Finite Group Databasebeta

← #20286532

Subgroups — #20286532

Order 6784.

462 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#16811331C211yesmaximal, characteristic, complemented
2D4×C22⋊D53C211yesmaximal, characteristic
3#16810968C211yesmaximal, characteristic, complemented
4#16810942C211yesmaximal, characteristic, complemented
5#16810940C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810849C211yesmaximal, characteristic, complemented
9#16810821C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11SD16×Dic53C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13C2×#13133138C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16C22×C22⋊D53—21no—
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic, complemented
19C22×#88322C2211yescharacteristic, complemented
20C2×C8×D53C2211yescharacteristic, complemented
21#13133216C2211yescharacteristic, complemented
22#13133213C2211yescharacteristic
23#13133207—21no—
24#13133197—21no—
25#13133194—21no—
26#13133193C2211yescharacteristic, complemented
27#13133191C2211yescharacteristic
28#13133189—21no—
29#13133188C2211yescharacteristic
30#13133186—21no—
31#13133185—21no—
32Q8×Dic53C2211yescharacteristic, complemented
33D4×Dic53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic, complemented
38#13133149C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic, complemented
55C22⋊Q8×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C24×C53—21no—
60C23×D53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—22no—
63C2×D4×C53—21no—
64C2×D4×C53—41no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—44no—
74C2×C22⋊D53D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77D4×D53—22no—
78D4×D53—41no—
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83SD16×C53—22no—
84SD16×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719924—41no—
94#11719921—21no—
95#11719921—21no—
96#11719921D411yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322—41no—
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—81no—
117C23×C53C2×D411yescharacteristic
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—42no—
129D4×C53#4112yes—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2411yescharacteristic
136C22⋊D53—42no—
137C22⋊D53—42no—
138C22⋊D53—42no—
139C22⋊D53—42no—
140C22⋊D53—81no—
141C22⋊D53SD1614yescomplemented
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—42no—
146C2×Dic53—81no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150D212—21no—
151D212—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—81no—
155Dic106—81no—
156C8×C53—21no—
157C8×C53—21no—
158#88322—21no—
159#88322—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—84no—
169C22×C53C2×SD1612yes—
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—84no—
177C2×D53C2×SD1612yes—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53C2×#4111yescharacteristic
182C4×C53H(5,𝔽2)11yescharacteristic
183Dic53—21no—
184Dic53—21no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53C2×SD1612yes—
189#3295—531nomaximal, Sylow
190C2×C53—22no—
191C2×C53—42no—
192C2×C53—81no—
193C2×C53#54611yescharacteristic
194C2×C53C22×SD1611yescharacteristic
195C2×C53#57611yescharacteristic
196D53—22no—
197D53—81no—
198#554—531no—
199D42—531no—
200#486—531no—
201#481—531no—
202#481—531no—
203#476—531no—
204#465—531no—
205#465—531no—
206#460—531no—
207#450—531no—
208#450D5311yescharacteristic, complemented
209C4×SD16—531no—
210C8×D4—531no—
211C2×#102—531no—
212C2×#102—531no—
213C53#329511yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C22×D4—1061no—
216C22×D4C2×D5311yescharacteristic
217C2×SD16—531no—
218C2×SD16C2×D5311yescharacteristic
219C2×SD16C2×D5311yescharacteristic, complemented
220C22×C8—531no—
221C22×C8—531no—
222C4⋊Q8—531no—
223#127—531no—
224#125—1061no—
225C22⋊Q8—531no—
226C22⋊Q8—1061no—
227C22⋊Q8—1061no—
228C22⋊Q8C2×D5311yescharacteristic, complemented
229#121—531no—
230#121—531no—
231#121—1061no—
232#120—1061no—
233#120—1061no—
234C4×Q8—531no—
235C4×D4—531no—
236C4×D4—531no—
237C4×D4—1061no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240#106—531no—
241#106—531no—
242#106—531no—
243C4⋊C8—531no—
244#103—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252#102C2×D5311yescharacteristic
253#102C2×D5311yescharacteristic, complemented
254C22⋊C8—531no—
255C22⋊C8C2×D5311yescharacteristic, complemented
256C4×C8—531no—
257C24—21no—
258C24—1061no—
259C2×Q8—1061no—
260C2×Q8C22×D5311yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1062no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—2121no—
273C2×D4—2124no—
274C2×D4C22×D5311yescharacteristic
275C2×D4C22×D5311yescharacteristic
276C22×C4—531no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4C22×D5311yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—2121no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8C22×D5311yescharacteristic
289C2×C8C22×D5311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—531no—
292C4⋊C4—531no—
293C4⋊C4—531no—
294C4⋊C4—531no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4C22×D5311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—531no—
303C22⋊C4—531no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—531no—
309C42—1061no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—531no—
316C23—1061no—
317C23—4241no—
318C23—2122no—
319C23—2122no—
320C23D4×D5311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—4241no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—534no—
329D4—1061no—
330D4—1061no—
331D4—4241no—
332D4—2122no—
333D4—2122no—
334D4—2122no—
335D4—2122no—
336D4#1171994712yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2122no—
352C2×C4D4×D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—21no—
355C8—21no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—532no—
368C22—1061no—
369C22—1061no—
370C22—4244no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22SD16×D5312yes—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—532no—
380C4—1061no—
381C4—1061no—
382C4—2121no—
383C4—2121no—
384C4—2121no—
385C4C2×#1171994711yescharacteristic
386C4#1313326211yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—1062no—
391C2—4241no—
392C2#1681126011yescharacteristic
393C2C2×SD16×D5311yescharacteristic
394C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3295
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811331C2semidirect
3. #16810968C2semidirect
4. #16810942C2semidirect
5. #16810940C2semidirect
6. #16810911C2semidirect
7. #16810854C2semidirect
8. #16810849C2semidirect
9. #16810821C2semidirect
10. #16810772C2semidirect
11. SD16×Dic53C2semidirect
12. #16810686C2semidirect
13. C2×#13133138C2semidirect
14. #102×D53C2semidirect
15. C53×#450C2semidirect
18. C2×#11719924C22semidirect
19. C22×#88322C22semidirect
20. C2×C8×D53C22semidirect
21. #13133216C22semidirect
26. #13133193C22semidirect
32. Q8×Dic53C22semidirect
36. C2×C4⋊Dic53C22semidirect
37. C4⋊C4×D53C22semidirect
38. #13133149C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133141C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
49. #13133137C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
54. C2×SD16×C53C22semidirect
55. C22⋊Q8×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
141. C22⋊D53SD16semidirect
208. #450D53semidirect
213. C53#3295semidirect
219. C2×SD16C2×D53semidirect
228. C22⋊Q8C2×D53semidirect
253. #102C2×D53semidirect
255. C22⋊C8C2×D53semidirect