Finite Group Databasebeta

← #20285573

Subgroups — #20285573

Order 6784.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811203C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810872C211yesmaximal, characteristic, complemented
8#16810860C211yesmaximal, characteristic, complemented
9#16810855C211yesmaximal, characteristic, complemented
10#16810773C211yesmaximal, characteristic, complemented
11#16810771C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic, complemented
13#16810759C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C2×#11719945C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×D424C2211yescharacteristic
22C2×D424C2211yescharacteristic
23C2×#11719912C2211yescharacteristic, complemented
24C2×C8×D53C2211yescharacteristic, complemented
25#13133214—21no—
26#13133205—21no—
27#13133195—21no—
28#13133193C2211yescharacteristic, complemented
29#13133192—21no—
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34#13133192C2211yescharacteristic, complemented
35#13133191C2211yescharacteristic
36#13133190—21no—
37#13133190—21no—
38#13133187—21no—
39#13133187—21no—
40C4⋊C4×D53C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
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×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53D411yescharacteristic
64C2×C22⋊D53D411yescharacteristic
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×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212D411yescharacteristic
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×Dic106D411yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C42×C53—21no—
96#11719924—41no—
97#11719924—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111D424—41no—
112#11719912—41no—
113C8×D53—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—41no—
134C22⋊D53—41no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—41no—
146D212—42no—
147D212—42no—
148D212—81no—
149D212—81no—
150D212—81no—
151D212—81no—
152D212—81no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—42no—
158C4×D53—81no—
159Dic106—41no—
160C8×C53—21no—
161C8×C53—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53H(5,𝔽2)11yescharacteristic
190C4×C53H(5,𝔽2)11yescharacteristic
191Dic53—21no—
192Dic53—21no—
193Dic53—41no—
194#3198—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53#53411yescharacteristic
197C2×C53#57611yescharacteristic
198C2×C53#57711yescharacteristic
199D53—41no—
200D53—41no—
201D53—81no—
202D53—81no—
203D53—81no—
204#537—531no—
205#535—531no—
206#485D5311yescharacteristic, complemented
207#482—531no—
208#481—531no—
209#480—531no—
210#469—531no—
211#468—531no—
212#466—531no—
213#465—531no—
214#450—531no—
215#449—531no—
216#449—531no—
217#447—531no—
218#408—531no—
219C53#319811yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41—531no—
222C22×D4—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×D8—531no—
226C2×D8—531no—
227C2×M16—531no—
228C22×C8—531no—
229#127—1061no—
230#124—1061no—
231C22⋊Q8—531no—
232C22⋊Q8—1061no—
233C22⋊Q8C2×D5311yescharacteristic
234#121—531no—
235#121—531no—
236#121—531no—
237#121—531no—
238#121—531no—
239#121—531no—
240#121—1061no—
241#121—1061no—
242#121—1061no—
243#120—1061no—
244#120—1061no—
245#117C2×D5311yescharacteristic
246C2×C4⋊C4—531no—
247#107—531no—
248#107C2×D5311yescharacteristic, complemented
249#106—531no—
250#106C2×D5311yescharacteristic, complemented
251#103C2×D5311yescharacteristic, complemented
252#103C2×D5311yescharacteristic, complemented
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259C22⋊C8—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262C22⋊C8C2×D5311yescharacteristic
263C24—1061no—
264#41—2121no—
265#41—2121no—
266#41—2121no—
267#41—2121no—
268C2×Q8—531no—
269C2×Q8C22×D5311yescharacteristic
270C2×D4—531no—
271C2×D4—531no—
272C2×D4—531no—
273C2×D4—531no—
274C2×D4—531no—
275C2×D4—531no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—2121no—
285C2×D4—2121no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4C22×D5311yescharacteristic
291SD16—2121no—
292SD16—2121no—
293D8—2121no—
294D8—2121no—
295M16—2121no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—2121no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—531no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4C22×D5311yescharacteristic
306C4⋊C4C22×D5311yescharacteristic
307C4⋊C4C22×D5311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—21no—
318C23—531no—
319C23—531no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326C23D4×D5311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—2121no—
330D4—1061no—
331D4—1061no—
332D4—1061no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—4241no—
337D4—4241no—
338D4—4241no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—2121no—
343D4—2122no—
344D4—2122no—
345D4—2121no—
346D4—2121no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—531no—
354C2×C4—531no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—4241no—
361C2×C4—2122no—
362C2×C4D4×D5311yescharacteristic
363C2×C4C23×D5311yescharacteristic
364C8—21no—
365C8—21no—
366C8—1061no—
367C8—1061no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—1061no—
372C22—1061no—
373C22—1061no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—4244no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C2×D4×D5311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1061no—
394C4—1061no—
395C4—2121no—
396C4#1313326311yescharacteristic
397C4#1313326311yescharacteristic
398C2—41no—
399C2—4241no—
400C2—4241no—
401C2—4241no—
402C2—2121no—
403C2—2121no—
404C2#1681118411yescharacteristic
405C2#1681142511yescharacteristic
406C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3198
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. #16811203C2semidirect
2. #16811192C2semidirect
3. #16810947C2semidirect
4. #16810943C2semidirect
5. #16810938C2semidirect
6. #16810877C2semidirect
7. #16810872C2semidirect
8. #16810860C2semidirect
9. #16810855C2semidirect
10. #16810773C2semidirect
11. #16810771C2semidirect
12. #16810771C2semidirect
13. #16810759C2semidirect
14. #16810588C2semidirect
15. C53×#485C2semidirect
16. C2×#11719949C22semidirect
23. C2×#11719912C22semidirect
24. C2×C8×D53C22semidirect
28. #13133193C22semidirect
34. #13133192C22semidirect
40. C4⋊C4×D53C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
206. #485D53semidirect
219. C53#3198semidirect
248. #107C2×D53semidirect
250. #106C2×D53semidirect
251. #103C2×D53semidirect
252. #103C2×D53semidirect