Finite Group Databasebeta

← #20286196

Subgroups — #20286196

Order 6784.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133242C211yesmaximal, characteristic, complemented
2C22×#11719922C211yesmaximal, characteristic, complemented
3D4×D212C211yesmaximal, characteristic, complemented
4#16811013C211yesmaximal, characteristic, complemented
5#16810874C211yesmaximal, characteristic, complemented
6#16810854C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810818C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810757C211yesmaximal, characteristic, complemented
12#16810756C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic, complemented
18C2×D4×D53—21no—
19#13133242—24no—
20C2×#11719924C2211yescharacteristic, complemented
21C2×#11719924C2211yescharacteristic, complemented
22C2×#11719924C2211yescharacteristic, complemented
23C2×#11719922—24no—
24C2×#11719922C2211yescharacteristic
25C2×#11719922C2211yescharacteristic
26C2×#11719922C2211yescharacteristic, complemented
27C2×#11719922C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic, complemented
29C2×#11719922C2211yescharacteristic, complemented
30C2×#11719917C2211yescharacteristic, complemented
31C22×#88322C2211yescharacteristic, complemented
32#13133214C2211yescharacteristic, complemented
33#13133192—21no—
34#13133191C2211yescharacteristic
35#13133191C2211yescharacteristic, complemented
36#13133187—21no—
37#13133187—21no—
38C4×D212C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133138C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic, complemented
44#13133137C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47C22⋊#88322C2211yescharacteristic, complemented
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C2×#41×C53C2211yescharacteristic, complemented
51C22×D4×C53C2211yescharacteristic, complemented
52#124×C53C2211yescharacteristic, complemented
53#123×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53—21no—
56#121×C53C2211yescharacteristic, complemented
57#120×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C4×D4×C53C2211yescharacteristic, complemented
60C2×C22⋊C4×C53—21no—
61C24×C53—21no—
62C23×D53—21no—
63C23×D53—21no—
64#41×C53—22no—
65#41×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—41no—
72C2×D4×C53D411yescharacteristic
73C2×D4×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81D4×D53—44no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212C2311yescharacteristic
91C2×D212C2311yescharacteristic
92C2×D212C2311yescharacteristic
93C2×C4×D53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53D411yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C22⋊C4×C53D411yescharacteristic
103C22⋊C4×C53D411yescharacteristic
104C42×C53C2311yescharacteristic
105#11719924—22no—
106#11719924—22no—
107#11719924—41no—
108#11719922—22no—
109#11719922—22no—
110#11719922—22no—
111#11719922—22no—
112#11719922—41no—
113#11719922D414yescomplemented
114#11719921—21no—
115#11719921—21no—
116#11719921—21no—
117C4⋊Dic53C2311yescharacteristic
118#11719917—22no—
119C2×#88322—22no—
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C2×#88322C2311yescharacteristic
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—81no—
129C23×C53C2×D411yescharacteristic
130C23×C53C2×D411yescharacteristic
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—81no—
139C22×D53—81no—
140Q8×C53—21no—
141Q8×C53—21no—
142D4×C53—22no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—21no—
146D4×C53—42no—
147D4×C53—42no—
148D4×C53—81no—
149D4×C53C2×D412yescomplemented
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—42no—
157C2×C4×C53—81no—
158C2×C4×C53C2×D411yescharacteristic
159C2×C4×C53C2×D411yescharacteristic
160C2×C4×C53C2×D411yescharacteristic
161C2×C4×C53C2×D411yescharacteristic
162C2×C4×C53C2411yescharacteristic
163C22⋊D53—42no—
164C22⋊D53—42no—
165C2×Dic53—21no—
166D212—21no—
167D212—21no—
168D212—21no—
169D212—21no—
170D212—21no—
171D212—42no—
172D212—42no—
173D212—42no—
174D212—81no—
175D212C2×D412yes—
176C4×D53—42no—
177#88322—21no—
178#88322—21no—
179#88322—21no—
180#88322C2×D412yescomplemented
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—22no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—84no—
193C22×C53AGL(1,ℤ/8)12yescomplemented
194C22×C53C22×D411yescharacteristic
195C2×D53—22no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—84no—
205C2×D53—84no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53C22×D411yescharacteristic
213C4×C53H(5,𝔽2)11yescharacteristic
214Dic53—41no—
215#3264—531nomaximal, Sylow
216C2×C53—22no—
217C2×C53—41no—
218C2×C53—42no—
219C2×C53—81no—
220C2×C53D4211yescharacteristic
221C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C53#57611yescharacteristic
223D53—42no—
224D53—81no—
225D53—81no—
226C2×AGL(1,ℤ/8)—531no—
227C22×D8—531no—
228#546D5311yescharacteristic, complemented
229D42—531no—
230#496—531no—
231#469—531no—
232#465—531no—
233#463—531no—
234#459—531no—
235#450—531no—
236#449—531no—
237#447—531no—
238#447—531no—
239#442—531no—
240#435—531no—
241C53#326411yescharacteristic, complemented, Sylow
242C2×#41C2×D5311yescharacteristic, complemented
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—1061no—
246C22×D4C2×D5311yescharacteristic, complemented
247AGL(1,ℤ/8)—1064no—
248C2×SD16—531no—
249C2×SD16—531no—
250C2×SD16—531no—
251C2×D8—531no—
252C2×D8—531no—
253C2×D8—531no—
254C2×D8—531no—
255C2×D8—531no—
256C2×D8—531no—
257C2×D8—1064no—
258C2×M16—531no—
259C22×C8—531no—
260#127—531no—
261#124C2×D5311yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D5311yescharacteristic, complemented
264#121—21no—
265#121—531no—
266#121—531no—
267#121—1061no—
268#121C2×D5311yescharacteristic, complemented
269#120—21no—
270#120—1061no—
271#120—1061no—
272C4×D4—531no—
273C4×D4C2×D5311yescharacteristic
274C4×D4C2×D5311yescharacteristic, complemented
275C2×C22⋊C4—21no—
276#106—531no—
277C4⋊C8—531no—
278#103—531no—
279#102—531no—
280#102—531no—
281#102—531no—
282#102—531no—
283#102—531no—
284C22⋊C8—531no—
285C22⋊C8—531no—
286#97—531no—
287C24—21no—
288C24—1061no—
289C24—1061no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D5311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—531no—
299C2×D4—531no—
300C2×D4—531no—
301C2×D4—1062no—
302C2×D4—1062no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2124no—
312C2×D4C22⋊D5311yescharacteristic
313C2×D4C22×D5311yescharacteristic
314C2×D4C22×D5311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—1061no—
318C22×C4C22×D5311yescharacteristic
319C22×C4C22×D5311yescharacteristic
320SD16—1062no—
321SD16—1062no—
322SD16—2121no—
323D8—534no—
324D8—1062no—
325D8—1062no—
326D8—1062no—
327D8—1062no—
328D8—2121no—
329M16—1062no—
330C2×C8—531no—
331C2×C8—531no—
332C2×C8—531no—
333C2×C8—531no—
334C2×C8—1062no—
335C4⋊C4—21no—
336C4⋊C4—531no—
337C4⋊C4C22⋊D5311yescharacteristic
338C4⋊C4C22×D5311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4C22⋊D5311yescharacteristic
348C22⋊C4C22⋊D5311yescharacteristic
349C42C22×D5311yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—1061no—
356C23—1061no—
357C23—1061no—
358C23—4241no—
359C23—4241no—
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23C2×C22⋊D5311yescharacteristic
365C23C2×C22⋊D5311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—532no—
376D4—1061no—
377D4—1061no—
378D4—1061no—
379D4—1061no—
380D4—1061no—
381D4—4241no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387D4D4×D5312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—1061no—
397C2×C4—2122no—
398C2×C4C2×C22⋊D5311yescharacteristic
399C2×C4C2×C22⋊D5311yescharacteristic
400C2×C4C2×C22⋊D5311yescharacteristic
401C2×C4C2×C22⋊D5311yescharacteristic
402C2×C4C23×D5311yescharacteristic
403C8—532no—
404C8—1061no—
405C8—1061no—
406C8—1061no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—1062no—
420C22—4244no—
421C22—4244no—
422C22—2121no—
423C22—2121no—
424C22—2121no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22#1313324212yescomplemented
431C22C22×C22⋊D5311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—2121no—
439C4C2×D4×D5311yescharacteristic
440C4#1313326211yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—4241no—
446C2—4241no—
447C2—2122no—
448C2D4×C22⋊D5311yescharacteristic
449C2C2×#1313324211yescharacteristic
450C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3264
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#13133242C2semidirect
2. C22×#11719922C2semidirect
3. D4×D212C2semidirect
4. #16811013C2semidirect
5. #16810874C2semidirect
6. #16810854C2semidirect
7. #16810835C2semidirect
8. #16810818C2semidirect
9. #16810773C2semidirect
10. #16810768C2semidirect
11. #16810757C2semidirect
12. #16810756C2semidirect
13. #16810733C2semidirect
14. #16810695C2semidirect
15. C53×#546C2semidirect
17. C22×D212C22semidirect
20. C2×#11719924C22semidirect
21. C2×#11719924C22semidirect
22. C2×#11719924C22semidirect
26. C2×#11719922C22semidirect
27. C2×#11719922C22semidirect
28. C2×#11719922C22semidirect
29. C2×#11719922C22semidirect
30. C2×#11719917C22semidirect
31. C22×#88322C22semidirect
32. #13133214C22semidirect
35. #13133191C22semidirect
38. C4×D212C22semidirect
39. #13133148C22semidirect
40. C4⋊#88322C22semidirect
41. #13133141C22semidirect
42. #13133138C22semidirect
43. #13133138C22semidirect
44. #13133137C22semidirect
45. #13133137C22semidirect
46. #13133137C22semidirect
47. C22⋊#88322C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
50. C2×#41×C53C22semidirect
51. C22×D4×C53C22semidirect
52. #124×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #121×C53C22semidirect
59. C4×D4×C53C22semidirect
113. #11719922D4semidirect
149. D4×C53C2×D4semidirect
180. #88322C2×D4semidirect
193. C22×C53AGL(1,ℤ/8)semidirect
228. #546D53semidirect
241. C53#3264semidirect
242. C2×#41C2×D53semidirect
246. C22×D4C2×D53semidirect
261. #124C2×D53semidirect
263. C22⋊Q8C2×D53semidirect
268. #121C2×D53semidirect
274. C4×D4C2×D53semidirect
387. D4D4×D53semidirect
430. C22#13133242semidirect