Finite Group Databasebeta

← #20286251

Subgroups — #20286251

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×#13133243C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3#16811294C211yesmaximal, characteristic, complemented
4#16811013C211yesmaximal, characteristic, complemented
5#16810871C211yesmaximal, characteristic, complemented
6#16810857C211yesmaximal, characteristic, complemented
7#16810842C211yesmaximal, characteristic, complemented
8#16810818C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810766C211yesmaximal, characteristic, complemented
12#16810757C211yesmaximal, characteristic, complemented
13#16810723C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic, complemented
18#13133243—24no—
19#13133237—24no—
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719924C2211yescharacteristic, complemented
24C2×#11719924C2211yescharacteristic, complemented
25C2×#11719922C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719917C2211yescharacteristic
30C22×#88322C2211yescharacteristic, complemented
31#13133214C2211yescharacteristic, complemented
32#13133213—21no—
33#13133191C2211yescharacteristic
34#13133191C2211yescharacteristic, complemented
35#13133190—21no—
36#13133190—21no—
37C4×Dic106C2211yescharacteristic, complemented
38#13133179—21no—
39#13133150C2211yescharacteristic, complemented
40C4⋊#88322C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133140C2211yescharacteristic, complemented
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133137C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic, complemented
49#13133128C2211yescharacteristic, complemented
50C2×#41×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic
52#124×C53C2211yescharacteristic
53#123×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53—21no—
56#121×C53C2211yescharacteristic
57#120×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C4×D4×C53C2211yescharacteristic
60C2×C22⋊C4×C53—21no—
61C24×C53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—41no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53—21no—
80C2×C22⋊D53—21no—
81#11719945—22no—
82#11719945—22no—
83#11719945—41no—
84#11719945—41no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212C2311yescharacteristic
88C2×D212C2311yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×Dic106C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53D411yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53D411yescharacteristic
101C22⋊C4×C53D411yescharacteristic
102C42×C53C2311yescharacteristic
103#11719925—22no—
104#11719924—22no—
105#11719924—41no—
106#11719923—22no—
107#11719923D414yescomplemented
108#11719922—22no—
109#11719922—22no—
110#11719922—22no—
111#11719922—41no—
112#11719921—21no—
113#11719921—21no—
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116C4×Dic53—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—81no—
128C23×C53C2×D411yescharacteristic
129C23×C53C2×D411yescharacteristic
130C22×D53—21no—
131C22×D53—21no—
132Q8×C53—21no—
133Q8×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—41no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—81no—
141D4×C53C2×D412yes—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—42no—
149C2×C4×C53—81no—
150C2×C4×C53C2×D411yescharacteristic
151C2×C4×C53C2×D411yescharacteristic
152C2×C4×C53C2×D411yescharacteristic
153C2×C4×C53C2×D411yescharacteristic
154C2×C4×C53C2411yescharacteristic
155C22⋊D53—42no—
156C22⋊D53—42no—
157C22⋊D53—81no—
158C22⋊D53—81no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161D212—21no—
162D212—21no—
163D212—21no—
164D212—21no—
165D212—42no—
166D212—81no—
167C4×D53—42no—
168C4×D53—42no—
169Dic106—21no—
170Dic106C2×D412yes—
171#88322—21no—
172#88322—21no—
173#88322—21no—
174#88322C2×D412yes—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—22no—
178C22×C53—21no—
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—84no—
188C22×C53C22×D411yescharacteristic
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53C22×D411yescharacteristic
202C4×C53H(5,𝔽2)11yescharacteristic
203Dic53—22no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—41no—
207C2×C53—41no—
208C2×C53—42no—
209C2×C53—81no—
210C2×C53D4211yescharacteristic
211C2×C53#57511yescharacteristic
212C2×C53#57611yescharacteristic
213D53—81no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
217#550—531no—
218#546D5311yescharacteristic, complemented
219#496—531no—
220#468—531no—
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×#41C2×D5311yescharacteristic
234C22×D4C2×D5311yescharacteristic
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×D8—531no—
245C2×D8—531no—
246C2×M16—531no—
247C22×C8—531no—
248#127—531no—
249#127—1061no—
250#124C2×D5311yescharacteristic
251#123—21no—
252C22⋊Q8C2×D5311yescharacteristic
253#121—21no—
254#121—531no—
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121C2×D5311yescharacteristic
259#120—21no—
260C4×Q8—531no—
261C4×D4—1061no—
262C4×D4C2×D5311yescharacteristic
263C4×D4C2×D5311yescharacteristic
264C2×C22⋊C4—21no—
265#107—531no—
266C4⋊C8—531no—
267#103—531no—
268#103—531no—
269#102—531no—
270#102—531no—
271#102—531no—
272#102—531no—
273C22⋊C8—531no—
274C22⋊C8—531no—
275#97—531no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—1062no—
280#41—1062no—
281#41—2121no—
282#41—2121no—
283C2×Q8—531no—
284C2×Q8C22×D5311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—531no—
291C2×D4—531no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4C22⋊D5311yescharacteristic
299C2×D4C22×D5311yescharacteristic
300C2×D4C22×D5311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307Q16—1062no—
308SD16—534no—
309SD16—1062no—
310SD16—1062no—
311SD16—2121no—
312D8—1062no—
313D8—1062no—
314D8—1062no—
315D8—2121no—
316M16—1062no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—531no—
321C2×C8—1062no—
322C4⋊C4—21no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4C22⋊D5311yescharacteristic
326C4⋊C4C22×D5311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4C22⋊D5311yescharacteristic
335C22⋊C4C22⋊D5311yescharacteristic
336C42—1061no—
337C42C22×D5311yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—1061no—
344C23—1061no—
345C23C2×C22⋊D5311yescharacteristic
346C23C2×C22⋊D5311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—532no—
350Q8—1061no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—1061no—
359D4—1061no—
360D4—1061no—
361D4—1061no—
362D4—4241no—
363D4—4241no—
364D4—4241no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368D4D4×D5312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4C2×C22⋊D5311yescharacteristic
382C2×C4C2×C22⋊D5311yescharacteristic
383C2×C4C2×C22⋊D5311yescharacteristic
384C2×C4C2×C22⋊D5311yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—532no—
387C8—1061no—
388C8—1061no—
389C8—1061no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—2121no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C22×C22⋊D5311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—1062no—
417C4—2121no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326211yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—4241no—
425C2—4241no—
426C2D4×C22⋊D5311yescharacteristic
427C2#1681142111yescharacteristic
428C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3268
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#13133243C2semidirect
2. C2×#13133237C2semidirect
3. #16811294C2semidirect
4. #16811013C2semidirect
5. #16810871C2semidirect
6. #16810857C2semidirect
7. #16810842C2semidirect
8. #16810818C2semidirect
9. #16810783C2semidirect
10. #16810768C2semidirect
11. #16810766C2semidirect
12. #16810757C2semidirect
13. #16810723C2semidirect
14. #16810695C2semidirect
15. C53×#546C2semidirect
17. C2×#11719945C22semidirect
20. C2×#11719925C22semidirect
23. C2×#11719924C22semidirect
24. C2×#11719924C22semidirect
30. C22×#88322C22semidirect
31. #13133214C22semidirect
34. #13133191C22semidirect
37. C4×Dic106C22semidirect
39. #13133150C22semidirect
40. C4⋊#88322C22semidirect
41. #13133141C22semidirect
42. #13133140C22semidirect
45. #13133137C22semidirect
46. #13133137C22semidirect
48. C22⋊#88322C22semidirect
49. #13133128C22semidirect
107. #11719923D4semidirect
218. #546D53semidirect
230. C53#3268semidirect