Finite Group Databasebeta

← #20286262

Subgroups — #20286262

Order 6784.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133245C211yesmaximal, characteristic, complemented
2C2×#13133238C211yesmaximal, characteristic, complemented
3#16811291C211yesmaximal, characteristic, complemented
4#16811264C211yesmaximal, characteristic, complemented
5#16811009C211yesmaximal, characteristic, complemented
6#16810860C211yesmaximal, characteristic, complemented
7#16810848C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810769C211yesmaximal, characteristic, complemented
12#16810756C211yesmaximal, characteristic, complemented
13#16810722C211yesmaximal, characteristic, complemented
14#16810692C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719947C2211yescharacteristic, complemented
19C22×D212C2211yescharacteristic
20#13133245—24no—
21#13133238—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719913C2211yescharacteristic
25C2×#11719924C2211yescharacteristic, complemented
26C2×D424C2211yescharacteristic
27C2×D424C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719922C2211yescharacteristic
30C2×C8×D53C2211yescharacteristic, complemented
31#13133213—21no—
32#13133213C2211yescharacteristic, complemented
33#13133206C2211yescharacteristic
34#13133202—21no—
35#13133192—21no—
36#13133192C2211yescharacteristic
37#13133192C2211yescharacteristic, complemented
38#13133190—21no—
39#13133189—21no—
40#13133187—21no—
41#13133182C2211yescharacteristic, complemented
42#13133179C2211yescharacteristic
43C4×C22⋊D53C2211yescharacteristic
44#13133177—21no—
45C22⋊C4×D53—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic
52#13133129C2211yescharacteristic, complemented
53#13133127C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×M16×C53C2211yescharacteristic
56C22⋊Q8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71C22×Dic53—21no—
72#11719949—22no—
73#11719949—41no—
74#11719947—22no—
75#11719947—41no—
76#11719945—22no—
77#11719945—41no—
78C2×D212—22no—
79C2×D212—21no—
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×Dic106C2311yescharacteristic
92SD16×C53—22no—
93M16×C53—22no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C4⋊C4×C53—21no—
97C4⋊C4×C53C2311yescharacteristic
98C4⋊C4×C53C2311yescharacteristic
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊Dic53D411yescharacteristic
102#11719925—22no—
103#11719924—41no—
104#11719922—22no—
105#11719922—41no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118D424—22no—
119D424—22no—
120#11719913—22no—
121#11719913D414yescomplemented
122C8×D53—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—81no—
131C22×D53C2×D411yescharacteristic
132Q8×C53—21no—
133Q8×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—81no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2411yescharacteristic
144C22⋊D53—41no—
145C22⋊D53—42no—
146C22⋊D53—42no—
147C22⋊D53—42no—
148C22⋊D53—81no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53C2×D411yescharacteristic
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—21no—
155D212—21no—
156D212—21no—
157D212—21no—
158D212—21no—
159D212—42no—
160D212—42no—
161D212—81no—
162D212—81no—
163D212C2×D412yes—
164C4×D53—21no—
165C4×D53—21no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—81no—
169Dic106—21no—
170Dic106C2×D412yes—
171C8×C53—21no—
172C8×C53C2×D412yes—
173#88322—21no—
174#88322—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53C22×D411yescharacteristic
182C2×D53—21no—
183C2×D53—22no—
184C2×D53—21no—
185C2×D53—21no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C4×C53—21no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53C22×D411yescharacteristic
200C4×C53H(5,𝔽2)11yescharacteristic
201Dic53—22no—
202Dic53—21no—
203Dic53—21no—
204Dic53—21no—
205#3268—531nomaximal, Sylow
206C2×C53—41no—
207C2×C53—81no—
208C2×C53D4211yescharacteristic
209C2×C53#57511yescharacteristic
210C2×C53#57611yescharacteristic
211D53—41no—
212D53—42no—
213D53—81no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
217#550—531no—
218#546—531no—
219#496—531no—
220#468D5311yescharacteristic, complemented
221#466—531no—
222#464—531no—
223#459—531no—
224#452—531no—
225#449—531no—
226#449—531no—
227#447—531no—
228#440—531no—
229#435—531no—
230C53#326811yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×#41—531no—
233C2×#41—531no—
234C22×D4—531no—
235AGL(1,ℤ/8)—1064no—
236#135—1064no—
237C2×Q16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16C2×D5311yescharacteristic, complemented
242C2×D8—531no—
243C2×D8—531no—
244C2×D8—531no—
245C2×D8—531no—
246C2×M16C2×D5311yescharacteristic
247C22×C8—531no—
248#127—531no—
249#127—1061no—
250#124—531no—
251#123—1061no—
252C22⋊Q8C2×D5311yescharacteristic
253#121—531no—
254#121—531no—
255#121—1061no—
256#121—1061no—
257#121—1061no—
258#121C2×D5311yescharacteristic
259#120—1061no—
260C4×Q8—531no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4—1061no—
264C2×C22⋊C4—1061no—
265#107C2×D5311yescharacteristic, complemented
266C4⋊C8—531no—
267#103—531no—
268#103C2×D5311yescharacteristic, complemented
269#102—531no—
270#102—531no—
271#102—531no—
272#102C2×D5311yescharacteristic, complemented
273C22⋊C8—531no—
274C22⋊C8—531no—
275#97—531no—
276C24—1061no—
277#41—1062no—
278#41—1062no—
279#41—1062no—
280#41—2121no—
281#41—2121no—
282#41—2121no—
283C2×Q8—531no—
284C2×Q8C22×D5311yescharacteristic
285C2×D4—21no—
286C2×D4—531no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—531no—
290C2×D4—1062no—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—2121no—
300C2×D4C22×D5311yescharacteristic
301C22×C4—531no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4C22×D5311yescharacteristic
307Q16—1062no—
308SD16—22no—
309SD16—534no—
310SD16—1062no—
311SD16—2121no—
312D8—1062no—
313D8—1062no—
314D8—1062no—
315D8—2121no—
316M16—22no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—1062no—
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4C22×D5311yescharacteristic
326C4⋊C4C22×D5311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—531no—
330C22⋊C4—531no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C42—531no—
337C42—1061no—
338C23—21no—
339C23—531no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—4241no—
344C23—2122no—
345C23—2122no—
346C23D4×D5311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—532no—
350Q8—1061no—
351D4—21no—
352D4—21no—
353D4—81no—
354D4—532no—
355D4—1061no—
356D4—1061no—
357D4—1061no—
358D4—1061no—
359D4—1061no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—2122no—
364D4—2122no—
365D4—2121no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—531no—
375C2×C4—531no—
376C2×C4—531no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—4241no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4D4×D5311yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—21no—
387C8—1061no—
388C8—1061no—
389C8D4×D5312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—1061no—
397C22—1061no—
398C22—1061no—
399C22—1062no—
400C22—4244no—
401C22—2121no—
402C22—2121no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—1061no—
415C4—1061no—
416C4—1061no—
417C4—1062no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326311yescharacteristic
420C2—41no—
421C2—81no—
422C2—4241no—
423C2—4241no—
424C2—2121no—
425C2—2122no—
426C2#1681125711yescharacteristic
427C2#1681142211yescharacteristic
428C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3268
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×#13133238C2semidirect
3. #16811291C2semidirect
4. #16811264C2semidirect
5. #16811009C2semidirect
6. #16810860C2semidirect
7. #16810848C2semidirect
8. #16810817C2semidirect
9. #16810783C2semidirect
10. #16810771C2semidirect
11. #16810769C2semidirect
12. #16810756C2semidirect
13. #16810722C2semidirect
14. #16810692C2semidirect
15. C53×#468C2semidirect
18. C2×#11719947C22semidirect
22. C2×#11719925C22semidirect
25. C2×#11719924C22semidirect
30. C2×C8×D53C22semidirect
32. #13133213C22semidirect
37. #13133192C22semidirect
41. #13133182C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
50. #13133137C22semidirect
52. #13133129C22semidirect
53. #13133127C22semidirect
54. C2×SD16×C53C22semidirect
58. #107×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
121. #11719913D4semidirect
220. #468D53semidirect
230. C53#3268semidirect
241. C2×SD16C2×D53semidirect
265. #107C2×D53semidirect
268. #103C2×D53semidirect
272. #102C2×D53semidirect