Finite Group Databasebeta

← #20285625

Subgroups — #20285625

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811204C211yesmaximal, characteristic, complemented
2#16811203C211yesmaximal, characteristic, complemented
3#16810955C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810941C211yesmaximal, characteristic, complemented
6#16810880C211yesmaximal, characteristic, complemented
7#16810877C211yesmaximal, characteristic, complemented
8#16810855C211yesmaximal, characteristic, complemented
9#16810853C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810783C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic, complemented
13#16810770C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#483C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719945C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×D424C2211yescharacteristic
23C2×#11719912C2211yescharacteristic
24C2×C8×D53C2211yescharacteristic
25#13133197—21no—
26#13133197C2211yescharacteristic
27#13133197C2211yescharacteristic
28#13133195—21no—
29#13133193C2211yescharacteristic
30#13133192—21no—
31#13133192—21no—
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34#13133192C2211yescharacteristic
35#13133191—21no—
36#13133191—21no—
37#13133191C2211yescharacteristic
38#13133190—21no—
39#13133190—21no—
40C4⋊C4×D53C2211yescharacteristic
41#13133148C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#106×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53D411yescharacteristic
65C2×C22⋊D53D411yescharacteristic
66#11719949—41no—
67#11719949—41no—
68#11719945—41no—
69#11719945—41no—
70#11719945—41no—
71#11719945—41no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212D411yescharacteristic
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×C4×D53C2311yescharacteristic
84C2×Dic106D411yescharacteristic
85C2×Dic106C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94#11719925—41no—
95#11719924—41no—
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—
107#11719919—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111#11719913—41no—
112#11719912—41no—
113C8×D53—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53C2×D411yescharacteristic
118C22×D53C2×D411yescharacteristic
119Q8×C53—21no—
120Q8×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—41no—
131C22⋊D53—41no—
132C22⋊D53—42no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—41no—
143D212—42no—
144D212—81no—
145D212—81no—
146D212—81no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—42no—
152C4×D53—42no—
153C4×D53—81no—
154Dic106—21no—
155Dic106—21no—
156Dic106—41no—
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—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53H(5,𝔽2)11yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188#3204—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53#53411yescharacteristic
191C2×C53#57711yescharacteristic
192C2×C53#57711yescharacteristic
193D53—41no—
194D53—41no—
195D53—81no—
196D53—81no—
197#537—531no—
198#537—531no—
199#483—531no—
200#483D5311yescharacteristic, complemented
201#481—531no—
202#481—531no—
203#470—531no—
204#469—531no—
205#465—531no—
206#465—531no—
207#452—531no—
208#452—531no—
209#449—531no—
210#449—531no—
211#408—531no—
212C53#320411yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41—531no—
215C2×#41—531no—
216C2×Q16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×D8—531no—
220C2×M16—531no—
221C22×C8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8C2×D5311yescharacteristic
228#121—531no—
229#121—531no—
230#121—531no—
231#121—531no—
232#121—1061no—
233#121—1061no—
234#121—1061no—
235#121—1061no—
236#121—1061no—
237#121—1061no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4C2×D5311yescharacteristic
240#106—531no—
241#106—531no—
242#106C2×D5311yescharacteristic
243#106C2×D5311yescharacteristic
244#103—531no—
245#103—531no—
246#103C2×D5311yescharacteristic
247#103C2×D5311yescharacteristic
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8C2×D5311yescharacteristic
256#41—2121no—
257#41—2121no—
258#41—2121no—
259#41—2121no—
260#41—2121no—
261#41—2121no—
262C2×Q8—531no—
263C2×Q8—531no—
264C2×Q8C22×D5311yescharacteristic
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—531no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C22×C4—21no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4C22×D5311yescharacteristic
283Q16—2121no—
284SD16—2121no—
285SD16—2121no—
286D8—2121no—
287M16—2121no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—2121no—
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—531no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4C22×D5311yescharacteristic
301C4⋊C4C22×D5311yescharacteristic
302C4⋊C4C22×D5311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C23—531no—
312C23—531no—
313C23—1061no—
314C23—1061no—
315C23D4×D5311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—1061no—
319Q8—1061no—
320Q8—2121no—
321D4—1061no—
322D4—1061no—
323D4—1061no—
324D4—1061no—
325D4—4241no—
326D4—4241no—
327D4—4241no—
328D4—4241no—
329D4—2121no—
330D4—2122no—
331D4—2121no—
332D4—2121no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—81no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4D4×D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—21no—
355C8—21no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—1061no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1061no—
380C4—1061no—
381C4—2121no—
382C4—2121no—
383C4#1313326311yescharacteristic
384C4#1313326311yescharacteristic
385C2—41no—
386C2—4241no—
387C2—4241no—
388C2—2121no—
389C2—2121no—
390C2#1681118411yescharacteristic
391C2#1681142911yescharacteristic
392C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#483C2
socleC22×C53C22×D4
hypercenter#483D53
hypocenterC53#3204
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #16811204C2semidirect
2. #16811203C2semidirect
3. #16810955C2semidirect
4. #16810943C2semidirect
5. #16810941C2semidirect
6. #16810880C2semidirect
7. #16810877C2semidirect
8. #16810855C2semidirect
9. #16810853C2semidirect
10. #16810788C2semidirect
11. #16810783C2semidirect
12. #16810771C2semidirect
13. #16810770C2semidirect
14. #16810588C2semidirect
15. C53×#483C2semidirect
200. #483D53semidirect
212. C53#3204semidirect