Finite Group Databasebeta

← #20286237

Subgroups — #20286237

Order 6784.

500 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×#13133244C211yesmaximal, characteristic, complemented
2C2×#13133239C211yesmaximal, characteristic, complemented
3#16811275C211yesmaximal, characteristic, complemented
4#16811270C211yesmaximal, characteristic, complemented
5#16811014C211yesmaximal, characteristic, complemented
6#16810882C211yesmaximal, characteristic, complemented
7#16810837C211yesmaximal, characteristic, complemented
8#16810827C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810775C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810734C211yesmaximal, characteristic, complemented
14#16810694C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719945C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133244—24no—
21#13133239—24no—
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719925C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic, complemented
26C2×#11719923C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×D424C2211yescharacteristic
29C2×#11719922C2211yescharacteristic
30C2×#11719912C2211yescharacteristic
31#13133215C2211yescharacteristic, complemented
32#13133204C2211yescharacteristic
33#13133202—21no—
34#13133199—21no—
35#13133199—21no—
36#13133197C2211yescharacteristic
37#13133197C2211yescharacteristic
38#13133190—21no—
39#13133188—21no—
40#13133185—21no—
41#13133179C2211yescharacteristic
42#13133179C2211yescharacteristic, complemented
43C4×C22⋊D53C2211yescharacteristic
44C4⋊C4×D53—21no—
45C22⋊C4×D53—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53#13133127C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C22×C8×C53C2211yescharacteristic, complemented
56C22⋊Q8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#106×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70#11719949—22no—
71#11719949—41no—
72D4×D53—22no—
73D4×D53—41no—
74#11719945—22no—
75#11719945—22no—
76#11719945—41no—
77#11719945—41no—
78C2×D212—21no—
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88C2×Dic106C2311yescharacteristic
89SD16×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53C2311yescharacteristic
95C4⋊C4×C53C2311yescharacteristic
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53D411yescharacteristic
101#11719925—41no—
102#11719924—22no—
103#11719923—41no—
104#11719922—22no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921D411yescharacteristic
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112#11719919—21no—
113#11719919—21no—
114#11719919D411yescharacteristic
115C4×Dic53C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118Dic212—22no—
119D424D414yescomplemented
120#11719913—22no—
121#11719913—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—81no—
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—
145C22⋊D53—42no—
146C22⋊D53—42no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53C2×D411yescharacteristic
150C2×Dic53C2×D411yescharacteristic
151C2×Dic53C2×D411yescharacteristic
152D212—21no—
153D212—21no—
154D212—42no—
155D212C2×D412yes—
156D212C2×D412yes—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—81no—
164Dic106—21no—
165Dic106—21no—
166Dic106—21no—
167Dic106—21no—
168C8×C53—21no—
169C8×C53C2×D412yes—
170#88322—21no—
171#88322—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53C22×D411yescharacteristic
179C2×D53—21no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—21no—
183C2×D53—21no—
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×C53C22×D411yescharacteristic
194C4×C53H(5,𝔽2)11yescharacteristic
195Dic53—21no—
196Dic53—21no—
197Dic53—21no—
198Dic53—41no—
199Dic53—41no—
200#3267—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—81no—
203C2×C53D4211yescharacteristic
204C2×C53#57511yescharacteristic
205C2×C53#57711yescharacteristic
206D53—41no—
207D53—42no—
208D53—42no—
209C2×AGL(1,ℤ/8)—531no—
210C2×#135—531no—
211#547—531no—
212#546—531no—
213#497—531no—
214#470—531no—
215#465D5311yescharacteristic, complemented
216#463—531no—
217#461—531no—
218#452—531no—
219#452—531no—
220#450—531no—
221#449—531no—
222#442—531no—
223#435—531no—
224C53#326711yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41—531no—
227C2×#41—531no—
228C22×D4—531no—
229AGL(1,ℤ/8)—1064no—
230#135—1064no—
231C2×Q16—531no—
232C2×Q16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—531no—
237C2×SD16C2×D5311yescharacteristic
238C2×D8—531no—
239C2×D8—531no—
240C2×M16—531no—
241C22×C8C2×D5311yescharacteristic, complemented
242C4⋊Q8—531no—
243#124—531no—
244#123—1061no—
245#123—1061no—
246#123—1061no—
247C22⋊Q8—531no—
248C22⋊Q8—531no—
249C22⋊Q8C2×D5311yescharacteristic, complemented
250#121—1061no—
251#121—1061no—
252#121C2×D5311yescharacteristic
253#120—1061no—
254C4×D4—531no—
255C4×D4—531no—
256C4×D4—531no—
257C2×C4⋊C4—1061no—
258C2×C22⋊C4—1061no—
259#106C2×D5311yescharacteristic, complemented
260C4⋊C8—531no—
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—
269#97—531no—
270C24—1061no—
271#41—1062no—
272#41—1062no—
273#41—1062no—
274#41—2121no—
275#41—2121no—
276#41—2121no—
277C2×Q8—531no—
278C2×Q8—531no—
279C2×Q8C22×D5311yescharacteristic
280C2×D4—21no—
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1062no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—2121no—
290C2×D4C22×D5311yescharacteristic
291C22×C4—531no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4C22×D5311yescharacteristic
298Q16—1062no—
299Q16—2121no—
300SD16—22no—
301SD16—1062no—
302SD16—1062no—
303SD16—1062no—
304SD16—2121no—
305D8—534no—
306D8—1062no—
307M16—1062no—
308C2×C8—22no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—531no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C4⋊C4C22×D5311yescharacteristic
320C4⋊C4C22×D5311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—531no—
324C22⋊C4—531no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C22⋊C4—1061no—
331C42—531no—
332C23—21no—
333C23—531no—
334C23—1061no—
335C23—1061no—
336C23—4241no—
337C23—2122no—
338C23—2122no—
339C23D4×D5311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—1061no—
343Q8—1061no—
344Q8—1061no—
345Q8—1061no—
346D4—21no—
347D4—21no—
348D4—81no—
349D4—532no—
350D4—532no—
351D4—1061no—
352D4—1061no—
353D4—2122no—
354D4—2121no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—531no—
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×C4—2122no—
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—1061no—
379C8—1061no—
380C8D4×D5312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—1061no—
388C22—1061no—
389C22—1061no—
390C22—1062no—
391C22—1062no—
392C22—4244no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—1061no—
403C4—1061no—
404C4—1061no—
405C4—2121no—
406C4—2121no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326311yescharacteristic
409C2—41no—
410C2—81no—
411C2—2121no—
412C2—2122no—
413C2—2122no—
414C2#1681125711yescharacteristic
415C2#1681142011yescharacteristic
416C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3267
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. C2×#13133244C2semidirect
2. C2×#13133239C2semidirect
3. #16811275C2semidirect
4. #16811270C2semidirect
5. #16811014C2semidirect
6. #16810882C2semidirect
7. #16810837C2semidirect
8. #16810827C2semidirect
9. #16810788C2semidirect
10. #16810783C2semidirect
11. #16810775C2semidirect
12. #16810766C2semidirect
13. #16810734C2semidirect
14. #16810694C2semidirect
15. C53×#465C2semidirect
18. C2×#11719945C22semidirect
22. C2×Dic212C22semidirect
23. C2×#11719925C22semidirect
25. C2×#11719913C22semidirect
31. #13133215C22semidirect
42. #13133179C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
48. #13133140C22semidirect
50. #13133137C22semidirect
51. C22⋊#88322C22semidirect
53. #13133127C22semidirect
55. C22×C8×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #106×C53C22semidirect
59. #103×C53C22semidirect
119. D424D4semidirect
215. #465D53semidirect
224. C53#3267semidirect
241. C22×C8C2×D53semidirect
249. C22⋊Q8C2×D53semidirect
259. #106C2×D53semidirect
263. #103C2×D53semidirect