Finite Group Databasebeta

← #20285586

Subgroups — #20285586

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811196C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic
3#16810964C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810949C211yesmaximal, characteristic, complemented
6#16810872C211yesmaximal, characteristic, complemented
7#16810870C211yesmaximal, characteristic, complemented
8#16810855C211yesmaximal, characteristic, complemented
9#16810853C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810773C211yesmaximal, characteristic, complemented
12#16810763C211yesmaximal, characteristic, complemented
13#16810763C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×#11719912C2211yescharacteristic, complemented
24C2×C8×D53C2211yescharacteristic, complemented
25#13133216—21no—
26#13133214—21no—
27#13133205—21no—
28#13133203—21no—
29#13133202—21no—
30#13133199—21no—
31#13133197C2211yescharacteristic, complemented
32#13133196C2211yescharacteristic
33#13133193C2211yescharacteristic, complemented
34#13133192C2211yescharacteristic
35#13133192C2211yescharacteristic
36#13133192C2211yescharacteristic
37#13133192C2211yescharacteristic
38#13133187—21no—
39#13133187—21no—
40#13133175C2211yescharacteristic, 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
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⋊D53D411yescharacteristic
63C2×C22⋊D53D411yescharacteristic
64Q8×D53—41no—
65Q8×D53—41no—
66#11719945—41no—
67#11719945—41no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—41no—
74C2×D212—41no—
75C2×D212D411yescharacteristic
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106D411yescharacteristic
83C2×Dic106C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53—21no—
93C22⋊Dic53—21no—
94#11719924—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⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719913—41no—
111#11719913—41no—
112#11719912—41no—
113C8×D53—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53C2×D411yescharacteristic
121C22×D53C2×D411yescharacteristic
122Q8×C53—21no—
123Q8×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—41no—
133C22⋊D53—41no—
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—
147D212—81no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—42no—
153Dic106—21no—
154Dic106—21no—
155Dic106—41no—
156Dic106—42no—
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—
177C2×D53—84no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53H(5,𝔽2)11yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—41no—
188Dic53—41no—
189#3200—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53#53411yescharacteristic
192C2×C53#57711yescharacteristic
193C2×C53#57711yescharacteristic
194D53—41no—
195D53—41no—
196D53—81no—
197D53—81no—
198#536—531no—
199#535—531no—
200#485—531no—
201#485D5311yescharacteristic, complemented
202#482—531no—
203#482—531no—
204#468—531no—
205#468—531no—
206#465—531no—
207#465—531no—
208#450—531no—
209#450—531no—
210#448—531no—
211#448—531no—
212#408—531no—
213C53#320011yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C22×Q8—531no—
216C22×D4—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×SD16—531no—
221C2×M16—531no—
222C22×C8—531no—
223C4⋊Q8—1061no—
224#127—1061no—
225#124—1061no—
226#124—1061no—
227#123—1061no—
228#123—1061no—
229C22⋊Q8—531no—
230C22⋊Q8—531no—
231C22⋊Q8—531no—
232C22⋊Q8C2×D5311yescharacteristic
233#121—531no—
234#121—531no—
235#121—531no—
236#121—531no—
237#120—1061no—
238#120—1061no—
239#117—531no—
240#117C2×D5311yescharacteristic
241#107—531no—
242#107C2×D5311yescharacteristic, complemented
243#106—531no—
244#106C2×D5311yescharacteristic, complemented
245#103—531no—
246#103—531no—
247#103C2×D5311yescharacteristic, complemented
248#103C2×D5311yescharacteristic, complemented
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253C22⋊C8—531no—
254C22⋊C8—531no—
255C22⋊C8—531no—
256C22⋊C8C2×D5311yescharacteristic
257C24—1061no—
258#41—2121no—
259#41—2121no—
260C2×Q8—531no—
261C2×Q8—531no—
262C2×Q8—1061no—
263C2×Q8—2121no—
264C2×Q8—2121no—
265C2×Q8C22×D5311yescharacteristic
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4—531no—
270C2×D4—531no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—2121no—
277C2×D4—2121no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—1061no—
281C22×C4C22×D5311yescharacteristic
282SD16—2121no—
283SD16—2121no—
284SD16—2121no—
285SD16—2121no—
286M16—2121no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—2121no—
290C2×C8C22×D5311yescharacteristic
291C2×C8C22×D5311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—531no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4C22×D5311yescharacteristic
298C4⋊C4C22×D5311yescharacteristic
299C4⋊C4C22×D5311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C42—21no—
311C42—1061no—
312C23—531no—
313C23—531no—
314C23—1061no—
315C23—1061no—
316C23—4241no—
317C23—4241no—
318C23—2122no—
319C23D4×D5311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1061no—
323Q8—1061no—
324Q8—2121no—
325Q8—2122no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—1061no—
330D4—4241no—
331D4—4241no—
332D4—4241no—
333D4—4241no—
334D4—2121no—
335D4—2122no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—2122no—
353C2×C4D4×D5311yescharacteristic
354C2×C4C23×D5311yescharacteristic
355C8—21no—
356C8—21no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—4244no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1061no—
382C4—1061no—
383C4—2121no—
384C4—2121no—
385C4#1313326311yescharacteristic
386C4#1313326311yescharacteristic
387C2—41no—
388C2—4241no—
389C2—4241no—
390C2—2121no—
391C2—2121no—
392C2#1681118411yescharacteristic
393C2#1681143011yescharacteristic
394C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3200
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. #16811196C2semidirect
3. #16810964C2semidirect
4. #16810950C2semidirect
5. #16810949C2semidirect
6. #16810872C2semidirect
7. #16810870C2semidirect
8. #16810855C2semidirect
9. #16810853C2semidirect
10. #16810776C2semidirect
11. #16810773C2semidirect
12. #16810763C2semidirect
13. #16810763C2semidirect
14. #16810588C2semidirect
15. C53×#485C2semidirect
17. C2×Q8×D53C22semidirect
23. C2×#11719912C22semidirect
24. C2×C8×D53C22semidirect
31. #13133197C22semidirect
33. #13133193C22semidirect
40. #13133175C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
43. #13133140C22semidirect
44. #13133140C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
201. #485D53semidirect
213. C53#3200semidirect
242. #107C2×D53semidirect
244. #106C2×D53semidirect
247. #103C2×D53semidirect
248. #103C2×D53semidirect