Finite Group Databasebeta

← #20286210

Subgroups — #20286210

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×#13133244C211yesmaximal, characteristic, complemented
2#16811284C211yesmaximal, characteristic, complemented
3D4×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16810866C211yesmaximal, characteristic, complemented
6#16810859C211yesmaximal, characteristic, complemented
7#16810840C211yesmaximal, characteristic, complemented
8#16810821C211yesmaximal, characteristic, complemented
9#16810772C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810762C211yesmaximal, characteristic, complemented
12#16810757C211yesmaximal, characteristic, complemented
13#16810724C211yesmaximal, characteristic, complemented
14#16810694C211yesmaximal, characteristic, complemented
15C22×SD16×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C22×C22⋊D53—21no—
18C2×D4×D53C2211yescharacteristic
19#13133244—24no—
20C2×#11719924C2211yescharacteristic, complemented
21C2×#11719924C2211yescharacteristic, complemented
22C2×D424C2211yescharacteristic, complemented
23C2×#11719922C2211yescharacteristic
24C2×#11719922C2211yescharacteristic, complemented
25C2×#11719912C2211yescharacteristic, complemented
26#13133213C2211yescharacteristic
27#13133206—21no—
28#13133206C2211yescharacteristic, complemented
29#13133193—21no—
30#13133193C2211yescharacteristic, complemented
31#13133191—21no—
32#13133191C2211yescharacteristic
33#13133191C2211yescharacteristic, complemented
34#13133189—21no—
35#13133188C2211yescharacteristic
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#13133141C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic, complemented
47#13133139C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic, complemented
50C22⋊#88322C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#13133127C2211yescharacteristic, complemented
53C22×Q8×C53C2211yescharacteristic, complemented
54C22×D4×C53C2211yescharacteristic
55C2×SD16×C53—24no—
56C2×SD16×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic, complemented
59C2×SD16×C53C2211yescharacteristic, complemented
60C22×C8×C53C2211yescharacteristic, complemented
61C24×C53—21no—
62C23×D53—21no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53—22no—
69C2×D4×C53—21no—
70C2×D4×C53—41no—
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—
84#11719949—22no—
85#11719949—41no—
86D4×D53—22no—
87D4×D53—41no—
88C2×D212—21no—
89C2×D212C2311yescharacteristic
90C2×D212C2311yescharacteristic
91C2×C4×D53—21no—
92C2×C4×D53C2311yescharacteristic
93SD16×C53—22no—
94SD16×C53—22no—
95SD16×C53—22no—
96SD16×C53D414yescomplemented
97C2×C8×C53—22no—
98C2×C8×C53C2311yescharacteristic
99C2×C8×C53C2311yescharacteristic
100C22⋊Dic53—21no—
101C22⋊Dic53—21no—
102C22⋊Dic53D411yescharacteristic
103#11719924—22no—
104#11719924—41no—
105#11719922—22no—
106#11719922—41no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
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
122D424—22no—
123#11719912—22no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—81no—
129C23×C53C2×D411yescharacteristic
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—81no—
135C22×D53C2×D411yescharacteristic
136Q8×C53—21no—
137Q8×C53—21no—
138Q8×C53—21no—
139Q8×C53—42no—
140Q8×C53C2×D412yescomplemented
141D4×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—42no—
145D4×C53C2×D412yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—42no—
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2411yescharacteristic
153C22⋊D53—22no—
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×Dic53—42no—
163C2×Dic53C2×D411yescharacteristic
164C2×Dic53C2×D411yescharacteristic
165C2×Dic53C2×D411yescharacteristic
166D212—21no—
167D212—21no—
168D212—21no—
169D212—21no—
170D212—42no—
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—42no—
174C8×C53—21no—
175C8×C53C2×D412yescomplemented
176#88322—21no—
177#88322—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—84no—
187C22×C53AGL(1,ℤ/8)12yescomplemented
188C22×C53C22×D411yescharacteristic
189C2×D53—21no—
190C2×D53—21no—
191C2×D53—21no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—84no—
199C4×C53—22no—
200C4×C53—21no—
201C4×C53—41no—
202C4×C53C22×D411yescharacteristic
203C4×C53H(5,𝔽2)11yescharacteristic
204Dic53—21no—
205Dic53—21no—
206Dic53—21no—
207Dic53—41no—
208Dic53—41no—
209#3265—531nomaximal, Sylow
210C2×C53—22no—
211C2×C53—42no—
212C2×C53—81no—
213C2×C53D4211yescharacteristic
214C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C53#57711yescharacteristic
216D53—41no—
217D53—81no—
218D53—81no—
219C2×AGL(1,ℤ/8)—531no—
220C22×SD16D5311yescharacteristic, complemented
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×#41—531no—
236C22×Q8C2×D5311yescharacteristic, complemented
237C22×D4—531no—
238C22×D4—1061no—
239C22×D4C2×D5311yescharacteristic
240AGL(1,ℤ/8)—1064no—
241C2×SD16—24no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16C2×D5311yescharacteristic
245C2×SD16C2×D5311yescharacteristic
246C2×SD16C2×D5311yescharacteristic, complemented
247C2×SD16C2×D5311yescharacteristic, complemented
248C2×D8—531no—
249C2×D8—531no—
250C2×D8—531no—
251C2×M16—531no—
252C22×C8C2×D5311yescharacteristic, complemented
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—1062no—
283#41—2121no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D5311yescharacteristic
288C2×Q8C22×D5311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—531no—
293C2×D4—531no—
294C2×D4—531no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1062no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4—2124no—
305C2×D4C22×D5311yescharacteristic
306C2×D4C22×D5311yescharacteristic
307C22×C4—21no—
308C22×C4—531no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4C22×D5311yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—22no—
315SD16—1062no—
316SD16—2121no—
317SD16C22⋊D5314yescomplemented
318D8—1062no—
319D8—1062no—
320D8—2121no—
321M16—1062no—
322C2×C8—22no—
323C2×C8—531no—
324C2×C8—531no—
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—42no—
346C23—42no—
347C23—81no—
348C23—531no—
349C23—1061no—
350C23—1061no—
351C23—4241no—
352C23—2122no—
353C23—2122no—
354C23D4×D5311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8C2×C22⋊D5312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—1062no—
365D4—1061no—
366D4—1061no—
367D4—1061no—
368D4—1061no—
369D4—4241no—
370D4—2122no—
371D4—2122no—
372D4—2122no—
373D4—2122no—
374D4—2122no—
375D4—2122no—
376D4C2×C22⋊D5312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—531no—
383C2×C4—531no—
384C2×C4—531no—
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—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—84no—
406C22—1061no—
407C22—1061no—
408C22—1061no—
409C22—4244no—
410C22—2121no—
411C22—2121no—
412C22—2121no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22#1313324412yescomplemented
417C22C2×D4×D5311yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—1061no—
422C4—1061no—
423C4—1061no—
424C4—2121no—
425C4—2121no—
426C4C22×C22⋊D5311yescharacteristic
427C4#1313326211yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—4241no—
432C2—4241no—
433C2—2121no—
434C2D4×C22⋊D5311yescharacteristic
435C2C2×#1313324411yescharacteristic
436C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×SD16×C53C2
socleC22×C53C22×D4
hypercenterC22×SD16D53
hypocenterC53#3265
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #16811284C2semidirect
3. D4×C22⋊D53C2semidirect
4. #16811009C2semidirect
5. #16810866C2semidirect
6. #16810859C2semidirect
7. #16810840C2semidirect
8. #16810821C2semidirect
9. #16810772C2semidirect
10. #16810767C2semidirect
11. #16810762C2semidirect
12. #16810757C2semidirect
13. #16810724C2semidirect
14. #16810694C2semidirect
15. C22×SD16×C53C2semidirect
16. C2×#11719949C22semidirect
20. C2×#11719924C22semidirect
21. C2×#11719924C22semidirect
22. C2×D424C22semidirect
24. C2×#11719922C22semidirect
25. C2×#11719912C22semidirect
28. #13133206C22semidirect
30. #13133193C22semidirect
33. #13133191C22semidirect
38. Q8×Dic53C22semidirect
42. #13133151C22semidirect
43. #13133146C22semidirect
44. #13133141C22semidirect
45. #13133141C22semidirect
46. #13133139C22semidirect
47. #13133139C22semidirect
49. #13133138C22semidirect
50. C22⋊#88322C22semidirect
51. #13133129C22semidirect
52. #13133127C22semidirect
53. C22×Q8×C53C22semidirect
58. C2×SD16×C53C22semidirect
59. C2×SD16×C53C22semidirect
60. C22×C8×C53C22semidirect
96. SD16×C53D4semidirect
140. Q8×C53C2×D4semidirect
175. C8×C53C2×D4semidirect
187. C22×C53AGL(1,ℤ/8)semidirect
220. C22×SD16D53semidirect
234. C53#3265semidirect
236. C22×Q8C2×D53semidirect
246. C2×SD16C2×D53semidirect
247. C2×SD16C2×D53semidirect
252. C22×C8C2×D53semidirect
317. SD16C22⋊D53semidirect
359. Q8C2×C22⋊D53semidirect
396. C8C2×C22⋊D53semidirect
416. C22#13133244semidirect