Finite Group Databasebeta

← #20286739

Subgroups — #20286739

Order 6784.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811312C211yesmaximal, characteristic, complemented
2#16811257C211yesmaximal, characteristic
3#16810986C211yesmaximal, characteristic, complemented
4#16810938C211yesmaximal, characteristic, complemented
5#16810922C211yesmaximal, characteristic, complemented
6#16810867C211yesmaximal, characteristic, complemented
7#16810853C211yesmaximal, characteristic, complemented
8#16810822C211yesmaximal, characteristic, complemented
9#16810776C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810697C211yesmaximal, characteristic, complemented
13#16810627C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic, complemented
22C2×#11719917C2211yescharacteristic, complemented
23C2×C8×D53C2211yescharacteristic, complemented
24#13133213C2211yescharacteristic
25#13133210—21no—
26#13133209—21no—
27#13133208C2211yescharacteristic, complemented
28#13133199—21no—
29#13133197C2211yescharacteristic, complemented
30#13133196C2211yescharacteristic, complemented
31#13133192C2211yescharacteristic
32#13133190—21no—
33#13133187—21no—
34#13133185—21no—
35#13133182C2211yescharacteristic, complemented
36#13133179C2211yescharacteristic
37C4×C22⋊D53C2211yescharacteristic
38#13133174—21no—
39C4⋊C4×D53C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68D4×D53—22no—
69D4×D53—41no—
70D4×D53—44no—
71C2×D212—22no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—41no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719923—41no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—22no—
110#11719913—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53C2×D411yescharacteristic
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—22no—
134C22⋊D53—42no—
135C22⋊D53—42no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142D212—21no—
143D212—21no—
144D212—21no—
145D212—42no—
146D212—42no—
147D212#4112yes—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—42no—
151C4×D53—42no—
152Dic106—21no—
153Dic106—21no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—22no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—84no—
174C2×D53—84no—
175C2×D53AGL(1,ℤ/8)12yes—
176C4×C53—21no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53C2×#4111yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—21no—
183Dic53—21no—
184Dic53—21no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—41no—
189C2×C53—81no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—22no—
194D53—42no—
195D53—81no—
196#552—531no—
197D42—531no—
198#490—531no—
199#482D5311yescharacteristic, complemented
200#480—531no—
201#478—531no—
202#468—531no—
203#465—531no—
204#460—531no—
205#450—531no—
206#450—531no—
207#440—531no—
208#435—531no—
209#418—531no—
210C2×#102—531no—
211C53#331011yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C22×D4—531no—
214C22×D4—1061no—
215C2×SD16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×M16—531no—
219C22×C8—531no—
220#127—531no—
221#126—1061no—
222#125—531no—
223#125—1061no—
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227#121—531no—
228#121—1061no—
229#121C2×D5311yescharacteristic
230#120—1061no—
231#120—1061no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4—531no—
235#117—1061no—
236#117C2×D5311yescharacteristic, complemented
237C2×C4⋊C4—531no—
238#107—531no—
239#107C2×D5311yescharacteristic, complemented
240#106C2×D5311yescharacteristic, complemented
241C4⋊C8—531no—
242#103—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102C2×D5311yescharacteristic, complemented
251#102C2×D5311yescharacteristic, complemented
252C22⋊C8—531no—
253C22⋊C8C2×D5311yescharacteristic, complemented
254#97—531no—
255C24—1061no—
256C24—1061no—
257C2×Q8—531no—
258C2×D4—21no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—1062no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1062no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2124no—
272C2×D4C22×D5311yescharacteristic
273C22×C4—531no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4C22×D5311yescharacteristic
277SD16—1062no—
278SD16—2121no—
279SD16—2121no—
280M16—2121no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—2121no—
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4C22×D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
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—
305C42—21no—
306C42—531no—
307C42—1061no—
308C23—21no—
309C23—531no—
310C23—1061no—
311C23—1061no—
312C23—4241no—
313C23—4241no—
314C23—2122no—
315C23—2122no—
316C23—2122no—
317C23—2122no—
318C23D4×D5311yescharacteristic
319Q8—1061no—
320Q8—1061no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—532no—
325D4—1062no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—4241no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—531no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—532no—
361C22—1061no—
362C22—1061no—
363C22—1062no—
364C22—4244no—
365C22—4244no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22C2×D4×D5311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—1061no—
377C4—1061no—
378C4—1061no—
379C4—2121no—
380C4—2121no—
381C4#41×D5311yescharacteristic
382C4#1313326311yescharacteristic
383C2—41no—
384C2—81no—
385C2—1062no—
386C2—4241no—
387C2—2122no—
388C2#1681126511yescharacteristic
389C2AGL(1,ℤ/8)×D5311yescharacteristic
390C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811312C2semidirect
3. #16810986C2semidirect
4. #16810938C2semidirect
5. #16810922C2semidirect
6. #16810867C2semidirect
7. #16810853C2semidirect
8. #16810822C2semidirect
9. #16810776C2semidirect
10. #16810775C2semidirect
11. #16810722C2semidirect
12. #16810697C2semidirect
13. #16810627C2semidirect
14. #102×D53C2semidirect
15. C53×#482C2semidirect
20. C2×#11719913C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719917C22semidirect
23. C2×C8×D53C22semidirect
27. #13133208C22semidirect
29. #13133197C22semidirect
30. #13133196C22semidirect
35. #13133182C22semidirect
39. C4⋊C4×D53C22semidirect
40. #13133150C22semidirect
41. #13133146C22semidirect
42. #13133140C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
48. #13133137C22semidirect
49. #13133137C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
53. #117×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
199. #482D53semidirect
211. C53#3310semidirect
236. #117C2×D53semidirect
239. #107C2×D53semidirect
240. #106C2×D53semidirect
250. #102C2×D53semidirect
251. #102C2×D53semidirect
253. C22⋊C8C2×D53semidirect