Finite Group Databasebeta

← #20286312

Subgroups — #20286312

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×#13133244C211yesmaximal, characteristic, complemented
3#16811275C211yesmaximal, characteristic, complemented
4#16811257C211yesmaximal, characteristic, complemented
5#16810994C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810827C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810775C211yesmaximal, characteristic, complemented
10#16810773C211yesmaximal, characteristic, complemented
11#16810771C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810705C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic, complemented
17C2×#11719945C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C2×D4×D53—21no—
20C2×D4×D53C2211yescharacteristic
21#13133245—24no—
22#13133244—24no—
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic, complemented
25C2×#11719923C2211yescharacteristic, complemented
26C2×#11719924C2211yescharacteristic
27C2×#11719924C2211yescharacteristic, complemented
28C2×D424C2211yescharacteristic
29C2×D424C2211yescharacteristic, complemented
30C2×#11719922C2211yescharacteristic, complemented
31C2×#11719912C2211yescharacteristic, complemented
32#13133215C2211yescharacteristic, complemented
33#13133213C2211yescharacteristic
34#13133202—21no—
35#13133199—21no—
36#13133197C2211yescharacteristic, complemented
37#13133192C2211yescharacteristic
38#13133190—21no—
39#13133190—21no—
40#13133187—21no—
41#13133185—21no—
42#13133179C2211yescharacteristic
43#13133179C2211yescharacteristic, complemented
44C4×C22⋊D53C2211yescharacteristic
45C4⋊C4×D53—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×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
72#11719949—22no—
73#11719949—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×D53—21no—
91C2×C4×D53C2311yescharacteristic
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—41no—
106#11719922—22no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112C4⋊Dic53—21no—
113#11719919—21no—
114#11719919—21no—
115#11719919D411yescharacteristic
116C4×Dic53C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118C2×#88322C2311yescharacteristic
119D424—22no—
120D424D414yescomplemented
121#11719913—22no—
122#11719913—22no—
123#11719912—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—42no—
162D212—42no—
163D212—42no—
164D212C2×D412yes—
165D212C2×D412yescomplemented
166C4×D53—21no—
167C4×D53—21no—
168C4×D53—42no—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—42no—
172Dic106—21no—
173Dic106—21no—
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—22no—
187C2×D53—22no—
188C2×D53—21no—
189C2×D53—21no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—84no—
198C2×D53—84no—
199C4×C53—21no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53C22×D411yescharacteristic
204C4×C53H(5,𝔽2)11yescharacteristic
205Dic53—21no—
206Dic53—21no—
207Dic53—41no—
208Dic53AGL(1,ℤ/8)12yescomplemented
209#3272—531nomaximal, Sylow
210C2×C53—41no—
211C2×C53—81no—
212C2×C53D4211yescharacteristic
213C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C53#57711yescharacteristic
215D53—41no—
216D53—42no—
217D53—42no—
218D53—81no—
219C2×AGL(1,ℤ/8)—531no—
220C2×AGL(1,ℤ/8)—531no—
221#547—531no—
222D42—531no—
223#492—531no—
224#468—531no—
225#468D5311yescharacteristic, complemented
226#461—531no—
227#459—531no—
228#450—531no—
229#450—531no—
230#449—531no—
231#449—531no—
232C4×D8—531no—
233#436—531no—
234C53#327211yescharacteristic, complemented, Sylow
235C2×#41—531no—
236C2×#41—531no—
237C22×D4—531no—
238C22×D4—531no—
239C22×D4—1061no—
240AGL(1,ℤ/8)—1064no—
241AGL(1,ℤ/8)—1064no—
242C2×SD16—531no—
243C2×SD16—531no—
244C2×SD16—531no—
245C2×SD16—531no—
246C2×SD16—531no—
247C2×SD16C2×D5311yescharacteristic, complemented
248C2×D8—531no—
249C2×D8—531no—
250C2×D8—531no—
251C2×M16—531no—
252C2×M16C2×D5311yescharacteristic, complemented
253C4⋊Q8—531no—
254#127—531no—
255#123—1061no—
256#123—1061no—
257C22⋊Q8—531no—
258C22⋊Q8C2×D5311yescharacteristic, complemented
259#121—531no—
260#121—1061no—
261#121—1061no—
262#121C2×D5311yescharacteristic
263#120—1061no—
264#120—1061no—
265C4×D4—531no—
266C4×D4—531no—
267C4×D4—531no—
268C2×C4⋊C4—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—
279C4×C8—531no—
280C24—1061no—
281C24—1061no—
282#41—1062no—
283#41—1062no—
284#41—2121no—
285#41—2121no—
286C2×Q8—531no—
287C2×Q8C22×D5311yescharacteristic
288C2×D4—21no—
289C2×D4—531no—
290C2×D4—531no—
291C2×D4—531no—
292C2×D4—531no—
293C2×D4—1062no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1062no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2124no—
305C2×D4C22×D5311yescharacteristic
306C22×C4—531no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4C22×D5311yescharacteristic
312SD16—22no—
313SD16—1062no—
314SD16—1062no—
315SD16—1062no—
316SD16—2121no—
317SD16—2121no—
318D8—534no—
319D8—1062no—
320D8—1062no—
321M16—22no—
322M16—1062no—
323C2×C8—531no—
324C2×C8—531no—
325C2×C8C22×D5311yescharacteristic
326C2×C8C22×D5311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—531no—
329C4⋊C4—1061no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4C22×D5311yescharacteristic
333C4⋊C4C22×D5311yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—531no—
337C22⋊C4—531no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C22⋊C4—1061no—
342C42—531no—
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—1061no—
358Q8—1061no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—532no—
363D4—532no—
364D4—1062no—
365D4—1061no—
366D4—1061no—
367D4—1061no—
368D4—1061no—
369D4—4241no—
370D4—2122no—
371D4—2122no—
372D4—2122no—
373D4—2122no—
374D4—2122no—
375D4—2122no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—531no—
382C2×C4—531no—
383C2×C4—531no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—2122no—
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—1061no—
404C22—1061no—
405C22—1061no—
406C22—1062no—
407C22—1062no—
408C22—4244no—
409C22—4244no—
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—532no—
423C4—1061no—
424C4—1061no—
425C4—2121no—
426C4C2×D4×D5311yescharacteristic
427C4#1313326311yescharacteristic
428C2—41no—
429C2—81no—
430C2—4241no—
431C2—2121no—
432C2—2122no—
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#3272
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×#13133244C2semidirect
3. #16811275C2semidirect
4. #16811257C2semidirect
5. #16810994C2semidirect
6. #16810870C2semidirect
7. #16810827C2semidirect
8. #16810817C2semidirect
9. #16810775C2semidirect
10. #16810773C2semidirect
11. #16810771C2semidirect
12. #16810766C2semidirect
13. #16810705C2semidirect
14. #16810699C2semidirect
15. C53×#468C2semidirect
16. C2×#11719949C22semidirect
17. C2×#11719945C22semidirect
24. C2×#11719913C22semidirect
25. C2×#11719923C22semidirect
27. C2×#11719924C22semidirect
29. C2×D424C22semidirect
30. C2×#11719922C22semidirect
31. C2×#11719912C22semidirect
32. #13133215C22semidirect
36. #13133197C22semidirect
43. #13133179C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
49. #13133137C22semidirect
50. #13133137C22semidirect
51. C22⋊#88322C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C2×SD16×C53C22semidirect
55. C2×M16×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #107×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
120. D424D4semidirect
165. D212C2×D4semidirect
175. C8×C53C2×D4semidirect
208. Dic53AGL(1,ℤ/8)semidirect
225. #468D53semidirect
234. C53#3272semidirect
247. C2×SD16C2×D53semidirect
252. C2×M16C2×D53semidirect
258. C22⋊Q8C2×D53semidirect
269. #107C2×D53semidirect
272. #103C2×D53semidirect
276. #102C2×D53semidirect
396. C8D4×D53semidirect