Finite Group Databasebeta

← #20286534

Subgroups — #20286534

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
2#16811257C211yesmaximal, characteristic
3#16810968C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810939C211yesmaximal, characteristic, complemented
6#16810912C211yesmaximal, characteristic, complemented
7#16810855C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810708C211yesmaximal, characteristic, complemented
12C8×C22⋊D53C211yesmaximal, characteristic, complemented
13C2×#13133139C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133216C2211yescharacteristic, complemented
23#13133213C2211yescharacteristic
24#13133207—21no—
25#13133197—21no—
26#13133194—21no—
27#13133193C2211yescharacteristic, complemented
28#13133192C2211yescharacteristic
29#13133190—21no—
30#13133187—21no—
31#13133185—21no—
32Q8×Dic53C2211yescharacteristic, complemented
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic, complemented
38#13133148C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70D4×D53—22no—
71D4×D53—41no—
72D4×D53—44no—
73C2×D212—22no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—41no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83SD16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719924—22no—
94#11719924—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
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
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C22⋊D53SD1614yescomplemented
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—81no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212—42no—
150D212#4112yes—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—42no—
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—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53C2×SD1612yes—
166C22×C53C22×D411yescharacteristic
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—22no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—84no—
176C2×D53—84no—
177C2×D53C2×SD1612yes—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53C2×#4111yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—21no—
185Dic53—21no—
186Dic53—41no—
187Dic53—41no—
188Dic53C2×SD1612yes—
189#3295—531nomaximal, Sylow
190C2×C53—22no—
191C2×C53—81no—
192C2×C53#54611yescharacteristic
193C2×C53C22×SD1611yescharacteristic
194C2×C53#57611yescharacteristic
195D53—22no—
196D53—42no—
197D53—81no—
198#554—531no—
199D42—531no—
200#486—531no—
201#481—531no—
202#481—531no—
203#476—531no—
204#465—531no—
205#465D5311yescharacteristic, complemented
206#460—531no—
207#450—531no—
208#450—531no—
209C4×SD16—531no—
210C8×D4—531no—
211C2×#102—531no—
212C2×#102—531no—
213C53#329511yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C22×D4—531no—
216C22×D4—1061no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16C2×D5311yescharacteristic, complemented
220C22×C8—531no—
221C22×C8C2×D5311yescharacteristic, complemented
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—1061no—
231#121C2×D5311yescharacteristic
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#106C2×D5311yescharacteristic, complemented
243C4⋊C8—531no—
244#103C2×D5311yescharacteristic, complemented
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252#102—531no—
253#102C2×D5311yescharacteristic, complemented
254C22⋊C8—531no—
255C22⋊C8—531no—
256C4×C8—531no—
257C24—1061no—
258C24—1061no—
259C2×Q8—1061no—
260C2×Q8C22×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—1062no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1062no—
271C2×D4—1061no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4—2124no—
275C2×D4C22×D5311yescharacteristic
276C22×C4—531no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4C22×D5311yescharacteristic
281SD16—41no—
282SD16—1062no—
283SD16—2121no—
284C2×C8—41no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—2121no—
288C2×C8C22×D5311yescharacteristic
289C2×C8C22×D5311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—531no—
292C4⋊C4—531no—
293C4⋊C4—531no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1061no—
299C4⋊C4C22×D5311yescharacteristic
300C4⋊C4C22×D5311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C42—531no—
309C42—1061no—
310C23—21no—
311C23—531no—
312C23—1061no—
313C23—1061no—
314C23—4241no—
315C23—4241no—
316C23—2122no—
317C23—2122no—
318C23—2122no—
319C23—2122no—
320C23D4×D5311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—4241no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—534no—
328D4—532no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—4241no—
333D4—2122no—
334D4—2122no—
335D4—2122no—
336D4—2122no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
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—41no—
361C22—41no—
362C22—41no—
363C22—532no—
364C22—1061no—
365C22—1061no—
366C22—1062no—
367C22—4244no—
368C22—4244no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22SD16×D5312yes—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—532no—
381C4—1061no—
382C4—1061no—
383C4—2121no—
384C4—2121no—
385C4#41×D5311yescharacteristic
386C4#1313326211yescharacteristic
387C2—22no—
388C2—81no—
389C2—1062no—
390C2—4241no—
391C2—2122no—
392C2#1681126211yescharacteristic
393C2C2×SD16×D5311yescharacteristic
394C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3295
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811331C2semidirect
3. #16810968C2semidirect
4. #16810943C2semidirect
5. #16810939C2semidirect
6. #16810912C2semidirect
7. #16810855C2semidirect
8. #16810821C2semidirect
9. #16810773C2semidirect
10. #16810772C2semidirect
11. #16810708C2semidirect
12. C8×C22⋊D53C2semidirect
13. C2×#13133139C2semidirect
14. #102×D53C2semidirect
15. C53×#465C2semidirect
20. C2×#11719924C22semidirect
21. C2×C8×D53C22semidirect
22. #13133216C22semidirect
27. #13133193C22semidirect
32. Q8×Dic53C22semidirect
36. C2×C4⋊Dic53C22semidirect
37. C4⋊C4×D53C22semidirect
38. #13133148C22semidirect
39. #13133148C22semidirect
40. #13133146C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
48. #13133137C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
52. C2×SD16×C53C22semidirect
53. C22×C8×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #106×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
138. C22⋊D53SD16semidirect
205. #465D53semidirect
213. C53#3295semidirect
219. C2×SD16C2×D53semidirect
221. C22×C8C2×D53semidirect
228. C22⋊Q8C2×D53semidirect
242. #106C2×D53semidirect
244. #103C2×D53semidirect
253. #102C2×D53semidirect