Finite Group Databasebeta

← #20286217

Subgroups — #20286217

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×#13133245C211yesmaximal, characteristic, complemented
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3#16811283C211yesmaximal, characteristic, complemented
4#16811257C211yesmaximal, characteristic, complemented
5#16811009C211yesmaximal, characteristic, complemented
6#16810860C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810822C211yesmaximal, characteristic, complemented
9#16810773C211yesmaximal, characteristic, complemented
10#16810766C211yesmaximal, characteristic, complemented
11#16810763C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810722C211yesmaximal, characteristic, complemented
14#16810692C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C2×Q8×D53C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C2×D4×D53—21no—
20C2×D4×D53C2211yescharacteristic
21#13133245—24no—
22C2×#11719913C2211yescharacteristic
23C2×#11719913C2211yescharacteristic, complemented
24C2×#11719923C2211yescharacteristic, complemented
25C2×#11719924C2211yescharacteristic
26C2×#11719924C2211yescharacteristic, complemented
27SD16×D53—24no—
28C2×D424C2211yescharacteristic
29C2×D424C2211yescharacteristic, complemented
30C2×#11719922C2211yescharacteristic, complemented
31C2×C8×D53C2211yescharacteristic, complemented
32#13133213C2211yescharacteristic
33#13133206C2211yescharacteristic, complemented
34#13133203—21no—
35#13133195—21no—
36#13133192C2211yescharacteristic
37#13133192C2211yescharacteristic, complemented
38#13133190—21no—
39#13133190—21no—
40#13133187—21no—
41#13133185—21no—
42#13133182C2211yescharacteristic, complemented
43#13133179—21no—
44#13133179C2211yescharacteristic
45C4×C22⋊D53C2211yescharacteristic
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53#13133127C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×M16×C53C2211yescharacteristic, complemented
56C22⋊Q8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72Q8×D53—22no—
73Q8×D53—41no—
74D4×D53—22no—
75D4×D53—41no—
76D4×D53—44no—
77#11719945—22no—
78#11719945—41no—
79C2×D212—22no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—41no—
84C2×D212C2311yescharacteristic
85C2×D212C2311yescharacteristic
86C2×D212C2311yescharacteristic
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53C2311yescharacteristic
91C2×Dic106—21no—
92C2×Dic106C2311yescharacteristic
93SD16×C53—22no—
94M16×C53—22no—
95C2×C8×C53C2311yescharacteristic
96C2×C8×C53C2311yescharacteristic
97C4⋊C4×C53—21no—
98C4⋊C4×C53C2311yescharacteristic
99C4⋊C4×C53C2311yescharacteristic
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C22⋊Dic53D411yescharacteristic
103#11719924—22no—
104#11719924—41no—
105#11719923—22no—
106#11719922—41no—
107#11719921—21no—
108#11719921—21no—
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
123C8×D53—22no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—42no—
133C22×D53—81no—
134C22×D53—81no—
135C22×D53C2×D411yescharacteristic
136Q8×C53—21no—
137Q8×C53—21no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—42no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53C2×D411yescharacteristic
147C2×C4×C53C2411yescharacteristic
148C22⋊D53—22no—
149C22⋊D53—42no—
150C22⋊D53—42no—
151C22⋊D53—42no—
152C22⋊D53—81no—
153C2×Dic53—21no—
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156C2×Dic53C2×D411yescharacteristic
157D212—21no—
158D212—21no—
159D212—21no—
160D212—21no—
161D212—21no—
162D212—42no—
163D212—42no—
164D212—42no—
165D212C2×D412yes—
166C4×D53—21no—
167C4×D53—21no—
168C4×D53—42no—
169C4×D53—42no—
170C4×D53—42no—
171Dic106—21no—
172Dic106—42no—
173Dic106C2×D412yescomplemented
174C8×C53—21no—
175C8×C53C2×D412yescomplemented
176#88322—21no—
177#88322—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—22no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—84no—
197C2×D53—84no—
198C2×D53AGL(1,ℤ/8)12yescomplemented
199C4×C53—21no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53C22×D411yescharacteristic
204C4×C53H(5,𝔽2)11yescharacteristic
205Dic53—22no—
206Dic53—21no—
207Dic53—21no—
208Dic53—21no—
209#3265—531nomaximal, Sylow
210C2×C53—41no—
211C2×C53—81no—
212C2×C53D4211yescharacteristic
213C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C53#57711yescharacteristic
215D53—22no—
216D53—42no—
217D53—81no—
218D53—81no—
219C2×AGL(1,ℤ/8)—531no—
220C22×SD16—531no—
221#548—531no—
222D42—531no—
223#496—531no—
224#468D5311yescharacteristic, complemented
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—531no—
238C22×D4—531no—
239C22×D4—1061no—
240AGL(1,ℤ/8)—1064no—
241C2×SD16—531no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16—531no—
245C2×SD16—531no—
246C2×SD16—1064no—
247C2×SD16C2×D5311yescharacteristic, complemented
248C2×D8—531no—
249C2×D8—531no—
250C2×D8—531no—
251C2×M16C2×D5311yescharacteristic, complemented
252C22×C8—531no—
253#127—531no—
254#124—531no—
255#124—1061no—
256C22⋊Q8—1061no—
257C22⋊Q8C2×D5311yescharacteristic, complemented
258#121—531no—
259#121—531no—
260#121—1061no—
261#121—1061no—
262#121C2×D5311yescharacteristic
263#120—1061no—
264#120—1061no—
265C4×Q8—531no—
266C4×D4—531no—
267C4×D4—531no—
268C4×D4—1061no—
269#107C2×D5311yescharacteristic, complemented
270C4⋊C8—531no—
271#103—531no—
272#103C2×D5311yescharacteristic, complemented
273#102—531no—
274#102—531no—
275#102—531no—
276#102C2×D5311yescharacteristic, complemented
277C22⋊C8—531no—
278C22⋊C8—531no—
279#97—531no—
280C24—1061no—
281C24—1061no—
282#41—1062no—
283#41—2121no—
284C2×Q8—531no—
285C2×Q8—1061no—
286C2×Q8—1062no—
287C2×Q8—2121no—
288C2×Q8C22×D5311yescharacteristic
289C2×D4—21no—
290C2×D4—531no—
291C2×D4—531no—
292C2×D4—531no—
293C2×D4—531no—
294C2×D4—1062no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1062no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2124no—
306C2×D4C22×D5311yescharacteristic
307C22×C4—531no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4C22×D5311yescharacteristic
312SD16—22no—
313SD16—534no—
314SD16—1062no—
315SD16—1062no—
316SD16—1062no—
317SD16—2121no—
318D8—1062no—
319D8—1062no—
320D8—2121no—
321M16—22no—
322C2×C8—531no—
323C2×C8—531no—
324C2×C8—1062no—
325C2×C8C22×D5311yescharacteristic
326C2×C8C22×D5311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—531no—
329C4⋊C4—1061no—
330C4⋊C4C22×D5311yescharacteristic
331C4⋊C4C22×D5311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—531no—
335C22⋊C4—531no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C42—531no—
342C42—1061no—
343C23—21no—
344C23—531no—
345C23—1061no—
346C23—1061no—
347C23—1061no—
348C23—4241no—
349C23—4241no—
350C23—2122no—
351C23—2122no—
352C23—2122no—
353C23—2122no—
354C23D4×D5311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—532no—
358Q8—1061no—
359Q8—2122no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—532no—
364D4—1062no—
365D4—1061no—
366D4—1061no—
367D4—1061no—
368D4—1061no—
369D4—1061no—
370D4—4241no—
371D4—2122no—
372D4—2122no—
373D4—2122no—
374D4—2122no—
375D4—2122no—
376D4—2122no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—531no—
383C2×C4—531no—
384C2×C4—531no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4D4×D5311yescharacteristic
392C2×C4C23×D5311yescharacteristic
393C8—21no—
394C8—1061no—
395C8—1061no—
396C8D4×D5312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—532no—
404C22—1061no—
405C22—1061no—
406C22—1062no—
407C22—4244no—
408C22—4244no—
409C22—2121no—
410C22—2121no—
411C22—2121no—
412C22—2121no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22C2×D4×D5311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—1061no—
423C4—1061no—
424C4—1061no—
425C4—1062no—
426C4C2×D4×D5311yescharacteristic
427C4#1313326311yescharacteristic
428C2—41no—
429C2—81no—
430C2—1062no—
431C2—4241no—
432C2—4241no—
433C2—2122no—
434C2#1681125711yescharacteristic
435C2AGL(1,ℤ/8)×D5311yescharacteristic
436C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3265
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. C2×SD16×D53C2semidirect
3. #16811283C2semidirect
4. #16811257C2semidirect
5. #16811009C2semidirect
6. #16810860C2semidirect
7. #16810841C2semidirect
8. #16810822C2semidirect
9. #16810773C2semidirect
10. #16810766C2semidirect
11. #16810763C2semidirect
12. #16810758C2semidirect
13. #16810722C2semidirect
14. #16810692C2semidirect
15. C53×#468C2semidirect
16. C2×#11719945C22semidirect
17. C2×Q8×D53C22semidirect
23. C2×#11719913C22semidirect
24. C2×#11719923C22semidirect
26. C2×#11719924C22semidirect
29. C2×D424C22semidirect
30. C2×#11719922C22semidirect
31. C2×C8×D53C22semidirect
33. #13133206C22semidirect
37. #13133192C22semidirect
42. #13133182C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
49. #13133137C22semidirect
50. #13133137C22semidirect
51. C22⋊#88322C22semidirect
52. #13133129C22semidirect
53. #13133127C22semidirect
54. C2×SD16×C53C22semidirect
55. C2×M16×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #107×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
122. #11719913D4semidirect
173. Dic106C2×D4semidirect
175. C8×C53C2×D4semidirect
198. C2×D53AGL(1,ℤ/8)semidirect
224. #468D53semidirect
234. C53#3265semidirect
247. C2×SD16C2×D53semidirect
251. C2×M16C2×D53semidirect
257. C22⋊Q8C2×D53semidirect
269. #107C2×D53semidirect
272. #103C2×D53semidirect
276. #102C2×D53semidirect
396. C8D4×D53semidirect