Finite Group Databasebeta

← #20286194

Subgroups — #20286194

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×#13133244C211yesmaximal, characteristic, complemented
2C2×D8×D53C211yesmaximal, characteristic, complemented
3#16811266C211yesmaximal, characteristic, complemented
4#16811258C211yesmaximal, characteristic, complemented
5#16810873C211yesmaximal, characteristic, complemented
6#16810849C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810772C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810755C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810736C211yesmaximal, characteristic, complemented
14#16810696C211yesmaximal, characteristic, complemented
15C53×#496C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×D4×D53C2211yescharacteristic, complemented
20C2×D4×D53C2211yescharacteristic, complemented
21#13133244—24no—
22C2×#11719924C2211yescharacteristic, complemented
23C2×D424C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25C2×#11719922C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic, complemented
27D8×D53—24no—
28C2×#11719912C2211yescharacteristic, complemented
29C2×C8×D53C2211yescharacteristic, complemented
30#13133199—21no—
31#13133193C2211yescharacteristic, complemented
32#13133190—21no—
33#13133188—21no—
34#13133188C2211yescharacteristic
35#13133188C2211yescharacteristic, complemented
36#13133188C2211yescharacteristic, complemented
37#13133187—21no—
38#13133185—21no—
39#13133185—21no—
40C4×D212C2211yescharacteristic
41C4×D212C2211yescharacteristic, complemented
42C4×D212C2211yescharacteristic, complemented
43C22⋊C4×D53—21no—
44#13133149C2211yescharacteristic, complemented
45C4⋊#88322C2211yescharacteristic, complemented
46#13133141C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133138C2211yescharacteristic, complemented
50#13133138C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53#13133129C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic, complemented
56C2×D8×C53C2211yescharacteristic, complemented
57C2×D8×C53C2211yescharacteristic, complemented
58#127×C53C2211yescharacteristic, complemented
59#124×C53C2211yescharacteristic, complemented
60#97×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C23×D53—21no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53C2311yescharacteristic
68C2×D4×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76#11719949—22no—
77#11719949—41no—
78D4×D53—22no—
79D4×D53—22no—
80D4×D53—22no—
81D4×D53—41no—
82D4×D53—41no—
83D4×D53—41no—
84D4×D53—44no—
85C2×D212—21no—
86C2×D212D411yescharacteristic
87C2×D212C2311yescharacteristic
88C2×D212C2311yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53C2311yescharacteristic
93C2×C4×D53C2311yescharacteristic
94SD16×C53—22no—
95SD16×C53—41no—
96D8×C53—22no—
97D8×C53—41no—
98C2×C8×C53C2311yescharacteristic
99C2×C8×C53C2311yescharacteristic
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C42×C53C2311yescharacteristic
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—21no—
106#11719924—22no—
107#11719922—22no—
108#11719922—22no—
109#11719922—22no—
110#11719921—21no—
111#11719921—21no—
112#11719921—21no—
113#11719921D411yescharacteristic
114#11719921D411yescharacteristic
115C4⋊Dic53D411yescharacteristic
116C4⋊Dic53C2311yescharacteristic
117C4⋊Dic53C2311yescharacteristic
118#11719919—21no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121D424D414yescomplemented
122#11719912—22no—
123C8×D53—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—81no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53C2×D411yescharacteristic
139C22×D53C2×D411yescharacteristic
140Q8×C53—21no—
141Q8×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—21no—
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—21no—
148D4×C53—42no—
149D4×C53—81no—
150C2×C4×C53—21no—
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2411yescharacteristic
154C22⋊D53—42no—
155C22⋊D53—42no—
156C22⋊D53—42no—
157C22⋊D53—42no—
158C22⋊D53—42no—
159C22⋊D53—81no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53C2×D411yescharacteristic
163C2×Dic53C2×D411yescharacteristic
164D212—22no—
165D212—21no—
166D212—21no—
167D212—42no—
168D212C2×D412yes—
169D212C2×D412yescomplemented
170C4×D53—21no—
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—21no—
174C4×D53—42no—
175C4×D53—42no—
176C4×D53—81no—
177C8×C53—21no—
178C8×C53C2×D412yescomplemented
179#88322—21no—
180#88322—21no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53C22×D411yescharacteristic
191C2×D53—21no—
192C2×D53—21no—
193C2×D53—21no—
194C2×D53—22no—
195C2×D53—22no—
196C2×D53—21no—
197C2×D53—21no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—84no—
203C2×D53—84no—
204C2×D53—84no—
205C2×D53AGL(1,ℤ/8)12yescomplemented
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—41no—
209C4×C53C22×D411yescharacteristic
210C4×C53H(5,𝔽2)11yescharacteristic
211Dic53—21no—
212Dic53—21no—
213Dic53—41no—
214Dic53—41no—
215#3264—531nomaximal, Sylow
216C2×C53—81no—
217C2×C53—81no—
218C2×C53—81no—
219C2×C53D4211yescharacteristic
220C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C53#57611yescharacteristic
222D53—22no—
223D53—41no—
224D53—42no—
225D53—42no—
226C2×AGL(1,ℤ/8)—531no—
227C22×D8—531no—
228#546—531no—
229D42—531no—
230#496D5311yescharacteristic, complemented
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×#41—531no—
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—531no—
246C22×D4—1061no—
247AGL(1,ℤ/8)—1064no—
248C2×SD16—531no—
249C2×SD16C2×D5311yescharacteristic, complemented
250C2×SD16C2×D5311yescharacteristic, complemented
251C2×D8—531no—
252C2×D8—531no—
253C2×D8—531no—
254C2×D8—531no—
255C2×D8—1064no—
256C2×D8C2×D5311yescharacteristic, complemented
257C2×D8C2×D5311yescharacteristic, complemented
258C2×M16—531no—
259C22×C8—531no—
260#127C2×D5311yescharacteristic, complemented
261#124C2×D5311yescharacteristic, complemented
262#123—1061no—
263C22⋊Q8—531no—
264#121—531no—
265#121—531no—
266#121—531no—
267#121—1061no—
268#121—1061no—
269#120—1061no—
270#120—1061no—
271#120—1061no—
272C4×D4—531no—
273C4×D4—531no—
274C4×D4—531no—
275C2×C22⋊C4—1061no—
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#97C2×D5311yescharacteristic, complemented
287C24—1061no—
288C24—1061no—
289C24—1061no—
290#41—1062no—
291#41—2121no—
292C2×Q8C22×D5311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—531no—
296C2×D4—531no—
297C2×D4—531no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1062no—
303C2×D4—1062no—
304C2×D4—1062no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2124no—
312C2×D4C22×D5311yescharacteristic
313C2×D4C22×D5311yescharacteristic
314C2×D4C22×D5311yescharacteristic
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320SD16—22no—
321SD16—41no—
322SD16—1062no—
323D8—22no—
324D8—41no—
325D8—534no—
326D8—1062no—
327D8—1062no—
328D8—1062no—
329M16—1062no—
330C2×C8—531no—
331C2×C8—531no—
332C2×C8—1062no—
333C2×C8C22×D5311yescharacteristic
334C2×C8C22×D5311yescharacteristic
335C4⋊C4—531no—
336C4⋊C4—531no—
337C4⋊C4—531no—
338C4⋊C4—1061no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—531no—
342C22⋊C4—531no—
343C22⋊C4—1061no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C22⋊C4—1061no—
348C22⋊C4—1061no—
349C42C22×D5311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—531no—
354C23—531no—
355C23—1061no—
356C23—1061no—
357C23—4241no—
358C23—4241no—
359C23—4241no—
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23—2122no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—532no—
377D4—532no—
378D4—1062no—
379D4—1061no—
380D4—1061no—
381D4—4241no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387D4—2122no—
388C2×C4—21no—
389C2×C4—531no—
390C2×C4—531no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—4241no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4D4×D5311yescharacteristic
401C2×C4D4×D5311yescharacteristic
402C2×C4C23×D5311yescharacteristic
403C8—21no—
404C8—1061no—
405C8—1061no—
406C8D4×D5312yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—532no—
417C22—1061no—
418C22—1061no—
419C22—1061no—
420C22—1061no—
421C22—1061no—
422C22—1062no—
423C22—1062no—
424C22—4244no—
425C22—4244no—
426C22—4244no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22—2121no—
431C22C2×D4×D5311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—1061no—
436C4—1061no—
437C4—2121no—
438C4—2121no—
439C4C2×D4×D5311yescharacteristic
440C4#1313326211yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—1062no—
445C2—2121no—
446C2—2122no—
447C2—2122no—
448C2#1681125811yescharacteristic
449C2AGL(1,ℤ/8)×D5311yescharacteristic
450C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3264
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C2×D8×D53C2semidirect
3. #16811266C2semidirect
4. #16811258C2semidirect
5. #16810873C2semidirect
6. #16810849C2semidirect
7. #16810835C2semidirect
8. #16810815C2semidirect
9. #16810772C2semidirect
10. #16810767C2semidirect
11. #16810755C2semidirect
12. #16810755C2semidirect
13. #16810736C2semidirect
14. #16810696C2semidirect
15. C53×#496C2semidirect
16. C2×#11719949C22semidirect
19. C2×D4×D53C22semidirect
20. C2×D4×D53C22semidirect
22. C2×#11719924C22semidirect
25. C2×#11719922C22semidirect
26. C2×#11719922C22semidirect
28. C2×#11719912C22semidirect
29. C2×C8×D53C22semidirect
31. #13133193C22semidirect
35. #13133188C22semidirect
36. #13133188C22semidirect
41. C4×D212C22semidirect
42. C4×D212C22semidirect
44. #13133149C22semidirect
45. C4⋊#88322C22semidirect
46. #13133141C22semidirect
47. #13133139C22semidirect
48. #13133139C22semidirect
49. #13133138C22semidirect
50. #13133138C22semidirect
51. #13133138C22semidirect
52. #13133129C22semidirect
53. #13133129C22semidirect
54. C2×SD16×C53C22semidirect
55. C2×SD16×C53C22semidirect
56. C2×D8×C53C22semidirect
57. C2×D8×C53C22semidirect
58. #127×C53C22semidirect
59. #124×C53C22semidirect
60. #97×C53C22semidirect
121. D424D4semidirect
169. D212C2×D4semidirect
178. C8×C53C2×D4semidirect
205. C2×D53AGL(1,ℤ/8)semidirect
230. #496D53semidirect
241. C53#3264semidirect
249. C2×SD16C2×D53semidirect
250. C2×SD16C2×D53semidirect
256. C2×D8C2×D53semidirect
257. C2×D8C2×D53semidirect
260. #127C2×D53semidirect
261. #124C2×D53semidirect
286. #97C2×D53semidirect
406. C8D4×D53semidirect