Finite Group Databasebeta

← #20286341

Subgroups — #20286341

Order 6784.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133249C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3#16811285C211yesmaximal, characteristic, complemented
4#16811266C211yesmaximal, characteristic, complemented
5#16811005C211yesmaximal, characteristic, complemented
6#16810880C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810784C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810771C211yesmaximal, characteristic, complemented
12#16810760C211yesmaximal, characteristic, complemented
13C4×#11719924C211yesmaximal, characteristic, complemented
14#16810700C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133249—24no—
21#13133244—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719924C2211yescharacteristic
26C2×#11719924C2211yescharacteristic, complemented
27C2×D424C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719912C2211yescharacteristic
30C2×#11719912C2211yescharacteristic, complemented
31#13133205—21no—
32#13133205C2211yescharacteristic, complemented
33#13133199—21no—
34#13133197—21no—
35#13133197C2211yescharacteristic
36#13133193C2211yescharacteristic
37#13133192—21no—
38#13133192C2211yescharacteristic, complemented
39#13133190—21no—
40#13133188C2211yescharacteristic
41#13133187—21no—
42C4×D212—21no—
43C4×D212C2211yescharacteristic
44C4×D212C2211yescharacteristic
45C22⋊C4×D53—21no—
46#13133148C2211yescharacteristic, complemented
47#13133142C2211yescharacteristic, complemented
48#13133141C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic
51#13133129C2211yescharacteristic
52#13133129C2211yescharacteristic, complemented
53C4×#88322C2211yescharacteristic, complemented
54C2×Q16×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57C4×Q8×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67#11719949—22no—
68#11719949—22no—
69#11719949—41no—
70#11719949—41no—
71Q8×D53—22no—
72Q8×D53—41no—
73D4×D53—22no—
74D4×D53—41no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212D411yescharacteristic
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×C4×D53C2311yescharacteristic
87C2×Dic106—21no—
88C2×Dic106C2311yescharacteristic
89Q16×C53—22no—
90SD16×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C42×C53—21no—
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100#11719925—22no—
101#11719924—41no—
102#11719924D414yescomplemented
103#11719923—41no—
104#11719922—22no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112#11719921D411yescharacteristic
113C4⋊Dic53—21no—
114C4⋊Dic53D411yescharacteristic
115C4⋊Dic53C2311yescharacteristic
116#11719919—21no—
117C2×#88322C2311yescharacteristic
118C2×#88322C2311yescharacteristic
119D424—22no—
120#11719913—22no—
121#11719912—22no—
122#11719912—22no—
123C23×C53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—42no—
127C22×D53—42no—
128C22×D53—81no—
129C22×D53C2×D411yescharacteristic
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53C2×D412yes—
135D4×C53—21no—
136D4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2411yescharacteristic
143C22⋊D53—42no—
144C22⋊D53—81no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149D212—21no—
150D212—21no—
151D212—21no—
152D212—41no—
153D212—42no—
154D212—42no—
155D212—42no—
156D212C2×D412yes—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—81no—
165Dic106—21no—
166Dic106—21no—
167Dic106—42no—
168C8×C53—21no—
169C8×C53—21no—
170#88322—21no—
171#88322C2×D412yes—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53C22×D411yescharacteristic
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—22no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—84no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53C22×D411yescharacteristic
195C4×C53H(5,𝔽2)11yescharacteristic
196Dic53—21no—
197Dic53—21no—
198Dic53—41no—
199Dic53—41no—
200#3275—531nomaximal, Sylow
201C2×C53—81no—
202C2×C53D4211yescharacteristic
203C2×C53#57511yescharacteristic
204C2×C53#57711yescharacteristic
205D53—41no—
206D53—41no—
207D53—42no—
208D53—81no—
209C2×#137—531no—
210C2×AGL(1,ℤ/8)—531no—
211#548—531no—
212#546—531no—
213#495—531no—
214#470—531no—
215#469—531no—
216#464D5311yescharacteristic, complemented
217#463—531no—
218#452—531no—
219#450—531no—
220#449—531no—
221#448—531no—
222C4×SD16—531no—
223#436—531no—
224C53#327511yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41—531no—
227C22×Q8—531no—
228C22×D4—531no—
229#137—1064no—
230AGL(1,ℤ/8)—1064no—
231C2×Q16—531no—
232C2×Q16C2×D5311yescharacteristic, complemented
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×SD16C2×D5311yescharacteristic
238C2×D8—531no—
239C2×D8—531no—
240C2×M16—531no—
241C2×M16—531no—
242#124—531no—
243#124—1061no—
244#124C2×D5311yescharacteristic
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—531no—
248C22⋊Q8—1061no—
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8C2×D5311yescharacteristic, complemented
255C4×D4—531no—
256C4×D4—531no—
257C4×D4—1061no—
258C2×C22⋊C4—1061no—
259#106—531no—
260C4⋊C8C2×D5311yescharacteristic, complemented
261#103—531no—
262#103—531no—
263#103C2×D5311yescharacteristic, complemented
264#102—531no—
265#102—531no—
266#102C2×D5311yescharacteristic
267C22⋊C8—531no—
268C22⋊C8—531no—
269C4×C8—531no—
270C24—1061no—
271#41—1062no—
272#41—1062no—
273#41—2121no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—1061no—
277C2×Q8—1062no—
278C2×Q8—2121no—
279C2×Q8C22×D5311yescharacteristic
280C2×Q8C22×D5311yescharacteristic
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1062no—
289C2×D4—1061no—
290C2×D4—2121no—
291C2×D4C22×D5311yescharacteristic
292C22×C4—531no—
293C22×C4—531no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298Q16—22no—
299Q16—1062no—
300SD16—22no—
301SD16—534no—
302SD16—1062no—
303SD16—2121no—
304SD16—2121no—
305D8—1062no—
306D8—1062no—
307M16—1062no—
308M16—1062no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—531no—
315C4⋊C4—531no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4C22×D5311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—531no—
322C22⋊C4—531no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C42—21no—
331C42C22×D5311yescharacteristic
332C23—21no—
333C23—531no—
334C23—531no—
335C23—1061no—
336C23—1061no—
337C23—4241no—
338C23—2122no—
339C23—2122no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—1061no—
344Q8—1061no—
345Q8—2122no—
346Q8D4×D5312yes—
347D4—21no—
348D4—21no—
349D4—532no—
350D4—1061no—
351D4—1061no—
352D4—1061no—
353D4—4241no—
354D4—2121no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358D4—2122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—531no—
363C2×C4—531no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—4241no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4D4×D5311yescharacteristic
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—21no—
379C8—532no—
380C8—1061no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—1061no—
385C22—1061no—
386C22—1061no—
387C22—1061no—
388C22—1061no—
389C22—1061no—
390C22—1062no—
391C22—4244no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—1061no—
404C4—1061no—
405C4—2121no—
406C4—2121no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326311yescharacteristic
409C2—81no—
410C2—4241no—
411C2—2121no—
412C2—2121no—
413C2—2122no—
414C2#1681125711yescharacteristic
415C2#1681142011yescharacteristic
416C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3275
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
2. C2×#13133244C2semidirect
3. #16811285C2semidirect
4. #16811266C2semidirect
5. #16811005C2semidirect
6. #16810880C2semidirect
7. #16810875C2semidirect
8. #16810839C2semidirect
9. #16810784C2semidirect
10. #16810775C2semidirect
11. #16810771C2semidirect
12. #16810760C2semidirect
13. C4×#11719924C2semidirect
14. #16810700C2semidirect
15. C53×#464C2semidirect
18. C2×Q8×D53C22semidirect
22. C2×#11719925C22semidirect
26. C2×#11719924C22semidirect
30. C2×#11719912C22semidirect
32. #13133205C22semidirect
38. #13133192C22semidirect
46. #13133148C22semidirect
47. #13133142C22semidirect
48. #13133141C22semidirect
49. #13133139C22semidirect
52. #13133129C22semidirect
53. C4×#88322C22semidirect
54. C2×Q16×C53C22semidirect
57. C4×Q8×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
102. #11719924D4semidirect
216. #464D53semidirect
224. C53#3275semidirect
232. C2×Q16C2×D53semidirect
254. C4×Q8C2×D53semidirect
260. C4⋊C8C2×D53semidirect
263. #103C2×D53semidirect