Finite Group Databasebeta

← #20286343

Subgroups — #20286343

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×#13133245C211yesmaximal, characteristic, complemented
3#16811283C211yesmaximal, characteristic, complemented
4#16811264C211yesmaximal, characteristic, complemented
5#16811008C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810848C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810773C211yesmaximal, characteristic, complemented
11#16810771C211yesmaximal, characteristic, complemented
12#16810763C211yesmaximal, characteristic, complemented
13#16810710C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C53×#470C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19C22×D212C2211yescharacteristic
20#13133249—24no—
21#13133245—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×#11719924C2211yescharacteristic
26C2×#11719924C2211yescharacteristic
27C2×#11719924C2211yescharacteristic, complemented
28C2×D424C2211yescharacteristic
29C2×D424C2211yescharacteristic
30C2×#11719912C2211yescharacteristic, complemented
31#13133206C2211yescharacteristic
32#13133206C2211yescharacteristic, complemented
33#13133203—21no—
34#13133202—21no—
35#13133195—21no—
36#13133192—21no—
37#13133192C2211yescharacteristic
38#13133192C2211yescharacteristic, complemented
39#13133190—21no—
40#13133187—21no—
41#13133182C2211yescharacteristic, complemented
42#13133179—21no—
43#13133179C2211yescharacteristic
44C4×C22⋊D53C2211yescharacteristic
45C22⋊C4×D53—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic
52#13133129C2211yescharacteristic, complemented
53C8×Dic53C2211yescharacteristic, complemented
54C2×Q16×C53C2211yescharacteristic, complemented
55C2×M16×C53C2211yescharacteristic
56C22⋊Q8×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic
58#106×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#103×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53D411yescharacteristic
67#11719949—22no—
68#11719949—41no—
69Q8×D53—22no—
70Q8×D53—41no—
71#11719945—22no—
72#11719945—41no—
73C2×D212—22no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—41no—
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53C2311yescharacteristic
87C2×Dic106—21no—
88C2×Dic106C2311yescharacteristic
89Q16×C53—22no—
90M16×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53C2311yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊Dic53D411yescharacteristic
100#11719925—22no—
101#11719924—22no—
102#11719924—41no—
103#11719924—41no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118D424—22no—
119D424—22no—
120#11719913—22no—
121#11719913D414yescomplemented
122#11719912—22no—
123C23×C53C2×D411yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2411yescharacteristic
143C22⋊D53—41no—
144C22⋊D53—42no—
145C2×Dic53—21no—
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149D212—21no—
150D212—21no—
151D212—21no—
152D212—21no—
153D212—21no—
154D212—42no—
155D212—42no—
156D212—42no—
157D212—81no—
158D212C2×D412yes—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—81no—
165Dic106—21no—
166Dic106—42no—
167Dic106C2×D412yes—
168C8×C53—21no—
169C8×C53C2×D412yes—
170#88322—21no—
171#88322—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53C22×D411yescharacteristic
176C2×D53—21no—
177C2×D53—22no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—84no—
189C4×C53—21no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53C22×D411yescharacteristic
195C4×C53H(5,𝔽2)11yescharacteristic
196Dic53—22no—
197Dic53—21no—
198Dic53—21no—
199Dic53—21no—
200#3275—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53D4211yescharacteristic
203C2×C53#57511yescharacteristic
204C2×C53#57711yescharacteristic
205D53—41no—
206D53—42no—
207D53—81no—
208D53—81no—
209C2×#137—531no—
210C2×AGL(1,ℤ/8)—531no—
211#548—531no—
212#546—531no—
213#495—531no—
214#470D5311yescharacteristic, complemented
215#469—531no—
216#464—531no—
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×SD16—531no—
238C2×D8—531no—
239C2×D8—531no—
240C2×M16—531no—
241C2×M16C2×D5311yescharacteristic
242#124—531no—
243#124—531no—
244#124—1061no—
245#123—1061no—
246C22⋊Q8—1061no—
247C22⋊Q8C2×D5311yescharacteristic
248C22⋊Q8C2×D5311yescharacteristic
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8—531no—
255C4×D4—531no—
256C4×D4—531no—
257C4×D4—1061no—
258C2×C22⋊C4—1061no—
259#106C2×D5311yescharacteristic, complemented
260C4⋊C8—531no—
261#103—531no—
262#103C2×D5311yescharacteristic, complemented
263#103C2×D5311yescharacteristic, complemented
264#102—531no—
265#102—531no—
266#102—531no—
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—531no—
285C2×D4—1062no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—2121no—
292C22×C4—531no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4C22×D5311yescharacteristic
298Q16—22no—
299Q16—1062no—
300SD16—534no—
301SD16—1062no—
302SD16—1062no—
303SD16—2121no—
304SD16—2121no—
305D8—1062no—
306D8—1062no—
307M16—22no—
308M16—1062no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—531no—
316C4⋊C4—1061no—
317C4⋊C4C22×D5311yescharacteristic
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—531no—
331C42—1061no—
332C23—531no—
333C23—1061no—
334C23—1061no—
335C23—1061no—
336C23—4241no—
337C23—2122no—
338C23—2122no—
339C23D4×D5311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—532no—
345Q8—1061no—
346Q8—2122no—
347D4—532no—
348D4—1061no—
349D4—1061no—
350D4—1061no—
351D4—1061no—
352D4—1061no—
353D4—4241no—
354D4—2122no—
355D4—2122no—
356D4—2122no—
357D4—2121no—
358D4—2122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—531no—
366C2×C4—531no—
367C2×C4—531no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—4241no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—1061no—
379C8—1061no—
380C8D4×D5312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—1061no—
385C22—1061no—
386C22—1061no—
387C22—1062no—
388C22—4244no—
389C22—2121no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—1061no—
404C4—1061no—
405C4—1061no—
406C4—1062no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326311yescharacteristic
409C2—41no—
410C2—4241no—
411C2—4241no—
412C2—2121no—
413C2—2122no—
414C2#1681125711yescharacteristic
415C2#1681142211yescharacteristic
416C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#470C2
socleC22×C53C22×D4
hypercenter#470D53
hypocenterC53#3275
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
2. C2×#13133245C2semidirect
3. #16811283C2semidirect
4. #16811264C2semidirect
5. #16811008C2semidirect
6. #16810877C2semidirect
7. #16810848C2semidirect
8. #16810841C2semidirect
9. #16810783C2semidirect
10. #16810773C2semidirect
11. #16810771C2semidirect
12. #16810763C2semidirect
13. #16810710C2semidirect
14. #16810699C2semidirect
15. C53×#470C2semidirect
18. C2×Q8×D53C22semidirect
22. C2×#11719925C22semidirect
27. C2×#11719924C22semidirect
30. C2×#11719912C22semidirect
32. #13133206C22semidirect
38. #13133192C22semidirect
41. #13133182C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
50. #13133137C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C2×Q16×C53C22semidirect
58. #106×C53C22semidirect
59. #103×C53C22semidirect
60. #103×C53C22semidirect
121. #11719913D4semidirect
214. #470D53semidirect
224. C53#3275semidirect
232. C2×Q16C2×D53semidirect
259. #106C2×D53semidirect
262. #103C2×D53semidirect
263. #103C2×D53semidirect