Finite Group Databasebeta

← #20285565

Subgroups — #20285565

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#16811202C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic, complemented
3#16810964C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810875C211yesmaximal, characteristic, complemented
7#16810870C211yesmaximal, characteristic, complemented
8#16810858C211yesmaximal, characteristic, complemented
9#16810853C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810769C211yesmaximal, characteristic, complemented
12#16810769C211yesmaximal, characteristic, complemented
13#16810756C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719912C2211yescharacteristic, complemented
24C2×C8×D53C2211yescharacteristic, complemented
25#13133214—21no—
26#13133205—21no—
27#13133197C2211yescharacteristic, complemented
28#13133196C2211yescharacteristic
29#13133194—21no—
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34#13133190—21no—
35#13133189—21no—
36#13133188—21no—
37#13133188C2211yescharacteristic, complemented
38#13133187—21no—
39#13133187—21no—
40C4⋊C4×D53C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C2×C22⋊D53D411yescharacteristic
67C22×Dic53—21no—
68#11719947—41no—
69#11719947—41no—
70#11719945—41no—
71#11719945—41no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212D411yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53C2311yescharacteristic
84C2×C4×D53C2311yescharacteristic
85C2×Dic106D411yescharacteristic
86C2×Dic106C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C42×C53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719922—41no—
98#11719922—41no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719913—41no—
111#11719913—41no—
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122C22×D53C2×D411yescharacteristic
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—81no—
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—42no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—41no—
147D212—42no—
148D212—81no—
149D212—81no—
150D212—81no—
151D212—81no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—81no—
157Dic106—21no—
158Dic106—21no—
159Dic106—41no—
160C8×C53—21no—
161C8×C53—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
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×C53H(5,𝔽2)11yescharacteristic
189C4×C53H(5,𝔽2)11yescharacteristic
190Dic53—21no—
191Dic53—21no—
192Dic53—41no—
193Dic53—41no—
194#3198—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—81no—
197C2×C53#53411yescharacteristic
198C2×C53#57611yescharacteristic
199C2×C53#57711yescharacteristic
200D53—41no—
201D53—41no—
202D53—81no—
203D53—81no—
204#537—531no—
205#535—531no—
206#485—531no—
207#482D5311yescharacteristic, complemented
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—531no—
233C22⋊Q8—1061no—
234#121—531no—
235#121—531no—
236#121—531no—
237#121—531no—
238#121—531no—
239#121—1061no—
240#121—1061no—
241#121—1061no—
242#121C2×D5311yescharacteristic
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#103—531no—
252#103—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102C2×D5311yescharacteristic, complemented
258#102C2×D5311yescharacteristic, complemented
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×Q8—531no—
270C2×D4—21no—
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—2121no—
284C2×D4—2121no—
285C2×D4C22×D5311yescharacteristic
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—531no—
302C4⋊C4—1061no—
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—21no—
319C23—531no—
320C23—531no—
321C23—1061no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326C23D4×D5311yescharacteristic
327Q8—1061no—
328Q8—1061no—
329Q8—2121no—
330D4—21no—
331D4—21no—
332D4—81no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—4241no—
338D4—4241no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—2121no—
343D4—2122no—
344D4—2121no—
345D4—2121no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—1061no—
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—41no—
372C22—41no—
373C22—41no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—1061no—
380C22—4244no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C2×D4×D5311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—1061no—
393C4—1061no—
394C4—2121no—
395C4—2121no—
396C4#1313326311yescharacteristic
397C4#1313326311yescharacteristic
398C2—41no—
399C2—81no—
400C2—4241no—
401C2—4241no—
402C2—2121no—
403C2—2121no—
404C2#1681118411yescharacteristic
405C2#1681142711yescharacteristic
406C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3198
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811202C2semidirect
2. #16811192C2semidirect
3. #16810964C2semidirect
4. #16810943C2semidirect
5. #16810938C2semidirect
6. #16810875C2semidirect
7. #16810870C2semidirect
8. #16810858C2semidirect
9. #16810853C2semidirect
10. #16810776C2semidirect
11. #16810769C2semidirect
12. #16810769C2semidirect
13. #16810756C2semidirect
14. #16810588C2semidirect
15. C53×#482C2semidirect
17. C2×#11719947C22semidirect
23. C2×#11719912C22semidirect
24. C2×C8×D53C22semidirect
27. #13133197C22semidirect
37. #13133188C22semidirect
40. C4⋊C4×D53C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
43. #13133140C22semidirect
44. #13133140C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
207. #482D53semidirect
219. C53#3198semidirect
248. #107C2×D53semidirect
250. #106C2×D53semidirect
257. #102C2×D53semidirect
258. #102C2×D53semidirect