Finite Group Databasebeta

← #20286206

Subgroups — #20286206

Order 6784.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133243C211yesmaximal, characteristic, complemented
2C22×#11719923C211yesmaximal, characteristic, complemented
3#16811282C211yesmaximal, characteristic, complemented
4#16811010C211yesmaximal, characteristic, complemented
5#16810867C211yesmaximal, characteristic, complemented
6#16810857C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810820C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic, complemented
10#16810766C211yesmaximal, characteristic, complemented
11#16810761C211yesmaximal, characteristic, complemented
12#16810757C211yesmaximal, characteristic, complemented
13#16810723C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15D42×C53C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C22×Dic106C2211yescharacteristic, complemented
18#13133243—24no—
19C2×#11719923—24no—
20C2×#11719923C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719923C2211yescharacteristic, complemented
23C2×#11719923C2211yescharacteristic, complemented
24C2×#11719923C2211yescharacteristic, complemented
25C2×#11719923C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic
27C2×#11719922C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic, complemented
29C2×#11719917C2211yescharacteristic, complemented
30C22×#88322C2211yescharacteristic, complemented
31#13133205C2211yescharacteristic, complemented
32#13133203—21no—
33#13133196C2211yescharacteristic, complemented
34#13133195—21no—
35#13133191C2211yescharacteristic, complemented
36#13133189—21no—
37C4×Dic106C2211yescharacteristic, complemented
38#13133177—21no—
39#13133150C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133140C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47C22⋊#88322C2211yescharacteristic, complemented
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C22×D4×C53—21no—
51C22×D4×C53C2211yescharacteristic
52C22×D4×C53C2211yescharacteristic
53#127×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53C2211yescharacteristic
56#121×C53C2211yescharacteristic
57#120×C53—21no—
58#120×C53—21no—
59C4×D4×C53C2211yescharacteristic
60C4×D4×C53C2211yescharacteristic
61C24×C53—21no—
62C24×C53—21no—
63C2×D4×C53—22no—
64C2×D4×C53—22no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—44no—
74C2×D4×C53D411yescharacteristic
75C2×D4×C53C2311yescharacteristic
76C2×D4×C53C2311yescharacteristic
77C2×D4×C53C2311yescharacteristic
78C22×C4×C53—21no—
79C22×C4×C53C2311yescharacteristic
80C22×C4×C53C2311yescharacteristic
81C2×C22⋊D53—21no—
82C2×C22⋊D53—21no—
83C22×Dic53—21no—
84#11719945—22no—
85#11719945—41no—
86C2×D212C2311yescharacteristic
87C2×C4×D53—21no—
88C2×Dic106—22no—
89C2×Dic106—21no—
90C2×Dic106—41no—
91C2×Dic106C2311yescharacteristic
92C2×Dic106C2311yescharacteristic
93C4⋊C4×C53D411yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53D411yescharacteristic
99C22⋊C4×C53D411yescharacteristic
100C42×C53C2311yescharacteristic
101C22⋊Dic53—21no—
102#11719923—22no—
103#11719923—22no—
104#11719923—22no—
105#11719923—22no—
106#11719923—41no—
107#11719923D414yescomplemented
108#11719922—22no—
109#11719922—22no—
110#11719922—41no—
111#11719921—21no—
112#11719921—21no—
113#11719921—21no—
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116#11719919—21no—
117C4×Dic53—21no—
118#11719917—22no—
119C2×#88322—22no—
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C2×#88322C2311yescharacteristic
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—81no—
132C23×C53—81no—
133C23×C53C2×D411yescharacteristic
134C23×C53C2×D411yescharacteristic
135C22×D53—21no—
136D4×C53—22no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—21no—
141D4×C53—21no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53—42no—
146D4×C53—42no—
147D4×C53—81no—
148D4×C53C2×D412yes—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—42no—
155C2×C4×C53C2×D411yescharacteristic
156C2×C4×C53C2×D411yescharacteristic
157C2×C4×C53C2×D411yescharacteristic
158C2×C4×C53C2×D411yescharacteristic
159C2×C4×C53C2411yescharacteristic
160C22⋊D53—42no—
161C22⋊D53—42no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—42no—
166D212—21no—
167D212—21no—
168C4×D53—42no—
169Dic106—21no—
170Dic106—21no—
171Dic106—21no—
172Dic106—42no—
173Dic106C2×D412yescomplemented
174#88322—21no—
175#88322—21no—
176#88322—21no—
177#88322C2×D412yescomplemented
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—22no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—84no—
193C22×C53—84no—
194C22×C53AGL(1,ℤ/8)12yescomplemented
195C22×C53C22×D411yescharacteristic
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—41no—
204C4×C53C22×D411yescharacteristic
205C4×C53H(5,𝔽2)11yescharacteristic
206Dic53—22no—
207Dic53—41no—
208Dic53—41no—
209#3265—531nomaximal, Sylow
210C2×C53—22no—
211C2×C53—41no—
212C2×C53—42no—
213C2×C53—81no—
214C2×C53—81no—
215C2×C53D4211yescharacteristic
216C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
217C2×C53#57711yescharacteristic
218D53—81no—
219C2×AGL(1,ℤ/8)—531no—
220C22×SD16—531no—
221#548—531no—
222D42D5311yescharacteristic, complemented
223#496—531no—
224#468—531no—
225#466—531no—
226#463—531no—
227#460—531no—
228#450—531no—
229#449—531no—
230#448—531no—
231#447—531no—
232#440—531no—
233#435—531no—
234C53#326511yescharacteristic, complemented, Sylow
235C2×#41—531no—
236C22×Q8—531no—
237C22×D4—21no—
238C22×D4C2×D5311yescharacteristic
239C22×D4C2×D5311yescharacteristic
240AGL(1,ℤ/8)—1064no—
241C2×SD16—531no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16—531no—
245C2×SD16—531no—
246C2×SD16—531no—
247C2×SD16—1064no—
248C2×D8—531no—
249C2×D8—531no—
250C2×D8—531no—
251C2×M16—531no—
252C22×C8—531no—
253#127C2×D5311yescharacteristic
254#124—531no—
255#124—1061no—
256C22⋊Q8—531no—
257C22⋊Q8—1061no—
258#121—21no—
259#121—531no—
260#121—1061no—
261#121C2×D5311yescharacteristic
262#121C2×D5311yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—531no—
266C4×D4—1061no—
267C4×D4C2×D5311yescharacteristic
268C4×D4C2×D5311yescharacteristic
269#107—531no—
270C4⋊C8—531no—
271#103—531no—
272#103—531no—
273#102—531no—
274#102—531no—
275#102—531no—
276#102—531no—
277C22⋊C8—531no—
278C22⋊C8—531no—
279#97—531no—
280C24—21no—
281C24—21no—
282#41—1062no—
283#41—2121no—
284C2×Q8—531no—
285C2×Q8—531no—
286C2×Q8—1062no—
287C2×Q8—1061no—
288C2×Q8—2121no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—531no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4C22⋊D5311yescharacteristic
304C2×D4C22×D5311yescharacteristic
305C2×D4C22×D5311yescharacteristic
306C2×D4C22×D5311yescharacteristic
307C22×C4—21no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4C22×D5311yescharacteristic
311C22×C4C22×D5311yescharacteristic
312SD16—534no—
313SD16—1062no—
314SD16—1062no—
315SD16—1062no—
316SD16—1062no—
317SD16—2121no—
318D8—1062no—
319D8—1062no—
320D8—2121no—
321M16—1062no—
322C2×C8—531no—
323C2×C8—531no—
324C2×C8—531no—
325C2×C8—531no—
326C2×C8—1062no—
327C4⋊C4—531no—
328C4⋊C4—1061no—
329C4⋊C4—1061no—
330C4⋊C4C22⋊D5311yescharacteristic
331C4⋊C4C22×D5311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4C22⋊D5311yescharacteristic
340C22⋊C4C22⋊D5311yescharacteristic
341C42—1061no—
342C42C22×D5311yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—1061no—
353C23C2×C22⋊D5311yescharacteristic
354C23C2×C22⋊D5311yescharacteristic
355Q8—532no—
356Q8—1061no—
357Q8—1061no—
358Q8—1061no—
359Q8—2122no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—1061no—
373D4—1061no—
374D4—2122no—
375D4—2122no—
376D4D4×D5312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4C2×C22⋊D5311yescharacteristic
389C2×C4C2×C22⋊D5311yescharacteristic
390C2×C4C2×C22⋊D5311yescharacteristic
391C2×C4C2×C22⋊D5311yescharacteristic
392C2×C4C23×D5311yescharacteristic
393C8—532no—
394C8—1061no—
395C8—1061no—
396C8—1061no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22#1313324312yescomplemented
417C22C22×C22⋊D5311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—1062no—
424C4—2121no—
425C4—2121no—
426C4C2×D4×D5311yescharacteristic
427C4#1313326211yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—4241no—
434C2D4×C22⋊D5311yescharacteristic
435C2C2×#1313324311yescharacteristic
436C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD42×C53C2
socleC22×C53C22×D4
hypercenterD42D53
hypocenterC53#3265
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#13133243C2semidirect
2. C22×#11719923C2semidirect
3. #16811282C2semidirect
4. #16811010C2semidirect
5. #16810867C2semidirect
6. #16810857C2semidirect
7. #16810838C2semidirect
8. #16810820C2semidirect
9. #16810774C2semidirect
10. #16810766C2semidirect
11. #16810761C2semidirect
12. #16810757C2semidirect
13. #16810723C2semidirect
14. #16810695C2semidirect
15. D42×C53C2semidirect
16. C2×#11719945C22semidirect
17. C22×Dic106C22semidirect
22. C2×#11719923C22semidirect
23. C2×#11719923C22semidirect
24. C2×#11719923C22semidirect
25. C2×#11719923C22semidirect
27. C2×#11719922C22semidirect
28. C2×#11719922C22semidirect
29. C2×#11719917C22semidirect
30. C22×#88322C22semidirect
31. #13133205C22semidirect
33. #13133196C22semidirect
35. #13133191C22semidirect
37. C4×Dic106C22semidirect
39. #13133150C22semidirect
40. C4⋊#88322C22semidirect
41. #13133140C22semidirect
42. #13133140C22semidirect
44. #13133138C22semidirect
45. #13133138C22semidirect
46. #13133137C22semidirect
47. C22⋊#88322C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
107. #11719923D4semidirect
173. Dic106C2×D4semidirect
177. #88322C2×D4semidirect
194. C22×C53AGL(1,ℤ/8)semidirect
222. D42D53semidirect
234. C53#3265semidirect
416. C22#13133243semidirect