2×C22⋊C8—531no—
198C53#309111yescharacteristic, complemented, Sylow
199C2×#41—531no—
200C23×C4—531no—
201C22×C8—531no—
202C22×C8—531no—
203#123—1061no—
204C22⋊Q8—531no—
205C22⋊Q8C2×D5311yescharacteristic
206#121—531no—
207#121—531no—
208#120—1061no—
209C4×D4—1061no—
210C4×D4—1061no—
211#117—531no—
212#117—531no—
213#117—531no—
214#117—531no—
215#117—1061no—
216#117—1061no—
217#117C2×D5311yescharacteristic
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C22⋊C4—1061no—
221C2×C42—1061no—
222#107—531no—
223#107—531no—
224#107—531no—
225#107C2×D5311yescharacteristic, complemented
226#106—531no—
227#106—531no—
228#106—531no—
229#106C2×D5311yescharacteristic, complemented
230#103—531no—
231#103—531no—
232#103C2×D5311yescharacteristic, complemented
233#103C2×D5311yescharacteristic, complemented
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238C22⋊C8—531no—
239C22⋊C8—531no—
240C22⋊C8—531no—
241C22⋊C8C2×D5311yescharacteristic
242C24—531no—
243#41—1062no—
244#41—1062no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4C22×D5311yescharacteristic
261C2×C8—531no—
262C2×C8—531no—
263C2×C8—1062no—
264C2×C8—1062no—
265C2×C8C22×D5311yescharacteristic
266C2×C8C22×D5311yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4C22×D5311yescharacteristic
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—21no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C42—1061no—
291C23—531no—
292C23—531no—
293C23—1062no—
294C23—1062no—
295C23—1061no—
296C23—2121no—
297C23D4×D5311yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—1061no—
301D4—1061no—
302D4—2122no—
303D4—2122no—
304D4—2122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—531no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—531no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—532no—
336C22—532no—
337C22—1061no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22—2124no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22C2×D4×D5311yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—532no—
352C4—1061no—
353C4—2121no—
354C4—2121no—
355C4—2121no—
356C4#41×D5311yescharacteristic
357C4#41×D5311yescharacteristic
358C2—41no—
359C2—1062no—
360C2—1062no—
361C2—4241no—
362C2#123×D5311yescharacteristic
363C2#135×D5311yescharacteristic
364C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3091
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #117×D53C2semidirect
3. #16810961C2semidirect
4. #16810957C2semidirect
5. #16810952C2semidirect
6. #16810948C2semidirect
7. #16810947C2semidirect
8. #16810943C2semidirect
9. #16810938C2semidirect
10. #16810660C2semidirect
11. #16810658C2semidirect
12. #16810618C2semidirect
13. #16810618C2semidirect
14. C22⋊C8×D53C2semidirect
15. C53×#485C2semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. #13133141C22semidirect
44. #13133141C22semidirect
45. #13133139C22semidirect
46. #13133139C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
186. #485D53semidirect
198. C53#3091semidirect
225. #107C2×D53semidirect
229. #106C2×D53semidirect
232. #103C2×D53semidirect
233. #103C2×D53semidirect