Finite Group Databasebeta

← #20285816

Subgroups — #20285816

Order 6784.

408 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#16811245C211yesmaximal, characteristic, complemented
2#16811191C211yesmaximal, characteristic, complemented
3#16811009C211yesmaximal, characteristic, complemented
4#16811008C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810968C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810823C211yesmaximal, characteristic, complemented
9#16810819C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810771C211yesmaximal, characteristic, complemented
13#16810758C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719924C2211yescharacteristic, complemented
20C2×#11719924C2211yescharacteristic, complemented
21C2×D424C2211yescharacteristic
22C2×D424C2211yescharacteristic, complemented
23C2×#11719922C2211yescharacteristic
24#13133216C2211yescharacteristic, complemented
25#13133214C2211yescharacteristic
26#13133213C2211yescharacteristic
27#13133211—21no—
28#13133206C2211yescharacteristic
29#13133206C2211yescharacteristic, complemented
30#13133205—21no—
31#13133202—21no—
32#13133195—21no—
33#13133193C2211yescharacteristic
34#13133192C2211yescharacteristic
35#13133190—21no—
36#13133190—21no—
37#13133187—21no—
38#13133187—21no—
39C4×D212C2211yescharacteristic, complemented
40#13133174C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×Q16×C53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66#11719949—41no—
67#11719949—41no—
68D4×D53—41no—
69D4×D53—41no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×Dic106—21no—
81Q16×C53—41no—
82SD16×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91#11719924—41no—
92#11719924—41no—
93#11719924—41no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921D411yescharacteristic
103#11719921D411yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108#11719919D411yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112D424—41no—
113D424—41no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—81no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—42no—
148D212—81no—
149D212—81no—
150D212—81no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—42no—
154C4×D53—81no—
155Dic106—81no—
156C8×C53—21no—
157C8×C53—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—84no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53H(5,𝔽2)11yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—81no—
190C2×C53#53511yescharacteristic
191C2×C53#57611yescharacteristic
192C2×C53#57711yescharacteristic
193D53—41no—
194D53—81no—
195D53—81no—
196D53—81no—
197#542—531no—
198#535—531no—
199#496—531no—
200#495—531no—
201#489—531no—
202#486—531no—
203#464D5311yescharacteristic, complemented
204#463—531no—
205#460—531no—
206#459—531no—
207#452—531no—
208#450—531no—
209#449—531no—
210#447—531no—
211#433—531no—
212C53#322411yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C22×D4—531no—
215C2×Q16C2×D5311yescharacteristic, complemented
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16C2×D5311yescharacteristic
220C2×D8—531no—
221C2×D8—531no—
222C2×D8—531no—
223C4⋊Q8—531no—
224#127—531no—
225#127—531no—
226#126—1061no—
227#124—531no—
228#124—531no—
229#124—1061no—
230#124C2×D5311yescharacteristic
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234#121—531no—
235#121—1061no—
236#121—1061no—
237#120—1061no—
238#120—1061no—
239C4×Q8C2×D5311yescharacteristic, complemented
240C4×D4—531no—
241#117—531no—
242C4⋊C8—531no—
243C4⋊C8C2×D5311yescharacteristic, complemented
244#103—531no—
245#103C2×D5311yescharacteristic, complemented
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C4×C8—531no—
256C24—1061no—
257#41—2121no—
258#41—2121no—
259C2×Q8—1061no—
260C2×Q8C22×D5311yescharacteristic
261C2×Q8C22×D5311yescharacteristic
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4C22×D5311yescharacteristic
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278Q16—41no—
279SD16—41no—
280SD16—2121no—
281SD16—2121no—
282SD16—2121no—
283D8—2121no—
284D8—2121no—
285D8—2121no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8C22×D5311yescharacteristic
289C2×C8C22×D5311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—531no—
292C4⋊C4—531no—
293C4⋊C4—531no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—21no—
309C42—531no—
310C42C22×D5311yescharacteristic
311C23—21no—
312C23—531no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—4241no—
317C23—4241no—
318C23—2122no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—4241no—
324D4—21no—
325D4—21no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—4241no—
333D4—4241no—
334D4—4241no—
335D4—4241no—
336D4—4241no—
337D4—2122no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—4241no—
349C2×C4—2122no—
350C2×C4D4×D5311yescharacteristic
351C2×C4D4×D5311yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1061no—
356C8—1061no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—4244no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C2×D4×D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—2121no—
383C4#1313326211yescharacteristic
384C4#1313326311yescharacteristic
385C2—81no—
386C2—4241no—
387C2—4241no—
388C2—4241no—
389C2—2121no—
390C2#1681118911yescharacteristic
391C2#1681142511yescharacteristic
392C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #16811245C2semidirect
2. #16811191C2semidirect
3. #16811009C2semidirect
4. #16811008C2semidirect
5. #16810984C2semidirect
6. #16810968C2semidirect
7. #16810835C2semidirect
8. #16810823C2semidirect
9. #16810819C2semidirect
10. #16810784C2semidirect
11. #16810772C2semidirect
12. #16810771C2semidirect
13. #16810758C2semidirect
14. #16810683C2semidirect
15. C53×#464C2semidirect
19. C2×#11719924C22semidirect
20. C2×#11719924C22semidirect
22. C2×D424C22semidirect
24. #13133216C22semidirect
29. #13133206C22semidirect
39. C4×D212C22semidirect
41. C4⋊#88322C22semidirect
42. #13133141C22semidirect
44. #13133139C22semidirect
47. #13133137C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
52. C2×Q16×C53C22semidirect
55. C4×Q8×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
203. #464D53semidirect
212. C53#3224semidirect
215. C2×Q16C2×D53semidirect
239. C4×Q8C2×D53semidirect
243. C4⋊C8C2×D53semidirect
245. #103C2×D53semidirect