Finite Group Databasebeta

← #20285559

Subgroups — #20285559

Order 6784.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811136C211yesmaximal, characteristic, complemented
2#16811135C211yesmaximal, characteristic, complemented
3#16810967C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810947C211yesmaximal, characteristic, complemented
6#16810880C211yesmaximal, characteristic, complemented
7#16810877C211yesmaximal, characteristic, complemented
8#16810872C211yesmaximal, characteristic, complemented
9#16810870C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic, complemented
13#16810770C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C2×#11719925C2211yescharacteristic, complemented
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719924C2211yescharacteristic, complemented
22C2×D424C2211yescharacteristic, complemented
23C2×#11719912C2211yescharacteristic
24C2×#11719912C2211yescharacteristic
25#13133202—21no—
26#13133201—21no—
27#13133197C2211yescharacteristic
28#13133197C2211yescharacteristic
29#13133193C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133191C2211yescharacteristic
34#13133187—21no—
35#13133187—21no—
36C4×D212—21no—
37C4×D212—21no—
38#13133179—21no—
39#13133177—21no—
40#13133175C2211yescharacteristic
41#13133150C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, 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—
63C2×C22⋊D53—21no—
64C22×Dic53D411yescharacteristic
65#11719949—41no—
66#11719949—41no—
67#11719945—41no—
68#11719945—41no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53D411yescharacteristic
80C2×C4×D53D411yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×Dic106C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53—21no—
93#11719925—41no—
94#11719924—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107C4×Dic53—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111#11719913—41no—
112#11719912—41no—
113#11719912—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53C2×D411yescharacteristic
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—42no—
133C22⋊D53—42no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—41no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—42no—
144D212—42no—
145D212—42no—
146D212—42no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—42no—
154C4×D53—42no—
155Dic106—21no—
156Dic106—21no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—44no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53H(5,𝔽2)11yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—41no—
188Dic53—41no—
189#3197—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53#53411yescharacteristic
192C2×C53#57511yescharacteristic
193C2×C53#57511yescharacteristic
194D53—41no—
195D53—41no—
196D53—81no—
197D53—81no—
198#525—531no—
199#525—531no—
200#485—531no—
201#485D5311yescharacteristic, complemented
202#482—531no—
203#482—531no—
204#470—531no—
205#469—531no—
206#468—531no—
207#468—531no—
208#452—531no—
209#452—531no—
210#449—531no—
211#449—531no—
212#407—531no—
213C53#319711yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C2×#41—531no—
216C23×C4—531no—
217C2×Q16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×D8—531no—
221C2×M16—531no—
222C2×M16—531no—
223#123—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227C22⋊Q8—531no—
228C22⋊Q8C2×D5311yescharacteristic
229#121—531no—
230#121—531no—
231#121—531no—
232#121—531no—
233#120—1061no—
234#120—1061no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4—1061no—
239#117—531no—
240#117C2×D5311yescharacteristic
241#107—531no—
242#107C2×D5311yescharacteristic, complemented
243#106—531no—
244#106C2×D5311yescharacteristic, complemented
245#103—531no—
246#103—531no—
247#103C2×D5311yescharacteristic, complemented
248#103C2×D5311yescharacteristic, complemented
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8—531no—
256C22⋊C8C2×D5311yescharacteristic
257C24—531no—
258#41—2121no—
259#41—2121no—
260#41—2121no—
261#41—2121no—
262C2×Q8—531no—
263C2×Q8C22×D5311yescharacteristic
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—2121no—
280C22×C4—2121no—
281C22×C4C22×D5311yescharacteristic
282Q16—2121no—
283SD16—2121no—
284SD16—2121no—
285D8—2121no—
286M16—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8C22×D5311yescharacteristic
291C2×C8C22×D5311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—531no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4C22×D5311yescharacteristic
298C4⋊C4C22×D5311yescharacteristic
299C4⋊C4C22×D5311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C42—21no—
311C42—1061no—
312C23—531no—
313C23—531no—
314C23—1061no—
315C23—1061no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23D4×D5311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1061no—
323Q8—1061no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—1061no—
328D4—2122no—
329D4—2122no—
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—21no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4D4×D5311yescharacteristic
354C2×C4C23×D5311yescharacteristic
355C8—21no—
356C8—21no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—2124no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1061no—
382C4—1061no—
383C4—2121no—
384C4—2121no—
385C4#1313326311yescharacteristic
386C4#1313326311yescharacteristic
387C2—41no—
388C2—4241no—
389C2—4241no—
390C2—2121no—
391C2—2121no—
392C2#1681118411yescharacteristic
393C2#1681142011yescharacteristic
394C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3197
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
1. #16811136C2semidirect
2. #16811135C2semidirect
3. #16810967C2semidirect
4. #16810950C2semidirect
5. #16810947C2semidirect
6. #16810880C2semidirect
7. #16810877C2semidirect
8. #16810872C2semidirect
9. #16810870C2semidirect
10. #16810788C2semidirect
11. #16810783C2semidirect
12. #16810771C2semidirect
13. #16810770C2semidirect
14. #16810585C2semidirect
15. C53×#485C2semidirect
19. C2×#11719925C22semidirect
20. C2×#11719913C22semidirect
21. C2×#11719924C22semidirect
22. C2×D424C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
45. #13133139C22semidirect
46. #13133137C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
201. #485D53semidirect
213. C53#3197semidirect
242. #107C2×D53semidirect
244. #106C2×D53semidirect
247. #103C2×D53semidirect
248. #103C2×D53semidirect