Finite Group Databasebeta

← #20283761

Subgroups — #20283761

Order 6784.

412 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#16811135C211yesmaximal, characteristic
2#16810702C211yesmaximal, characteristic, complemented
3#16810700C211yesmaximal, characteristic, complemented
4#16810696C211yesmaximal, characteristic, complemented
5#16810696C211yesmaximal, characteristic, complemented
6#16810693C211yesmaximal, characteristic, complemented
7#16810693C211yesmaximal, characteristic, complemented
8C8×D212C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic, complemented
10#16810684C211yesmaximal, characteristic, complemented
11#16810588C211yesmaximal, characteristic, complemented
12#16810588C211yesmaximal, characteristic, complemented
13#16810585C211yesmaximal, characteristic, complemented
14C22⋊C8×D53C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21C2×#11719912C2211yescharacteristic, complemented
22C2×C8×D53C2211yescharacteristic
23C2×C8×D53C2211yescharacteristic
24C2×C8×D53C2211yescharacteristic, complemented
25C2×C8×D53C2211yescharacteristic, complemented
26#13133201C411yescharacteristic
27#13133201C411yescharacteristic
28#13133197C411yescharacteristic
29#13133197C411yescharacteristic
30#13133192C411yescharacteristic
31#13133192C411yescharacteristic
32#13133187C411yescharacteristic
33#13133187C411yescharacteristic
34C4×D212C2211yescharacteristic
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37C4×D212C2211yescharacteristic
38C4⋊#88322C2211yescharacteristic, complemented
39C4⋊#88322C2211yescharacteristic, complemented
40C22⋊#88322C2211yescharacteristic
41C22⋊#88322C2211yescharacteristic
42#13133129C2211yescharacteristic
43#13133129C2211yescharacteristic
44#13133129C2211yescharacteristic
45#13133129C2211yescharacteristic
46#13133129C2211yescharacteristic, complemented
47#13133129C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic, complemented
50#13133128C2211yescharacteristic, complemented
51C4×#88322C2211yescharacteristic, complemented
52#117×C53C2211yescharacteristic
53C4⋊C8×C53C2211yescharacteristic, complemented
54C4⋊C8×C53C2211yescharacteristic, complemented
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic, complemented
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53C2×C411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53C2×C411yescharacteristic
62C2×C22⋊D53C2×C411yescharacteristic
63C22×Dic53C2×C411yescharacteristic
64#11719945—41no—
65#11719945—41no—
66C2×D212C2×C411yescharacteristic
67C2×D212C2×C411yescharacteristic
68C2×D212C2×C411yescharacteristic
69C2×D212C2×C411yescharacteristic
70C2×D212C2×C411yescharacteristic
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53C2×C411yescharacteristic
74C2×C4×D53C2×C411yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×C4×D53C2311yescharacteristic
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×Dic106C2×C411yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2×C411yescharacteristic
85C4⋊C4×C53C2×C411yescharacteristic
86C22⋊C4×C53C2×C411yescharacteristic
87C22⋊C4×C53C2×C411yescharacteristic
88C42×C53C2311yescharacteristic
89C42×C53C2311yescharacteristic
90#11719921C2×C411yescharacteristic
91#11719921C2×C411yescharacteristic
92#11719921C2×C411yescharacteristic
93#11719921C2×C411yescharacteristic
94#11719921C2×C411yescharacteristic
95#11719921C2×C411yescharacteristic
96#11719921C2×C411yescharacteristic
97#11719921C2×C411yescharacteristic
98C4⋊Dic53C2×C411yescharacteristic
99C4⋊Dic53C2×C411yescharacteristic
100C4⋊Dic53C2×C411yescharacteristic
101C4⋊Dic53C2×C411yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—41no—
107#11719912—41no—
108#11719912—41no—
109#11719912—41no—
110C8×D53—41no—
111C8×D53—41no—
112C8×D53—41no—
113C8×D53—41no—
114C23×C53C22×C411yescharacteristic
115C22×D53—22no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53C22×C411yescharacteristic
119C22×D53C22×C411yescharacteristic
120C22×D53C22×C411yescharacteristic
121C22×D53C22×C411yescharacteristic
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C22×C411yescharacteristic
125C2×C4×C53C22×C411yescharacteristic
126C2×C4×C53C22×C411yescharacteristic
127C2×C4×C53C22×C411yescharacteristic
128C2×C4×C53C22×C411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—41no—
131C22⋊D53—41no—
132C2×Dic53—41no—
133C2×Dic53C22×C411yescharacteristic
134C2×Dic53C22×C411yescharacteristic
135C2×Dic53C22×C411yescharacteristic
136C2×Dic53C22×C411yescharacteristic
137D212—22no—
138D212—22no—
139D212—41no—
140D212—41no—
141D212—41no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—41no—
151C4×D53—41no—
152Dic106—41no—
153C8×C53—21no—
154C8×C53—21no—
155C8×C53—21no—
156C8×C53—21no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160#88322—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53C23×C411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—44no—
177C2×D53#13112yes—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53H(5,𝔽2)11yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189#2976—531nomaximal, Sylow
190C2×C53—41no—
191C2×C53#51811yescharacteristic
192C2×C53C2×#13111yescharacteristic
193C2×C53#56811yescharacteristic
194D53—22no—
195D53—41no—
196D53—41no—
197D53—41no—
198#525—531no—
199#436—531no—
200#436—531no—
201#435—531no—
202#435—531no—
203#435—531no—
204#435—531no—
205C8×D4—531no—
206C8×D4—531no—
207#433—531no—
208#433D5311yescharacteristic, complemented
209#408—531no—
210#408—531no—
211#407—531no—
212C2×C22⋊C8—531no—
213C53#297611yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C23×C4—531no—
216C2×M16—531no—
217C2×M16—531no—
218C2×M16—531no—
219C2×M16—531no—
220C22×C8—531no—
221C22×C8—531no—
222C22×C8—531no—
223C22×C8—531no—
224#123—531no—
225#123—531no—
226C22⋊Q8—531no—
227C22⋊Q8—531no—
228#121—531no—
229#121—531no—
230#120—531no—
231#120—531no—
232C4×D4—531no—
233C4×D4—531no—
234C4×D4—531no—
235C4×D4—531no—
236#117C2×D5311yescharacteristic
237C4⋊C8—531no—
238C4⋊C8—531no—
239C4⋊C8C2×D5311yescharacteristic, complemented
240C4⋊C8C2×D5311yescharacteristic, complemented
241C22⋊C8—531no—
242C22⋊C8—531no—
243C22⋊C8—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249C22⋊C8—531no—
250C22⋊C8—531no—
251C22⋊C8C2×D5311yescharacteristic
252C22⋊C8C2×D5311yescharacteristic
253#97—531no—
254#97C2×D5311yescharacteristic, complemented
255C4×C8—531no—
256C4×C8C2×D5311yescharacteristic, complemented
257C24—531no—
258#41—2121no—
259#41—2121no—
260C2×Q8—531no—
261C2×D4—531no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4C22×D5311yescharacteristic
278M16—2121no—
279M16—2121no—
280M16—2121no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8C22×D5311yescharacteristic
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C2×C8C22×D5311yescharacteristic
294C4⋊C4—531no—
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4C4×D5311yescharacteristic
299C4⋊C4C4×D5311yescharacteristic
300C22⋊C4—531no—
301C22⋊C4—531no—
302C22⋊C4—531no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—531no—
306C22⋊C4—531no—
307C22⋊C4—531no—
308C22⋊C4C4×D5311yescharacteristic
309C22⋊C4C4×D5311yescharacteristic
310C42C22×D5311yescharacteristic
311C42C22×D5311yescharacteristic
312C23—531no—
313C23—531no—
314C23—531no—
315C23—531no—
316C23—1062no—
317C23—2121no—
318C23—2121no—
319C23C2×C4×D5311yescharacteristic
320Q8—2121no—
321D4—1062no—
322D4—1062no—
323D4—2121no—
324D4—2121no—
325D4—2121no—
326D4—2121no—
327D4—2121no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4C2×C4×D5311yescharacteristic
346C2×C4C2×C4×D5311yescharacteristic
347C2×C4C2×C4×D5311yescharacteristic
348C2×C4C2×C4×D5311yescharacteristic
349C2×C4C2×C4×D5311yescharacteristic
350C2×C4C23×D5311yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—532no—
363C22—1061no—
364C22—1061no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22—1061no—
372C22—1061no—
373C22—1061no—
374C22—2124no—
375C22C22×C4×D5311yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—1061no—
382C4—1061no—
383C4—1061no—
384C4—1061no—
385C4#1313326311yescharacteristic
386C4#1313326311yescharacteristic
387C2—41no—
388C2—1062no—
389C2—2121no—
390C2—2121no—
391C2—2121no—
392C2#1681109711yescharacteristic
393C2#131×D5311yescharacteristic
394C2#1681138811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D53
commutator, socleC22×C53C23×C4
FittingC53×#433C2
hypercenter#433D53
hypocenterC53#2976
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #16810702C2semidirect
3. #16810700C2semidirect
4. #16810696C2semidirect
5. #16810696C2semidirect
6. #16810693C2semidirect
7. #16810693C2semidirect
8. C8×D212C2semidirect
9. #16810688C2semidirect
10. #16810684C2semidirect
11. #16810588C2semidirect
12. #16810588C2semidirect
13. #16810585C2semidirect
14. C22⋊C8×D53C2semidirect
15. C53×#433C2semidirect
20. C2×#11719912C22semidirect
21. C2×#11719912C22semidirect
24. C2×C8×D53C22semidirect
25. C2×C8×D53C22semidirect
38. C4⋊#88322C22semidirect
39. C4⋊#88322C22semidirect
46. #13133129C22semidirect
47. #13133129C22semidirect
48. #13133129C22semidirect
49. #13133129C22semidirect
50. #13133128C22semidirect
51. C4×#88322C22semidirect
53. C4⋊C8×C53C22semidirect
54. C4⋊C8×C53C22semidirect
57. #97×C53C22semidirect
58. C4×C8×C53C22semidirect
208. #433D53semidirect
213. C53#2976semidirect
239. C4⋊C8C2×D53semidirect
240. C4⋊C8C2×D53semidirect
254. #97C2×D53semidirect
256. C4×C8C2×D53semidirect