Finite Group Databasebeta

← #20286261

Subgroups — #20286261

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×#13133241C211yesmaximal, characteristic, complemented
2C2×#13133236C211yesmaximal, characteristic, complemented
3#16811293C211yesmaximal, characteristic, complemented
4#16811268C211yesmaximal, characteristic, complemented
5#16811010C211yesmaximal, characteristic, complemented
6#16810868C211yesmaximal, characteristic, complemented
7#16810858C211yesmaximal, characteristic, complemented
8#16810846C211yesmaximal, characteristic, complemented
9#16810786C211yesmaximal, characteristic, complemented
10#16810769C211yesmaximal, characteristic, complemented
11#16810765C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810723C211yesmaximal, characteristic, complemented
14#16810693C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×#11719947C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133241—24no—
21#13133236—24no—
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic, complemented
26C2×#11719923C2211yescharacteristic, complemented
27C2×#11719922C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719912C2211yescharacteristic
30C2×C8×D53C2211yescharacteristic, complemented
31#13133213—21no—
32#13133205C2211yescharacteristic
33#13133201—21no—
34#13133197C2211yescharacteristic
35#13133192C2211yescharacteristic
36#13133191—21no—
37#13133189—21no—
38#13133189—21no—
39#13133188C2211yescharacteristic
40#13133188C2211yescharacteristic, complemented
41#13133187—21no—
42C4×Dic106C2211yescharacteristic, complemented
43C4×D212C2211yescharacteristic
44D4×Dic53—21no—
45C2×#11719921—21no—
46#13133150C2211yescharacteristic, complemented
47#13133142C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic
50#13133138C2211yescharacteristic
51#13133129C2211yescharacteristic
52#13133129C2211yescharacteristic, complemented
53#13133128C2211yescharacteristic, complemented
54C2×D8×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic
56#127×C53C2211yescharacteristic, complemented
57C4×D4×C53C2211yescharacteristic
58C4⋊C8×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77#11719947—22no—
78#11719947—22no—
79#11719947—22no—
80#11719947—41no—
81#11719947—41no—
82#11719947—41no—
83D4×D53—22no—
84D4×D53—41no—
85C2×D212—21no—
86C2×D212D411yescharacteristic
87C2×D212C2311yescharacteristic
88C2×C4×D53C2311yescharacteristic
89C2×C4×D53C2311yescharacteristic
90C2×Dic106C2311yescharacteristic
91C2×Dic106C2311yescharacteristic
92D8×C53—22no—
93D8×C53—22no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101#11719923—22no—
102#11719923—41no—
103#11719923D414yescomplemented
104#11719922—22no—
105#11719922—41no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921D411yescharacteristic
111#11719921D411yescharacteristic
112C4⋊Dic53—21no—
113C4⋊Dic53D411yescharacteristic
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116C4×Dic53—21no—
117C2×#88322C2311yescharacteristic
118C2×#88322C2311yescharacteristic
119Dic212—22no—
120#11719913—22no—
121#11719912—22no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130C22×D53C2×D411yescharacteristic
131C22×D53C2×D411yescharacteristic
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—42no—
138D4×C53—81no—
139D4×C53C2×D412yes—
140C2×C4×C53—21no—
141C2×C4×C53—81no—
142C2×C4×C53C2×D411yescharacteristic
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2411yescharacteristic
145C22⋊D53—42no—
146C22⋊D53—42no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—81no—
150C22⋊D53—81no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159D212—21no—
160D212—21no—
161D212—41no—
162D212—81no—
163C4×D53—21no—
164C4×D53—21no—
165C4×D53—21no—
166C4×D53—21no—
167Dic106—21no—
168Dic106—21no—
169Dic106—21no—
170Dic106C2×D412yes—
171C8×C53—21no—
172C8×C53—21no—
173#88322—21no—
174#88322C2×D412yes—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—21no—
188C2×D53—21no—
189C2×D53—21no—
190C2×D53—21no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53C22×D411yescharacteristic
199C4×C53H(5,𝔽2)11yescharacteristic
200Dic53—22no—
201Dic53—21no—
202Dic53—21no—
203Dic53—41no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—42no—
207C2×C53—81no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—41no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
217#550—531no—
218#546—531no—
219#496—531no—
220#468—531no—
221#466—531no—
222#464—531no—
223#459D5311yescharacteristic, complemented
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×SD16—531no—
242C2×D8—531no—
243C2×D8—531no—
244C2×D8C2×D5311yescharacteristic
245C2×D8C2×D5311yescharacteristic
246C2×M16—531no—
247C22×C8—531no—
248#127—1061no—
249#127C2×D5311yescharacteristic, complemented
250#124—531no—
251#123—1061no—
252C22⋊Q8—531no—
253#121—531no—
254#121—531no—
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121—1061no—
259#120—1061no—
260C4×Q8—531no—
261C4×D4—531no—
262C4×D4—1061no—
263C4×D4C2×D5311yescharacteristic
264C2×C22⋊C4—1061no—
265#107—531no—
266C4⋊C8C2×D5311yescharacteristic, complemented
267#103—531no—
268#103—531no—
269#102—531no—
270#102—531no—
271#102C2×D5311yescharacteristic, complemented
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×Q8—531no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1062no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C2×D4C22×D5311yescharacteristic
299C2×D4C22×D5311yescharacteristic
300C2×D4C22×D5311yescharacteristic
301C22×C4—21no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—1061no—
307Q16—1062no—
308SD16—534no—
309SD16—1062no—
310SD16—1062no—
311SD16—2121no—
312D8—22no—
313D8—22no—
314D8—1062no—
315D8—2121no—
316M16—1062no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—1062no—
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4C22×D5311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—531no—
329C22⋊C4—531no—
330C22⋊C4—1061no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C42—1061no—
337C42C22×D5311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—531no—
342C23—531no—
343C23—1061no—
344C23—4241no—
345C23—2122no—
346C23—2122no—
347Q8—532no—
348Q8—1061no—
349Q8—1061no—
350Q8—1061no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—1061no—
359D4—1061no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—2121no—
364D4—2122no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368D4D4×D5312yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—531no—
372C2×C4—531no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4D4×D5311yescharacteristic
384C2×C4D4×D5311yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—21no—
387C8—21no—
388C8—532no—
389C8—1061no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—1061no—
400C22—1061no—
401C22—1061no—
402C22—1061no—
403C22—1061no—
404C22—1061no—
405C22—4244no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—1061no—
414C4—1061no—
415C4—1062no—
416C4—2121no—
417C4—2121no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326311yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—4241no—
424C2—2121no—
425C2—2121no—
426C2#1681125711yescharacteristic
427C2#1681141911yescharacteristic
428C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3268
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. C2×#13133236C2semidirect
3. #16811293C2semidirect
4. #16811268C2semidirect
5. #16811010C2semidirect
6. #16810868C2semidirect
7. #16810858C2semidirect
8. #16810846C2semidirect
9. #16810786C2semidirect
10. #16810769C2semidirect
11. #16810765C2semidirect
12. #16810755C2semidirect
13. #16810723C2semidirect
14. #16810693C2semidirect
15. C53×#459C2semidirect
18. C2×#11719947C22semidirect
22. C2×Dic212C22semidirect
25. C2×#11719923C22semidirect
26. C2×#11719923C22semidirect
30. C2×C8×D53C22semidirect
40. #13133188C22semidirect
42. C4×Dic106C22semidirect
46. #13133150C22semidirect
47. #13133142C22semidirect
48. #13133140C22semidirect
52. #13133129C22semidirect
53. #13133128C22semidirect
56. #127×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #102×C53C22semidirect
60. #102×C53C22semidirect
103. #11719923D4semidirect
223. #459D53semidirect
230. C53#3268semidirect
249. #127C2×D53semidirect
266. C4⋊C8C2×D53semidirect
271. #102C2×D53semidirect
272. #102C2×D53semidirect