Finite Group Databasebeta

← #20286735

Subgroups — #20286735

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
2#16811258C211yesmaximal, characteristic
3#16810945C211yesmaximal, characteristic, complemented
4#16810935C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810868C211yesmaximal, characteristic, complemented
7#16810851C211yesmaximal, characteristic, complemented
8#16810820C211yesmaximal, characteristic, complemented
9#16810775C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810725C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#490C211yesmaximal, characteristic, complemented
16C2×D4×D53—21no—
17C2×D4×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719923C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic, complemented
22C2×#11719912C2211yescharacteristic, complemented
23C2×C8×D53C2211yescharacteristic, complemented
24#13133212—21no—
25#13133208—21no—
26#13133202—21no—
27#13133197C2211yescharacteristic, complemented
28#13133197C2211yescharacteristic, complemented
29#13133188—21no—
30#13133188C2211yescharacteristic
31#13133188C2211yescharacteristic
32#13133185—21no—
33#13133185—21no—
34C4×Dic106C2211yescharacteristic, complemented
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37#13133175—21no—
38#13133175C2211yescharacteristic, complemented
39C4⋊C4×D53C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43C4⋊#88322C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#127×C53C2211yescharacteristic
53#125×C53C2211yescharacteristic, complemented
54#102×C53C2211yescharacteristic, complemented
55#102×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69D4×D53—22no—
70D4×D53—22no—
71D4×D53—41no—
72D4×D53—41no—
73D4×D53—44no—
74C2×D212D411yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
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
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719923—41no—
91#11719923—41no—
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
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—22no—
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—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—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—42no—
128D4×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—42no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—22no—
144D212—21no—
145D212#4112yes—
146C4×D53—21no—
147C4×D53—21no—
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—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—22no—
169C2×D53—21no—
170C2×D53—21no—
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×C53C2×#4111yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—21no—
183Dic53—21no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187#3310—531nomaximal, Sylow
188C2×C53—81no—
189C2×C53—81no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—22no—
194D53—41no—
195D53—42no—
196#552—531no—
197D42—531no—
198#490D5311yescharacteristic, complemented
199#482—531no—
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#127C2×D5311yescharacteristic
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×D4—531no—
235#117—531no—
236#117—1061no—
237C2×C4⋊C4—531no—
238#107—531no—
239#107—531no—
240#106—531no—
241C4⋊C8—531no—
242#103—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102C2×D5311yescharacteristic, complemented
249#102C2×D5311yescharacteristic, complemented
250#102C2×D5311yescharacteristic, complemented
251#102C2×D5311yescharacteristic, complemented
252C22⋊C8—531no—
253C22⋊C8—531no—
254#97C2×D5311yescharacteristic, complemented
255C24—1061no—
256C24—1061no—
257C2×Q8—531no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1062no—
265C2×D4—1062no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2124no—
271C2×D4C22×D5311yescharacteristic
272C2×D4C22×D5311yescharacteristic
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1061no—
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—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—531no—
298C22⋊C4—531no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C42—1061no—
306C42—1061no—
307C42C22×D5311yescharacteristic
308C23—21no—
309C23—21no—
310C23—531no—
311C23—531no—
312C23—1061no—
313C23—4241no—
314C23—4241no—
315C23—2122no—
316C23—2122no—
317C23—2122no—
318C23—2122no—
319Q8—1061no—
320Q8—1061no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—532no—
328D4—1062no—
329D4—1061no—
330D4—2122no—
331D4—2122no—
332D4—2122no—
333D4—2122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4D4×D5311yescharacteristic
348C2×C4D4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—1061no—
353C8—1061no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—532no—
361C22—1061no—
362C22—1061no—
363C22—1061no—
364C22—1062no—
365C22—1061no—
366C22—1061no—
367C22—4244no—
368C22—4244no—
369C22—2121no—
370C22—2121no—
371C22C2×D4×D5311yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—1061no—
377C4—1061no—
378C4—2121no—
379C4—2121no—
380C4—2121no—
381C4#41×D5311yescharacteristic
382C4#1313326311yescharacteristic
383C2—81no—
384C2—81no—
385C2—1062no—
386C2—2121no—
387C2—2122no—
388C2#1681126611yescharacteristic
389C2AGL(1,ℤ/8)×D5311yescharacteristic
390C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3310
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811311C2semidirect
3. #16810945C2semidirect
4. #16810935C2semidirect
5. #16810920C2semidirect
6. #16810868C2semidirect
7. #16810851C2semidirect
8. #16810820C2semidirect
9. #16810775C2semidirect
10. #16810775C2semidirect
11. #16810725C2semidirect
12. #16810696C2semidirect
13. #16810626C2semidirect
14. #102×D53C2semidirect
15. C53×#490C2semidirect
20. C2×#11719923C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719912C22semidirect
23. C2×C8×D53C22semidirect
27. #13133197C22semidirect
28. #13133197C22semidirect
34. C4×Dic106C22semidirect
38. #13133175C22semidirect
39. C4⋊C4×D53C22semidirect
40. #13133151C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
43. C4⋊#88322C22semidirect
44. #13133142C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
53. #125×C53C22semidirect
54. #102×C53C22semidirect
55. #102×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. #97×C53C22semidirect
198. #490D53semidirect
211. C53#3310semidirect
223. #125C2×D53semidirect
248. #102C2×D53semidirect
249. #102C2×D53semidirect
250. #102C2×D53semidirect
251. #102C2×D53semidirect
254. #97C2×D53semidirect