Finite Group Databasebeta

← #20284392

Subgroups — #20284392

Order 6784.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133241C211yesmaximal, characteristic, complemented
2C2×#13133225C211yesmaximal, characteristic, complemented
3#16811202C211yesmaximal, characteristic, complemented
4#16811189C211yesmaximal, characteristic, complemented
5#16810875C211yesmaximal, characteristic, complemented
6#16810869C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810857C211yesmaximal, characteristic, complemented
9#16810856C211yesmaximal, characteristic, complemented
10#16810853C211yesmaximal, characteristic, complemented
11#16810850C211yesmaximal, characteristic, complemented
12#16810662C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#469C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133241—24no—
20C2×#11719913C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133225—24no—
24C2×#11719912C2211yescharacteristic
25C2×#11719912C2211yescharacteristic, complemented
26C22×#88322C2211yescharacteristic, complemented
27C2×C8×D53C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic, complemented
29#13133213—21no—
30#13133203—21no—
31#13133197C2211yescharacteristic, complemented
32#13133196C2211yescharacteristic, complemented
33#13133194—21no—
34#13133192C2211yescharacteristic
35#13133191C2211yescharacteristic
36#13133190—21no—
37#13133189—21no—
38#13133188—21no—
39#13133188C2211yescharacteristic
40#13133188C2211yescharacteristic, complemented
41#13133187—21no—
42#13133185—21no—
43#13133175C2211yescharacteristic
44C4⋊C4×D53C2211yescharacteristic
45#13133151C2211yescharacteristic, complemented
46#13133150C2211yescharacteristic, complemented
47#13133148C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic
51#13133138C2211yescharacteristic
52#13133138C2211yescharacteristic, complemented
53#13133137C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55C2×M16×C53C2211yescharacteristic, complemented
56#121×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic, complemented
58#106×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53D411yescharacteristic
73C2×C22⋊D53D411yescharacteristic
74C22×Dic53—21no—
75#11719947—22no—
76#11719947—22no—
77D4×D53—22no—
78D4×D53—22no—
79#11719945—22no—
80#11719945—22no—
81C2×D212—21no—
82C2×D212D411yescharacteristic
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×C4×D53C2311yescharacteristic
87C2×Dic106D411yescharacteristic
88C2×Dic106C2311yescharacteristic
89D8×C53—22no—
90M16×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100#11719923—22no—
101#11719922—22no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53C2311yescharacteristic
108C4⋊Dic53C2311yescharacteristic
109#11719919—21no—
110#11719919—21no—
111C4×Dic53—21no—
112C2×#88322—22no—
113C2×#88322C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115#11719913—22no—
116#11719912—22no—
117#11719912D414yescomplemented
118C8×D53—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C2×D411yescharacteristic
123C22×D53—21no—
124C22×D53—42no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53C2×D411yescharacteristic
128C22×D53C2×D411yescharacteristic
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—81no—
134D4×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53—22no—
142C22⋊D53—22no—
143C22⋊D53—42no—
144C22⋊D53—42no—
145C22⋊D53—81no—
146C22⋊D53—81no—
147C22⋊D53—81no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—22no—
155D212—21no—
156D212—21no—
157D212—81no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—81no—
161C4×D53C2×D412yes—
162C4×D53#4112yes—
163Dic106—22no—
164Dic106—21no—
165Dic106—21no—
166C8×C53—21no—
167C8×C53C2×D412yes—
168#88322—21no—
169#88322C2×D412yes—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53C22×D411yescharacteristic
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—84no—
189C2×D53#12112yes—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53C22×D411yescharacteristic
194C4×C53C2×#4111yescharacteristic
195Dic53—21no—
196Dic53—41no—
197Dic53—41no—
198Dic53—41no—
199Dic53#12112yes—
200#3062—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—81no—
203C2×C53—81no—
204C2×C53C2×#12111yescharacteristic
205C2×C53#57611yescharacteristic
206C2×C53#57711yescharacteristic
207D53—22no—
208D53—41no—
209D53—81no—
210C2×AGL(1,ℤ/8)—531no—
211C2×#131—531no—
212#537—531no—
213#535—531no—
214#469—531no—
215#469D5311yescharacteristic, complemented
216#468—531no—
217#468—531no—
218#466—531no—
219#466—531no—
220#465—531no—
221#465—531no—
222#428—531no—
223#416—531no—
224C2×#102—531no—
225C53#306211yescharacteristic, complemented, Sylow
226C2×#41—531no—
227C2×#41—531no—
228C22×D4—531no—
229AGL(1,ℤ/8)—1064no—
230C2×SD16—531no—
231C2×SD16—531no—
232C2×D8—531no—
233C2×D8C2×D5311yescharacteristic
234#131—1064no—
235C2×M16—531no—
236C2×M16—531no—
237C2×M16C2×D5311yescharacteristic, complemented
238C22×C8—531no—
239C22×C8—531no—
240C22×C8—531no—
241#127—1061no—
242#124—1061no—
243C22⋊Q8—531no—
244C22⋊Q8—531no—
245C22⋊Q8—1061no—
246#121—531no—
247#121—531no—
248#121—531no—
249#121—531no—
250#121—1061no—
251#121—1061no—
252#121—1061no—
253#121C2×D5311yescharacteristic
254#121C2×D5311yescharacteristic, complemented
255#120—1061no—
256#120—1061no—
257#117—531no—
258C2×C4⋊C4—531no—
259#107—531no—
260#107—531no—
261#106—531no—
262#106C2×D5311yescharacteristic, complemented
263#103—531no—
264#103—531no—
265#102—531no—
266#102—531no—
267#102—531no—
268#102—531no—
269#102C2×D5311yescharacteristic
270#102C2×D5311yescharacteristic, complemented
271C24—1061no—
272#41—1062no—
273#41—1062no—
274#41—1062no—
275#41—1062no—
276C2×Q8—531no—
277C2×Q8—531no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—531no—
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1061no—
285C2×D4—1062no—
286C2×D4—1062no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4C22×D5311yescharacteristic
293C2×D4C22×D5311yescharacteristic
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4C22×D5311yescharacteristic
299SD16—1062no—
300SD16—1062no—
301D8—22no—
302D8—1062no—
303M16—22no—
304M16—534no—
305M16—1062no—
306C2×C8—531no—
307C2×C8—531no—
308C2×C8—1062no—
309C2×C8—1062no—
310C2×C8—1062no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—531no—
314C4⋊C4—531no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4C22×D5311yescharacteristic
319C4⋊C4C22×D5311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C42—1061no—
330C23—21no—
331C23—21no—
332C23—531no—
333C23—531no—
334C23—1061no—
335C23—4241no—
336C23—4241no—
337C23—2122no—
338C23D4×D5311yescharacteristic
339Q8—1062no—
340Q8—1061no—
341Q8—1061no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—1062no—
349D4—1062no—
350D4—1062no—
351D4—1061no—
352D4—1061no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—2122no—
358D4—2122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—531no—
364C2×C4—532no—
365C2×C4—532no—
366C2×C4—531no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—4241no—
373C2×C4—2122no—
374C2×C4D4×D5311yescharacteristic
375C2×C4C23×D5311yescharacteristic
376C8—21no—
377C8—532no—
378C8—1061no—
379C8D4×D5312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—532no—
390C22—1061no—
391C22—1061no—
392C22—1061no—
393C22—1061no—
394C22—1061no—
395C22—4244no—
396C22—2121no—
397C22—2121no—
398C22—2121no—
399C22C2×D4×D5311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—532no—
404C4—1061no—
405C4—2121no—
406C4—2121no—
407C4—2121no—
408C4C2×D4×D5311yescharacteristic
409C4#41×D5311yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—1062no—
414C2—4241no—
415C2—2121no—
416C2#121×D5311yescharacteristic
417C2#1681142711yescharacteristic
418C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3062
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. C2×#13133225C2semidirect
3. #16811202C2semidirect
4. #16811189C2semidirect
5. #16810875C2semidirect
6. #16810869C2semidirect
7. #16810868C2semidirect
8. #16810857C2semidirect
9. #16810856C2semidirect
10. #16810853C2semidirect
11. #16810850C2semidirect
12. #16810662C2semidirect
13. #16810616C2semidirect
14. #102×D53C2semidirect
15. C53×#469C2semidirect
25. C2×#11719912C22semidirect
26. C22×#88322C22semidirect
28. C2×C8×D53C22semidirect
31. #13133197C22semidirect
32. #13133196C22semidirect
40. #13133188C22semidirect
45. #13133151C22semidirect
46. #13133150C22semidirect
47. #13133148C22semidirect
48. #13133142C22semidirect
49. #13133140C22semidirect
52. #13133138C22semidirect
55. C2×M16×C53C22semidirect
57. #121×C53C22semidirect
58. #106×C53C22semidirect
60. #102×C53C22semidirect
117. #11719912D4semidirect
215. #469D53semidirect
225. C53#3062semidirect
237. C2×M16C2×D53semidirect
254. #121C2×D53semidirect
262. #106C2×D53semidirect
270. #102C2×D53semidirect