Finite Group Databasebeta

← #20286204

Subgroups — #20286204

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×SD16×D53C211yesmaximal, characteristic, complemented
2#16811280C211yesmaximal, characteristic, complemented
3D4×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811011C211yesmaximal, characteristic, complemented
5#16810869C211yesmaximal, characteristic, complemented
6#16810856C211yesmaximal, characteristic, complemented
7#16810836C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810760C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810692C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic, complemented
17C22×C22⋊D53—21no—
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic, complemented
20C2×#11719923C2211yescharacteristic
21C2×#11719923C2211yescharacteristic, complemented
22C2×#11719924C2211yescharacteristic, complemented
23SD16×D53—24no—
24C2×#11719922C2211yescharacteristic, complemented
25C2×C8×D53C2211yescharacteristic, complemented
26#13133213C2211yescharacteristic
27#13133204—21no—
28#13133204C2211yescharacteristic, complemented
29#13133196—21no—
30#13133196C2211yescharacteristic, complemented
31#13133191C2211yescharacteristic
32#13133189—21no—
33#13133188—21no—
34#13133188C2211yescharacteristic
35#13133188C2211yescharacteristic, complemented
36#13133186—21no—
37#13133185—21no—
38Q8×Dic53C2211yescharacteristic, complemented
39D4×Dic53C2211yescharacteristic
40C4×C22⋊D53—21no—
41C4×C22⋊D53C2211yescharacteristic
42#13133151C2211yescharacteristic, complemented
43#13133146C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic, complemented
49#13133138C2211yescharacteristic, complemented
50C22⋊#88322C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#13133127C2211yescharacteristic, complemented
53C2×#41×C53C2211yescharacteristic, complemented
54C22×D4×C53C2211yescharacteristic
55AGL(1,ℤ/8)×C53—24no—
56C2×SD16×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic, complemented
58C2×D8×C53C2211yescharacteristic
59C2×D8×C53C2211yescharacteristic, complemented
60C2×M16×C53C2211yescharacteristic, complemented
61C24×C53—21no—
62C23×D53—21no—
63#41×C53—22no—
64#41×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81C2×C22⋊D53—44no—
82C2×C22⋊D53D411yescharacteristic
83C22×Dic53—21no—
84Q8×D53—22no—
85Q8×D53—41no—
86D4×D53—22no—
87D4×D53—41no—
88C2×D212C2311yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53C2311yescharacteristic
91C2×Dic106—21no—
92C2×Dic106C2311yescharacteristic
93SD16×C53—22no—
94SD16×C53D414yescomplemented
95D8×C53—22no—
96D8×C53—22no—
97M16×C53—22no—
98C2×C8×C53C2311yescharacteristic
99C2×C8×C53C2311yescharacteristic
100C22⋊Dic53—21no—
101C22⋊Dic53—21no—
102C22⋊Dic53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—21no—
106C22⋊Dic53D411yescharacteristic
107#11719924—22no—
108#11719923—22no—
109#11719923—41no—
110#11719922—41no—
111#11719921—21no—
112#11719921D411yescharacteristic
113C4⋊Dic53—21no—
114C4⋊Dic53C2311yescharacteristic
115C4⋊Dic53C2311yescharacteristic
116#11719919—21no—
117#11719919D411yescharacteristic
118C4×Dic53—21no—
119C4×Dic53C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122#11719913—22no—
123C8×D53—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—81no—
130C23×C53C2×D411yescharacteristic
131C22×D53—21no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—81no—
135C22×D53C2×D411yescharacteristic
136Q8×C53—21no—
137Q8×C53C2×D412yescomplemented
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—21no—
141D4×C53—21no—
142D4×C53—21no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53C2×D412yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—42no—
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2411yescharacteristic
152C22⋊D53—22no—
153C22⋊D53—42no—
154C22⋊D53—42no—
155C22⋊D53—42no—
156C22⋊D53—42no—
157C22⋊D53—42no—
158C22⋊D53—81no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—42no—
163C2×Dic53C2×D411yescharacteristic
164C2×Dic53C2×D411yescharacteristic
165C2×Dic53C2×D411yescharacteristic
166D212—21no—
167D212—21no—
168C4×D53—21no—
169C4×D53—21no—
170C4×D53—42no—
171Dic106—21no—
172Dic106—21no—
173Dic106—42no—
174C8×C53—21no—
175C8×C53C2×D412yescomplemented
176#88322—21no—
177#88322—21no—
178C22×C53—22no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
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×C53—84no—
191C22×C53C22×D411yescharacteristic
192C2×D53—21no—
193C2×D53—21no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—84no—
198C2×D53AGL(1,ℤ/8)12yescomplemented
199C4×C53—22no—
200C4×C53—21no—
201C4×C53C22×D411yescharacteristic
202C4×C53H(5,𝔽2)11yescharacteristic
203Dic53—21no—
204Dic53—21no—
205Dic53—21no—
206Dic53—41no—
207Dic53—41no—
208Dic53—41no—
209#3265—531nomaximal, Sylow
210C2×C53—42no—
211C2×C53—41no—
212C2×C53—81no—
213C2×C53—81no—
214C2×C53D4211yescharacteristic
215C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C53#57711yescharacteristic
217D53—22no—
218D53—81no—
219C2×AGL(1,ℤ/8)D5311yescharacteristic, complemented
220C22×SD16—531no—
221#548—531no—
222D42—531no—
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×#41C2×D5311yescharacteristic, complemented
236C22×Q8—531no—
237C22×D4—531no—
238C22×D4—1061no—
239C22×D4C2×D5311yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—531no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16—531no—
245C2×SD16—1064no—
246C2×SD16C2×D5311yescharacteristic
247C2×SD16C2×D5311yescharacteristic, complemented
248C2×D8—531no—
249C2×D8C2×D5311yescharacteristic
250C2×D8C2×D5311yescharacteristic, complemented
251C2×M16C2×D5311yescharacteristic, complemented
252C22×C8—531no—
253#127—531no—
254#124—531no—
255#124—1061no—
256C22⋊Q8—531no—
257C22⋊Q8—1061no—
258#121—531no—
259#121—531no—
260#121—531no—
261#121—1061no—
262#121—1061no—
263#120—1061no—
264#120—1061no—
265C4×Q8—531no—
266C4×D4—531no—
267C4×D4—531no—
268C4×D4—1061no—
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—1061no—
282#41—22no—
283#41—41no—
284C2×Q8—531no—
285C2×Q8—1061no—
286C2×Q8—1062no—
287C2×Q8—2121no—
288C2×Q8C22×D5311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—531no—
294C2×D4—531no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1062no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—2121no—
303C2×D4—2124no—
304C2×D4C22×D5311yescharacteristic
305C2×D4C22×D5311yescharacteristic
306C2×D4C22×D5311yescharacteristic
307C22×C4—21no—
308C22×C4—531no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4C22×D5311yescharacteristic
312SD16—22no—
313SD16—1062no—
314SD16—1062no—
315SD16—1062no—
316SD16—2121no—
317SD16C22⋊D5314yescomplemented
318D8—22no—
319D8—22no—
320D8—2121no—
321M16—22no—
322C2×C8—531no—
323C2×C8—531no—
324C2×C8—1062no—
325C2×C8C22×D5311yescharacteristic
326C2×C8C22×D5311yescharacteristic
327C4⋊C4—531no—
328C4⋊C4—531no—
329C4⋊C4—531no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C22⋊C4—531no—
333C22⋊C4—531no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C42—531no—
342C42—1061no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—531no—
350C23—1061no—
351C23—4241no—
352C23—2122no—
353C23—2122no—
354C23D4×D5311yescharacteristic
355Q8—21no—
356Q8—1061no—
357Q8—1061no—
358Q8—2122no—
359Q8C2×C22⋊D5312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—1062no—
368D4—1061no—
369D4—1061no—
370D4—4241no—
371D4—2122no—
372D4—2122no—
373D4—2122no—
374D4—2122no—
375D4—2122no—
376D4C2×C22⋊D5312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—531no—
382C2×C4—531no—
383C2×C4—531no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4D4×D5311yescharacteristic
392C2×C4C23×D5311yescharacteristic
393C8—21no—
394C8—1061no—
395C8—1061no—
396C8C2×C22⋊D5312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—532no—
411C22—1061no—
412C22—1061no—
413C22—4244no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22C2×D4×D5311yescharacteristic
418C4—21no—
419C4—22no—
420C4—1061no—
421C4—1061no—
422C4—1061no—
423C4—2121no—
424C4—2121no—
425C4—2121no—
426C4C22×C22⋊D5311yescharacteristic
427C4#1313326211yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—1062no—
433C2—4241no—
434C2D4×C22⋊D5311yescharacteristic
435C2AGL(1,ℤ/8)×D5311yescharacteristic
436C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC22×C53C22×D4
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53#3265
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D53C2semidirect
2. #16811280C2semidirect
3. D4×C22⋊D53C2semidirect
4. #16811011C2semidirect
5. #16810869C2semidirect
6. #16810856C2semidirect
7. #16810836C2semidirect
8. #16810821C2semidirect
9. #16810774C2semidirect
10. #16810768C2semidirect
11. #16810760C2semidirect
12. #16810755C2semidirect
13. #16810724C2semidirect
14. #16810692C2semidirect
15. C2×AGL(1,ℤ/8)×C53C2semidirect
16. C2×Q8×D53C22semidirect
19. C2×#11719913C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719924C22semidirect
24. C2×#11719922C22semidirect
25. C2×C8×D53C22semidirect
28. #13133204C22semidirect
30. #13133196C22semidirect
35. #13133188C22semidirect
38. Q8×Dic53C22semidirect
42. #13133151C22semidirect
43. #13133146C22semidirect
44. #13133142C22semidirect
45. #13133141C22semidirect
46. #13133139C22semidirect
48. #13133138C22semidirect
49. #13133138C22semidirect
50. C22⋊#88322C22semidirect
51. #13133129C22semidirect
52. #13133127C22semidirect
53. C2×#41×C53C22semidirect
57. C2×SD16×C53C22semidirect
59. C2×D8×C53C22semidirect
60. C2×M16×C53C22semidirect
94. SD16×C53D4semidirect
137. Q8×C53C2×D4semidirect
175. C8×C53C2×D4semidirect
198. C2×D53AGL(1,ℤ/8)semidirect
219. C2×AGL(1,ℤ/8)D53semidirect
234. C53#3265semidirect
235. C2×#41C2×D53semidirect
247. C2×SD16C2×D53semidirect
250. C2×D8C2×D53semidirect
251. C2×M16C2×D53semidirect
317. SD16C22⋊D53semidirect
359. Q8C2×C22⋊D53semidirect
396. C8C2×C22⋊D53semidirect