Finite Group Databasebeta

← #20285530

Subgroups — #20285530

Order 6784.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811136C211yesmaximal, characteristic, complemented
2#121×D53C211yesmaximal, characteristic, complemented
3#16810952C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810946C211yesmaximal, characteristic, complemented
6#16810938C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810874C211yesmaximal, characteristic, complemented
9#16810871C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810771C211yesmaximal, characteristic, complemented
12#16810767C211yesmaximal, characteristic, complemented
13#16810758C211yesmaximal, characteristic, complemented
14#16810586C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C2×#11719924C2211yescharacteristic, complemented
21C2×D424C2211yescharacteristic, complemented
22C2×#11719922C2211yescharacteristic, complemented
23C2×#11719917C2211yescharacteristic, complemented
24#13133202—21no—
25#13133193C2211yescharacteristic
26#13133192C2211yescharacteristic
27#13133192C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic, complemented
30#13133190—21no—
31#13133189—21no—
32#13133188C2211yescharacteristic
33#13133187—21no—
34#13133179—21no—
35#13133177—21no—
36#13133175C2211yescharacteristic, complemented
37C4⋊C4×D53C2211yescharacteristic
38C22⋊C4×D53—21no—
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×M16×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53D411yescharacteristic
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⋊D53—21no—
69C22×Dic53D411yescharacteristic
70#11719949—41no—
71#11719949—41no—
72D4×D53—41no—
73D4×D53—41no—
74D4×D53—44no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53D411yescharacteristic
85C2×C4×D53D411yescharacteristic
86C2×C4×D53C2311yescharacteristic
87C2×C4×D53C2311yescharacteristic
88SD16×C53—41no—
89M16×C53—41no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊Dic53—21no—
98#11719924—41no—
99#11719922—41no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110C4×Dic53—21no—
111#11719917—41no—
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114D424—41no—
115C23×C53—21no—
116C23×C53C2×D411yescharacteristic
117C22×D53—22no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—42no—
123C22×D53—42no—
124C22×D53—42no—
125C22×D53C2×D411yescharacteristic
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—42no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2411yescharacteristic
139C22⋊D53—42no—
140C22⋊D53—42no—
141C22⋊D53—42no—
142C22⋊D53—42no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—41no—
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148D212—21no—
149D212—21no—
150D212—21no—
151D212—21no—
152D212—42no—
153D212—42no—
154D212—42no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—41no—
160C4×D53—41no—
161C4×D53—42no—
162C4×D53—42no—
163C8×C53—21no—
164C8×C53—21no—
165#88322—21no—
166#88322—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53C22×D411yescharacteristic
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—44no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—84no—
187C2×D53AGL(1,ℤ/8)12yes—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53H(5,𝔽2)11yescharacteristic
193C4×C53H(5,𝔽2)11yescharacteristic
194Dic53—21no—
195Dic53—21no—
196Dic53—41no—
197Dic53—41no—
198#3194—531nomaximal, Sylow
199C2×C53—41no—
200C2×C53—81no—
201C2×C53#53411yescharacteristic
202C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C53#57511yescharacteristic
204D53—22no—
205D53—41no—
206D53—81no—
207D53—81no—
208#525—531no—
209C2×#121—531no—
210#483—531no—
211#482—531no—
212#482—531no—
213#480—531no—
214#469—531no—
215#469—531no—
216#468—531no—
217#468D5311yescharacteristic, complemented
218#450—531no—
219#449—531no—
220#449—531no—
221#447—531no—
222#407—531no—
223C53#319411yescharacteristic, complemented, Sylow
224C2×#41—531no—
225C22×D4—531no—
226C22×D4—1061no—
227C23×C4—531no—
228C2×SD16—531no—
229C2×SD16C2×D5311yescharacteristic, complemented
230C2×D8—531no—
231C2×D8—531no—
232C2×M16—531no—
233C2×M16C2×D5311yescharacteristic, complemented
234#123—1061no—
235C22⋊Q8—531no—
236C22⋊Q8C2×D5311yescharacteristic, complemented
237#121—531no—
238#121—531no—
239#121—531no—
240#121—531no—
241#121—531no—
242#121—1061no—
243#121—1061no—
244#121C2×D5311yescharacteristic
245#120—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248#117—531no—
249C2×C4⋊C4—531no—
250C2×C22⋊C4—1061no—
251#107—531no—
252#107C2×D5311yescharacteristic, complemented
253#106—531no—
254#106—531no—
255#103—531no—
256#103C2×D5311yescharacteristic, complemented
257#102—531no—
258#102—531no—
259#102—531no—
260#102—531no—
261#102—531no—
262#102C2×D5311yescharacteristic, complemented
263C22⋊C8—531no—
264C22⋊C8—531no—
265C22⋊C8—531no—
266C22⋊C8—531no—
267C24—531no—
268C24—1061no—
269#41—2121no—
270#41—2121no—
271C2×Q8C22×D5311yescharacteristic
272C2×D4—21no—
273C2×D4—531no—
274C2×D4—531no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2124no—
285C2×D4C22×D5311yescharacteristic
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4C22×D5311yescharacteristic
296SD16—41no—
297SD16—2121no—
298D8—2121no—
299D8—2121no—
300M16—41no—
301M16—2121no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8C22×D5311yescharacteristic
305C2×C8C22×D5311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—531no—
308C4⋊C4—531no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4C22×D5311yescharacteristic
312C4⋊C4C22×D5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C42—1061no—
323C23—21no—
324C23—531no—
325C23—531no—
326C23—1062no—
327C23—1061no—
328C23—1061no—
329C23—2121no—
330C23—2121no—
331C23—2122no—
332C23—2122no—
333C23—2122no—
334C23D4×D5311yescharacteristic
335Q8—21no—
336Q8—21no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—1061no—
341D4—1061no—
342D4—1061no—
343D4—1061no—
344D4—2122no—
345D4—2122no—
346D4—2122no—
347D4—2122no—
348D4—2122no—
349D4—2122no—
350D4—2122no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—531no—
357C2×C4—531no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2121no—
369C2×C4D4×D5311yescharacteristic
370C2×C4C23×D5311yescharacteristic
371C8—21no—
372C8—21no—
373C8—1061no—
374C8—1061no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—532no—
382C22—1061no—
383C22—1061no—
384C22—1061no—
385C22—1061no—
386C22—1061no—
387C22—4244no—
388C22—2124no—
389C22—2121no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22C2×D4×D5311yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—1061no—
401C4—1061no—
402C4—2121no—
403C4—2121no—
404C4#1313326211yescharacteristic
405C4#1313326311yescharacteristic
406C2—41no—
407C2—81no—
408C2—1062no—
409C2—4241no—
410C2—4241no—
411C2—2121no—
412C2#1681118111yescharacteristic
413C2AGL(1,ℤ/8)×D5311yescharacteristic
414C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3194
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #16811136C2semidirect
2. #121×D53C2semidirect
3. #16810952C2semidirect
4. #16810950C2semidirect
5. #16810946C2semidirect
6. #16810938C2semidirect
7. #16810876C2semidirect
8. #16810874C2semidirect
9. #16810871C2semidirect
10. #16810772C2semidirect
11. #16810771C2semidirect
12. #16810767C2semidirect
13. #16810758C2semidirect
14. #16810586C2semidirect
15. C53×#468C2semidirect
16. C2×#11719949C22semidirect
20. C2×#11719924C22semidirect
21. C2×D424C22semidirect
22. C2×#11719922C22semidirect
23. C2×#11719917C22semidirect
29. #13133191C22semidirect
36. #13133175C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133141C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133137C22semidirect
47. #13133137C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
52. C2×SD16×C53C22semidirect
53. C2×M16×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #107×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
217. #468D53semidirect
223. C53#3194semidirect
229. C2×SD16C2×D53semidirect
233. C2×M16C2×D53semidirect
236. C22⋊Q8C2×D53semidirect
252. #107C2×D53semidirect
256. #103C2×D53semidirect
262. #102C2×D53semidirect