Finite Group Databasebeta

← #20285803

Subgroups — #20285803

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#16811192C211yesmaximal, characteristic, complemented
2#16811013C211yesmaximal, characteristic, complemented
3#16811005C211yesmaximal, characteristic, complemented
4#16810985C211yesmaximal, characteristic, complemented
5#16810971C211yesmaximal, characteristic, complemented
6#16810845C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810823C211yesmaximal, characteristic, complemented
9#16810818C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810773C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810756C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#542C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719923C2211yescharacteristic, complemented
20C2×#11719924C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×#11719924C2211yescharacteristic, complemented
23C2×#11719922C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25C2×#11719922C2211yescharacteristic, complemented
26#13133214C2211yescharacteristic
27#13133214C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133205C2211yescharacteristic
30#13133192—21no—
31#13133192—21no—
32#13133187—21no—
33#13133187—21no—
34C4⋊#88322C2211yescharacteristic, complemented
35C4⋊#88322C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133137C2211yescharacteristic
39#13133137C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic, complemented
43#13133137C2211yescharacteristic, complemented
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48C4⋊Q8×C53C2211yescharacteristic, complemented
49#126×C53—21no—
50#124×C53—21no—
51#124×C53C2211yescharacteristic, complemented
52#123×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic, complemented
57C4×D4×C53C2211yescharacteristic, complemented
58#117×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67#11719945—41no—
68#11719945—41no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×Dic106C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53D411yescharacteristic
85C4⋊C4×C53D411yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53D411yescharacteristic
94C22⋊C4×C53D411yescharacteristic
95C42×C53—21no—
96C42×C53C2311yescharacteristic
97C42×C53C2311yescharacteristic
98#11719925—41no—
99#11719924—41no—
100#11719924—41no—
101#11719924—41no—
102#11719923—41no—
103#11719922—41no—
104#11719922—41no—
105#11719922—41no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122Q8×C53—21no—
123Q8×C53—21no—
124Q8×C53—81no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—81no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—81no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—42no—
141C2×Dic53—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212—81no—
150D212—81no—
151D212—81no—
152D212—81no—
153C4×D53—42no—
154Dic106—21no—
155Dic106—21no—
156#88322—21no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53C22×D411yescharacteristic
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—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53H(5,𝔽2)11yescharacteristic
186C4×C53H(5,𝔽2)11yescharacteristic
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—81no—
191C2×C53#53511yescharacteristic
192C2×C53#57611yescharacteristic
193C2×C53#57711yescharacteristic
194D53—81no—
195D53—81no—
196D53—81no—
197#542D5311yescharacteristic, complemented
198#535—531no—
199#496—531no—
200#495—531no—
201#489—531no—
202#486—531no—
203#464—531no—
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×Q16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×D8—531no—
221C2×D8—531no—
222C2×D8—531no—
223C4⋊Q8C2×D5311yescharacteristic, complemented
224#127—531no—
225#127—531no—
226#126—21no—
227#124—21no—
228#124—531no—
229#124—531no—
230#124C2×D5311yescharacteristic, complemented
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×D5311yescharacteristic
234#121—1061no—
235#121—1061no—
236#121C2×D5311yescharacteristic
237#120—1061no—
238#120—1061no—
239C4×Q8C2×D5311yescharacteristic, complemented
240C4×D4C2×D5311yescharacteristic, complemented
241#117C2×D5311yescharacteristic
242C4⋊C8—531no—
243C4⋊C8—531no—
244#103—531no—
245#103—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C4×C8—531no—
256C24—1061no—
257#41—2121no—
258#41—2121no—
259C2×Q8—21no—
260C2×Q8—531no—
261C2×Q8C22×D5311yescharacteristic
262C2×D4—21no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
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—21no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278Q16—2121no—
279SD16—2121no—
280SD16—2121no—
281SD16—2121no—
282SD16—2121no—
283D8—2121no—
284D8—2121no—
285D8—2121no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—531no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D5311yescharacteristic
294C4⋊C4C22⋊D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4C22⋊D5311yescharacteristic
307C22⋊C4C22⋊D5311yescharacteristic
308C42—21no—
309C42C22×D5311yescharacteristic
310C42C22×D5311yescharacteristic
311C23—21no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—4241no—
316C23—4241no—
317C23—2122no—
318C23C2×C22⋊D5311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—1061no—
323Q8—1061no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—1061no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—1061no—
333D4—4241no—
334D4—4241no—
335D4—4241no—
336D4—4241no—
337D4—2122no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C2×C22⋊D5311yescharacteristic
349C2×C4C2×C22⋊D5311yescharacteristic
350C2×C4C2×C22⋊D5311yescharacteristic
351C2×C4C2×C22⋊D5311yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—1061no—
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—4244no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C22×C22⋊D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2121no—
383C4#1313326311yescharacteristic
384C4#1313326311yescharacteristic
385C2—41no—
386C2—81no—
387C2—4241no—
388C2—4241no—
389C2—4241no—
390C2#1681118711yescharacteristic
391C2#1681142611yescharacteristic
392C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#542C2
socleC22×C53C22×D4
hypercenter#542D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16811192C2semidirect
2. #16811013C2semidirect
3. #16811005C2semidirect
4. #16810985C2semidirect
5. #16810971C2semidirect
6. #16810845C2semidirect
7. #16810838C2semidirect
8. #16810823C2semidirect
9. #16810818C2semidirect
10. #16810783C2semidirect
11. #16810773C2semidirect
12. #16810766C2semidirect
13. #16810756C2semidirect
14. #16810684C2semidirect
15. C53×#542C2semidirect
18. C2×#11719925C22semidirect
19. C2×#11719923C22semidirect
22. C2×#11719924C22semidirect
25. C2×#11719922C22semidirect
34. C4⋊#88322C22semidirect
35. C4⋊#88322C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
42. #13133137C22semidirect
43. #13133137C22semidirect
46. #13133128C22semidirect
47. C4×#88322C22semidirect
48. C4⋊Q8×C53C22semidirect
51. #124×C53C22semidirect
56. C4×Q8×C53C22semidirect
57. C4×D4×C53C22semidirect
197. #542D53semidirect
212. C53#3224semidirect
223. C4⋊Q8C2×D53semidirect
230. #124C2×D53semidirect
239. C4×Q8C2×D53semidirect
240. C4×D4C2×D53semidirect