Finite Group Databasebeta

← #20286332

Subgroups — #20286332

Order 6784.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133245C211yesmaximal, characteristic, complemented
2C2×#13133241C211yesmaximal, characteristic, complemented
3#16811291C211yesmaximal, characteristic, complemented
4#16811257C211yesmaximal, characteristic, complemented
5#16810998C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810822C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810769C211yesmaximal, characteristic, complemented
10#16810766C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810756C211yesmaximal, characteristic, complemented
13#16810710C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C53×#469C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C2×#11719947C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C2×D4×D53—21no—
20C2×D4×D53C2211yescharacteristic
21#13133245—24no—
22#13133241—24no—
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic, complemented
25C2×#11719923C2211yescharacteristic, complemented
26C2×D424C2211yescharacteristic
27C2×D424C2211yescharacteristic, complemented
28C2×#11719922C2211yescharacteristic
29C2×#11719922C2211yescharacteristic, complemented
30C2×#11719922C2211yescharacteristic, complemented
31C2×#11719912C2211yescharacteristic, complemented
32#13133213—21no—
33#13133213C2211yescharacteristic
34#13133213C2211yescharacteristic, complemented
35#13133192C2211yescharacteristic
36#13133192C2211yescharacteristic, complemented
37#13133190—21no—
38#13133190—21no—
39#13133189—21no—
40#13133187—21no—
41#13133185—21no—
42#13133182C2211yescharacteristic, complemented
43#13133179C2211yescharacteristic
44C4×C22⋊D53C2211yescharacteristic
45#13133177—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53C8×Dic53C2211yescharacteristic, complemented
54C2×D8×C53C2211yescharacteristic, complemented
55C2×M16×C53C2211yescharacteristic, complemented
56#121×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic, complemented
58#106×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53D411yescharacteristic
76C22×Dic53—21no—
77#11719947—22no—
78#11719947—41no—
79D4×D53—22no—
80D4×D53—41no—
81D4×D53—44no—
82#11719945—22no—
83#11719945—41no—
84C2×D212—22no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—41no—
89C2×D212C2311yescharacteristic
90C2×D212C2311yescharacteristic
91C2×D212C2311yescharacteristic
92C2×C4×D53—21no—
93C2×C4×D53—21no—
94C2×C4×D53C2311yescharacteristic
95C2×Dic106C2311yescharacteristic
96D8×C53—22no—
97M16×C53—22no—
98C2×C8×C53C2311yescharacteristic
99C2×C8×C53C2311yescharacteristic
100C4⋊C4×C53C2311yescharacteristic
101C4⋊C4×C53C2311yescharacteristic
102C22⋊C4×C53—21no—
103C22⋊C4×C53—21no—
104C22⋊Dic53D411yescharacteristic
105#11719923—22no—
106#11719922—22no—
107#11719922—41no—
108#11719922—41no—
109#11719921—21no—
110#11719921—21no—
111#11719921—21no—
112#11719921D411yescharacteristic
113#11719919—21no—
114#11719919D411yescharacteristic
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118C2×#88322C2311yescharacteristic
119D424—22no—
120D424—22no—
121#11719913—22no—
122#11719913D414yescomplemented
123#11719912—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53C2×D411yescharacteristic
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—81no—
135C22×D53—81no—
136C22×D53C2×D411yescharacteristic
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—21no—
141D4×C53—42no—
142D4×C53—81no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53C2×D411yescharacteristic
148C2×C4×C53C2411yescharacteristic
149C22⋊D53—22no—
150C22⋊D53—42no—
151C22⋊D53—42no—
152C22⋊D53—42no—
153C22⋊D53—42no—
154C22⋊D53—42no—
155C22⋊D53—81no—
156C22⋊D53—81no—
157C2×Dic53—21no—
158C2×Dic53—42no—
159C2×Dic53C2×D411yescharacteristic
160C2×Dic53C2×D411yescharacteristic
161C2×Dic53C2×D411yescharacteristic
162D212—21no—
163D212—21no—
164D212—21no—
165D212—21no—
166D212—21no—
167D212—42no—
168D212—42no—
169D212—81no—
170D212C2×D412yes—
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—42no—
174C4×D53—42no—
175Dic106—21no—
176Dic106C2×D412yescomplemented
177C8×C53—21no—
178C8×C53C2×D412yescomplemented
179#88322—21no—
180#88322—21no—
181C22×C53—21no—
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×C53C22×D411yescharacteristic
191C2×D53—21no—
192C2×D53—22no—
193C2×D53—21no—
194C2×D53—21no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53—84no—
204C2×D53—84no—
205C4×C53—21no—
206C4×C53—41no—
207C4×C53—41no—
208C4×C53C22×D411yescharacteristic
209C4×C53H(5,𝔽2)11yescharacteristic
210Dic53—22no—
211Dic53—21no—
212Dic53—21no—
213Dic53AGL(1,ℤ/8)12yescomplemented
214#3274—531nomaximal, Sylow
215C2×C53—41no—
216C2×C53—81no—
217C2×C53—81no—
218C2×C53D4211yescharacteristic
219C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C53#57611yescharacteristic
221D53—41no—
222D53—42no—
223D53—81no—
224D53—81no—
225C2×AGL(1,ℤ/8)—531no—
226C2×AGL(1,ℤ/8)—531no—
227#550—531no—
228D42—531no—
229#493—531no—
230#469—531no—
231#469D5311yescharacteristic, complemented
232#460—531no—
233#459—531no—
234#449—531no—
235#449—531no—
236#447—531no—
237#447—531no—
238C4×SD16—531no—
239#436—531no—
240C53#327411yescharacteristic, complemented, Sylow
241C2×#41—531no—
242C2×#41—531no—
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—1061no—
246AGL(1,ℤ/8)—1064no—
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×D8C2×D5311yescharacteristic, complemented
257C2×M16—531no—
258C2×M16C2×D5311yescharacteristic, complemented
259#127—531no—
260#127—531no—
261#127—1061no—
262#121—531no—
263#121—531no—
264#121—1061no—
265#121—1061no—
266#121—1061no—
267#121C2×D5311yescharacteristic
268#121C2×D5311yescharacteristic, complemented
269#120—1061no—
270#120—1061no—
271C4×Q8—531no—
272C4×D4—531no—
273C4×D4—531no—
274C4×D4—1061no—
275#106C2×D5311yescharacteristic, complemented
276C4⋊C8—531no—
277#103—531no—
278#102—531no—
279#102—531no—
280#102—531no—
281#102C2×D5311yescharacteristic, complemented
282#102C2×D5311yescharacteristic, complemented
283C22⋊C8—531no—
284C22⋊C8—531no—
285C4×C8—531no—
286C24—1061no—
287C24—1061no—
288#41—1062no—
289#41—1062no—
290#41—2121no—
291#41—2121no—
292C2×Q8—531no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—531no—
296C2×D4—531no—
297C2×D4—531no—
298C2×D4—531no—
299C2×D4—1062no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1062no—
308C2×D4—1061no—
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2124no—
314C2×D4C22×D5311yescharacteristic
315C2×D4C22×D5311yescharacteristic
316C22×C4—531no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4C22×D5311yescharacteristic
321SD16—534no—
322SD16—1062no—
323SD16—1062no—
324D8—22no—
325D8—1062no—
326D8—1062no—
327D8—1062no—
328D8—2121no—
329D8—2121no—
330M16—22no—
331M16—1062no—
332C2×C8—531no—
333C2×C8—531no—
334C2×C8C22×D5311yescharacteristic
335C2×C8C22×D5311yescharacteristic
336C4⋊C4—531no—
337C4⋊C4—1061no—
338C4⋊C4C22×D5311yescharacteristic
339C4⋊C4C22×D5311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—531no—
343C22⋊C4—531no—
344C22⋊C4—1061no—
345C22⋊C4—1061no—
346C22⋊C4—1061no—
347C42—531no—
348C42—1061no—
349C23—21no—
350C23—21no—
351C23—531no—
352C23—1061no—
353C23—1061no—
354C23—1061no—
355C23—4241no—
356C23—4241no—
357C23—2122no—
358C23—2122no—
359C23—2122no—
360C23—2122no—
361C23D4×D5311yescharacteristic
362Q8—532no—
363Q8—1061no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—81no—
370D4—532no—
371D4—1062no—
372D4—1061no—
373D4—1061no—
374D4—1061no—
375D4—1061no—
376D4—1061no—
377D4—4241no—
378D4—4241no—
379D4—4241no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—531no—
392C2×C4—531no—
393C2×C4—531no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4D4×D5311yescharacteristic
401C2×C4C23×D5311yescharacteristic
402C8—21no—
403C8—1061no—
404C8—1061no—
405C8D4×D5312yescomplemented
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—1061no—
416C22—1061no—
417C22—1061no—
418C22—1062no—
419C22—4244no—
420C22—4244no—
421C22—2121no—
422C22—2121no—
423C22—2121no—
424C22—2121no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22—2121no—
429C22C2×D4×D5311yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—532no—
434C4—1061no—
435C4—1061no—
436C4—1062no—
437C4C2×D4×D5311yescharacteristic
438C4#1313326311yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—4241no—
443C2—4241no—
444C2—2121no—
445C2—2122no—
446C2#1681125711yescharacteristic
447C2AGL(1,ℤ/8)×D5311yescharacteristic
448C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3274
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×#13133245C2semidirect
2. C2×#13133241C2semidirect
3. #16811291C2semidirect
4. #16811257C2semidirect
5. #16810998C2semidirect
6. #16810877C2semidirect
7. #16810822C2semidirect
8. #16810817C2semidirect
9. #16810769C2semidirect
10. #16810766C2semidirect
11. #16810758C2semidirect
12. #16810756C2semidirect
13. #16810710C2semidirect
14. #16810699C2semidirect
15. C53×#469C2semidirect
16. C2×#11719945C22semidirect
17. C2×#11719947C22semidirect
24. C2×#11719913C22semidirect
25. C2×#11719923C22semidirect
27. C2×D424C22semidirect
29. C2×#11719922C22semidirect
30. C2×#11719922C22semidirect
31. C2×#11719912C22semidirect
34. #13133213C22semidirect
36. #13133192C22semidirect
42. #13133182C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
49. #13133137C22semidirect
50. #13133137C22semidirect
51. C22⋊#88322C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C2×D8×C53C22semidirect
55. C2×M16×C53C22semidirect
57. #121×C53C22semidirect
58. #106×C53C22semidirect
59. #102×C53C22semidirect
60. #102×C53C22semidirect
122. #11719913D4semidirect
176. Dic106C2×D4semidirect
178. C8×C53C2×D4semidirect
213. Dic53AGL(1,ℤ/8)semidirect
231. #469D53semidirect
240. C53#3274semidirect
256. C2×D8C2×D53semidirect
258. C2×M16C2×D53semidirect
268. #121C2×D53semidirect
275. #106C2×D53semidirect
281. #102C2×D53semidirect
282. #102C2×D53semidirect
405. C8D4×D53semidirect