Finite Group Databasebeta

← #20286733

Subgroups — #20286733

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#16811311C211yesmaximal, characteristic, complemented
2D4×D212C211yesmaximal, characteristic
3#16810989C211yesmaximal, characteristic, complemented
4#16810949C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810853C211yesmaximal, characteristic, complemented
8#16810824C211yesmaximal, characteristic, complemented
9#16810775C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#478C211yesmaximal, characteristic, complemented
16C22×D212—21no—
17C2×D4×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic, complemented
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic
22C2×#11719912C2211yescharacteristic, complemented
23C2×C8×D53C2211yescharacteristic, complemented
24#13133214C2211yescharacteristic
25#13133212—21no—
26#13133208—21no—
27#13133202—21no—
28#13133197C2211yescharacteristic, complemented
29#13133197C2211yescharacteristic, complemented
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133191—21no—
33#13133187—21no—
34#13133187—21no—
35C4×Dic106C2211yescharacteristic, complemented
36C4×D212C2211yescharacteristic
37#13133175—21no—
38#13133175C2211yescharacteristic, complemented
39C4⋊C4×D53C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133140C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic
45#13133137C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133128C2211yescharacteristic, complemented
52#125×C53C2211yescharacteristic, complemented
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65D4×D53—22no—
66D4×D53—22no—
67D4×D53—41no—
68D4×D53—41no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—44no—
74C2×D212D411yescharacteristic
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C42×C53C2311yescharacteristic
90#11719923—22no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921D411yescharacteristic
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—41no—
109#11719913—41no—
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122C22×D53C2×D411yescharacteristic
123D4×C53—21no—
124D4×C53#4112yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—42no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—42no—
133C22⋊D53—42no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—22no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—42no—
144D212—42no—
145D212—42no—
146D212—81no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—42no—
152Dic106—21no—
153Dic106—21no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
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—84no—
174C2×D53—84no—
175C2×D53AGL(1,ℤ/8)12yes—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53C2×#4111yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—21no—
184Dic53—21no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—42no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—22no—
193D53—41no—
194D53—81no—
195D53—81no—
196#552—531no—
197D42—531no—
198#490—531no—
199#482—531no—
200#480—531no—
201#478D5311yescharacteristic, complemented
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—1061no—
223#125C2×D5311yescharacteristic, complemented
224#123—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—531no—
227#121—531no—
228#121—531no—
229#121—1061no—
230#120—1061no—
231#120—1061no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4C2×D5311yescharacteristic
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—531no—
238#107—531no—
239#107C2×D5311yescharacteristic, complemented
240#106C2×D5311yescharacteristic, complemented
241C4⋊C8C2×D5311yescharacteristic, complemented
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
251#102C2×D5311yescharacteristic
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97—531no—
255C24—1061no—
256C24—1061no—
257C2×Q8—531no—
258C2×D4—531no—
259C2×D4—531no—
260C2×D4—531no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1062no—
266C2×D4—1062no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2124no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4C22×D5311yescharacteristic
273C22×C4—21no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1061no—
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—21no—
287C4⋊C4—21no—
288C4⋊C4—531no—
289C4⋊C4—531no—
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—531no—
299C22⋊C4—531no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C42—1061no—
306C42—1061no—
307C42C22×D5311yescharacteristic
308C23—21no—
309C23—531no—
310C23—531no—
311C23—1061no—
312C23—1061no—
313C23—4241no—
314C23—4241no—
315C23—2122no—
316C23—2122no—
317C23—2122no—
318C23—2122no—
319Q8—1061no—
320Q8—1061no—
321D4—21no—
322D4—1062no—
323D4—1061no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—4241no—
328D4—2122no—
329D4—2122no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4#1171994912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4D4×D5311yescharacteristic
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—532no—
358C22—1061no—
359C22—1061no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—4244no—
364C22—4244no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22C2×D4×D5311yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—1061no—
378C4—1061no—
379C4—2121no—
380C4—2121no—
381C4C2×#1171994911yescharacteristic
382C4#1313326311yescharacteristic
383C2—42no—
384C2—1062no—
385C2—4241no—
386C2—4241no—
387C2—2121no—
388C2#1681126411yescharacteristic
389C2AGL(1,ℤ/8)×D5311yescharacteristic
390C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811311C2semidirect
3. #16810989C2semidirect
4. #16810949C2semidirect
5. #16810938C2semidirect
6. #16810870C2semidirect
7. #16810853C2semidirect
8. #16810824C2semidirect
9. #16810775C2semidirect
10. #16810775C2semidirect
11. #16810723C2semidirect
12. #16810693C2semidirect
13. #16810626C2semidirect
14. #102×D53C2semidirect
15. C53×#478C2semidirect
19. C2×#11719913C22semidirect
20. C2×#11719913C22semidirect
22. C2×#11719912C22semidirect
23. C2×C8×D53C22semidirect
28. #13133197C22semidirect
29. #13133197C22semidirect
35. C4×Dic106C22semidirect
38. #13133175C22semidirect
39. C4⋊C4×D53C22semidirect
40. #13133150C22semidirect
41. #13133140C22semidirect
42. #13133139C22semidirect
43. #13133139C22semidirect
45. #13133137C22semidirect
46. #13133137C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. #13133129C22semidirect
50. #13133129C22semidirect
51. #13133128C22semidirect
52. #125×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. C4⋊C8×C53C22semidirect
201. #478D53semidirect
211. C53#3310semidirect
223. #125C2×D53semidirect
239. #107C2×D53semidirect
240. #106C2×D53semidirect
241. C4⋊C8C2×D53semidirect