Finite Group Databasebeta

← #20286521

Subgroups — #20286521

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811273C211yesmaximal, characteristic, complemented
2D4×C22⋊D53C211yesmaximal, characteristic, complemented
3#16810968C211yesmaximal, characteristic, complemented
4#16810937C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810856C211yesmaximal, characteristic, complemented
9#16810816C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11D8×Dic53C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13C2×#13133138C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#447C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C22×C22⋊D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719922C2211yescharacteristic, complemented
20C22×#88322C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133216C2211yescharacteristic, complemented
23#13133213C2211yescharacteristic
24#13133199—21no—
25#13133198—21no—
26#13133194—21no—
27#13133191C2211yescharacteristic
28#13133189—21no—
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic, complemented
31#13133186—21no—
32#13133185—21no—
33D4×Dic53C2211yescharacteristic
34D4×Dic53C2211yescharacteristic, complemented
35C4×C22⋊D53C2211yescharacteristic
36C2×C4⋊Dic53C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic, complemented
38#13133151C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C24×C53—21no—
60C23×D53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53—41no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—44no—
76C2×C22⋊D53D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80#11719947—44no—
81D4×D53—22no—
82D4×D53—41no—
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×Dic106—21no—
87D8×C53—22no—
88D8×C53—41no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53D411yescharacteristic
98#11719922—41no—
99#11719921—21no—
100#11719921D411yescharacteristic
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108C4×Dic53C2311yescharacteristic
109C2×#88322—41no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C8×D53—41no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53C2×D411yescharacteristic
120C22×D53—21no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—81no—
124C22×D53C2×D411yescharacteristic
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53#4112yescomplemented
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—42no—
140C22⋊D53—42no—
141C22⋊D53—42no—
142C22⋊D53—42no—
143C22⋊D53—81no—
144C22⋊D53D814yescomplemented
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53C2×D411yescharacteristic
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—21no—
155D212—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—42no—
159Dic106—42no—
160C8×C53—21no—
161C8×C53—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—84no—
176C22×C53C2×D812yes—
177C22×C53C22×D411yescharacteristic
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C2×D53C2×D812yes—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53C2×#4111yescharacteristic
188C4×C53H(5,𝔽2)11yescharacteristic
189Dic53—21no—
190Dic53—21no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194Dic53C2×D812yes—
195#3294—531nomaximal, Sylow
196C2×C53—22no—
197C2×C53—42no—
198C2×C53—81no—
199C2×C53—81no—
200C2×C53#54611yescharacteristic
201C2×C53C22×D811yescharacteristic
202C2×C53#57711yescharacteristic
203D53—22no—
204D53—81no—
205#547—531no—
206D42—531no—
207#486—531no—
208#480—531no—
209#480—531no—
210#474—531no—
211#466—531no—
212#466—531no—
213#459—531no—
214#447—531no—
215#447D5311yescharacteristic, complemented
216C4×D8—531no—
217C8×D4—531no—
218C2×#102—531no—
219C2×#102—531no—
220C53#329411yescharacteristic, complemented, Sylow
221C2×#41—1061no—
222C22×D4—531no—
223C22×D4—1061no—
224C22×D4C2×D5311yescharacteristic
225C2×D8—531no—
226C2×D8C2×D5311yescharacteristic
227C2×D8C2×D5311yescharacteristic, complemented
228C22×C8—531no—
229C22×C8—531no—
230C4⋊Q8—531no—
231#127—531no—
232#123—1061no—
233#123—1061no—
234C22⋊Q8—1061no—
235#121—531no—
236#121—531no—
237#121—531no—
238#121—1061no—
239#121C2×D5311yescharacteristic, complemented
240#120—1061no—
241#120—1061no—
242C4×D4—531no—
243C4×D4—531no—
244C4×D4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247#107—531no—
248#107—531no—
249#107—531no—
250C4⋊C8—531no—
251#102—531no—
252#102—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259#102C2×D5311yescharacteristic
260#102C2×D5311yescharacteristic, complemented
261C22⋊C8—531no—
262C22⋊C8C2×D5311yescharacteristic, complemented
263C4×C8—531no—
264C24—21no—
265C24—1061no—
266#41—2124no—
267C2×Q8—1061no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—531no—
273C2×D4—531no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1062no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—2121no—
282C2×D4—2124no—
283C2×D4C22×D5311yescharacteristic
284C2×D4C22×D5311yescharacteristic
285C2×D4C22×D5311yescharacteristic
286C22×C4—531no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4C22×D5311yescharacteristic
292D8—22no—
293D8—41no—
294D8—2121no—
295C2×C8—531no—
296C2×C8—531no—
297C2×C8—2121no—
298C2×C8—2121no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—531no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4C22×D5311yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—531no—
311C22⋊C4—531no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—531no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—531no—
325C23—1061no—
326C23—4241no—
327C23—2122no—
328C23—2122no—
329C23D4×D5311yescharacteristic
330Q8—2122no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—534no—
339D4—1061no—
340D4—1061no—
341D4—4241no—
342D4—2122no—
343D4—2122no—
344D4—2122no—
345D4—2122no—
346D4—2122no—
347D4#1171994712yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—531no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4D4×D5311yescharacteristic
364C2×C4C23×D5311yescharacteristic
365C8—21no—
366C8—21no—
367C8—1061no—
368C8—1061no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—532no—
382C22—1061no—
383C22—1061no—
384C22—4244no—
385C22—2121no—
386C22—2121no—
387C22—2121no—
388C22D8×D5312yes—
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—41no—
392C4—532no—
393C4—1061no—
394C4—1061no—
395C4—2121no—
396C4—2121no—
397C4—2121no—
398C4C2×#1171994711yescharacteristic
399C4#1313326211yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—1062no—
405C2—4241no—
406C2#1681126011yescharacteristic
407C2C2×D8×D5311yescharacteristic
408C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3294
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16811273C2semidirect
2. D4×C22⋊D53C2semidirect
3. #16810968C2semidirect
4. #16810937C2semidirect
5. #16810935C2semidirect
6. #16810901C2semidirect
7. #16810857C2semidirect
8. #16810856C2semidirect
9. #16810816C2semidirect
10. #16810755C2semidirect
11. D8×Dic53C2semidirect
12. #16810686C2semidirect
13. C2×#13133138C2semidirect
14. #102×D53C2semidirect
15. C53×#447C2semidirect
19. C2×#11719922C22semidirect
20. C22×#88322C22semidirect
21. C2×C8×D53C22semidirect
22. #13133216C22semidirect
30. #13133188C22semidirect
34. D4×Dic53C22semidirect
36. C2×C4⋊Dic53C22semidirect
37. C4⋊C4×D53C22semidirect
38. #13133151C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133146C22semidirect
42. #13133139C22semidirect
43. #13133139C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
49. #13133137C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
54. C2×D8×C53C22semidirect
55. #121×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
132. D4×C53#41semidirect
144. C22⋊D53D8semidirect
215. #447D53semidirect
220. C53#3294semidirect
227. C2×D8C2×D53semidirect
239. #121C2×D53semidirect
260. #102C2×D53semidirect
262. C22⋊C8C2×D53semidirect
347. D4#11719947semidirect