Finite Group Databasebeta

← #20285579

Subgroups — #20285579

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811192C211yesmaximal, characteristic, complemented
2#16811189C211yesmaximal, characteristic, complemented
3#16810950C211yesmaximal, characteristic, complemented
4#16810949C211yesmaximal, characteristic, complemented
5#16810947C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810860C211yesmaximal, characteristic, complemented
9#16810858C211yesmaximal, characteristic, complemented
10#16810759C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810756C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic, complemented
18C2×D4×D53C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719912C2211yescharacteristic, complemented
24C2×C8×D53C2211yescharacteristic, complemented
25#13133214—21no—
26#13133213—21no—
27#13133205—21no—
28#13133203—21no—
29#13133192C2211yescharacteristic
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic, complemented
32#13133192C2211yescharacteristic, complemented
33#13133192C2211yescharacteristic, complemented
34#13133191C2211yescharacteristic
35#13133188C2211yescharacteristic, complemented
36#13133187—21no—
37#13133187—21no—
38#13133187—21no—
39#13133185—21no—
40#13133175C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic, complemented
46#13133137C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68C2×C22⋊D53D411yescharacteristic
69D4×D53—41no—
70D4×D53—41no—
71#11719945—41no—
72#11719945—41no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212D411yescharacteristic
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×C4×D53C2311yescharacteristic
87C2×Dic106D411yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C2×C8×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C42×C53—21no—
96C22⋊Dic53—21no—
97#11719922—41no—
98#11719922—41no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53C2311yescharacteristic
107C4×Dic53—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111D424—41no—
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53C2×D411yescharacteristic
126C22×D53C2×D411yescharacteristic
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—41no—
138C22⋊D53—41no—
139C22⋊D53—42no—
140C22⋊D53—81no—
141C22⋊D53—81no—
142C2×Dic53—21no—
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212—21no—
150D212—21no—
151D212—41no—
152D212—42no—
153D212—81no—
154D212—81no—
155D212—81no—
156D212—81no—
157D212—81no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—21no—
162Dic106—41no—
163C8×C53—21no—
164C8×C53—21no—
165#88322—21no—
166#88322—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53C22×D411yescharacteristic
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—84no—
190C2×D53—84no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195C4×C53H(5,𝔽2)11yescharacteristic
196C4×C53H(5,𝔽2)11yescharacteristic
197Dic53—21no—
198Dic53—21no—
199Dic53—41no—
200#3199—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—81no—
203C2×C53#53411yescharacteristic
204C2×C53#57611yescharacteristic
205C2×C53#57611yescharacteristic
206D53—41no—
207D53—41no—
208D53—81no—
209D53—81no—
210D53—81no—
211#535—531no—
212#535—531no—
213#482—531no—
214#482—531no—
215#482—531no—
216#482D5311yescharacteristic, complemented
217#469—531no—
218#469—531no—
219#466—531no—
220#466—531no—
221#447—531no—
222#447—531no—
223#447—531no—
224#447—531no—
225#408—531no—
226C53#319911yescharacteristic, complemented, Sylow
227C2×#41—531no—
228C22×D4—531no—
229C22×D4—531no—
230C2×D8—531no—
231C2×D8—531no—
232C2×D8—531no—
233C2×D8—531no—
234C2×M16—531no—
235C22×C8—531no—
236#127—1061no—
237#127—1061no—
238#124—1061no—
239#124—1061no—
240#121—531no—
241#121—531no—
242#121—531no—
243#121—531no—
244#121—531no—
245#121—531no—
246#121—531no—
247#121C2×D5311yescharacteristic
248#120—1061no—
249#120—1061no—
250#120—1061no—
251#120—1061no—
252#117—531no—
253#117C2×D5311yescharacteristic, complemented
254#107—531no—
255#107C2×D5311yescharacteristic, complemented
256#106—531no—
257#106C2×D5311yescharacteristic, complemented
258#102—531no—
259#102—531no—
260#102—531no—
261#102—531no—
262#102—531no—
263#102—531no—
264#102C2×D5311yescharacteristic, complemented
265#102C2×D5311yescharacteristic, complemented
266C22⋊C8—531no—
267C22⋊C8—531no—
268C22⋊C8—531no—
269C22⋊C8C2×D5311yescharacteristic, complemented
270C24—1061no—
271C24—1061no—
272#41—2121no—
273#41—2121no—
274C2×Q8—531no—
275C2×D4—21no—
276C2×D4—531no—
277C2×D4—531no—
278C2×D4—531no—
279C2×D4—531no—
280C2×D4—531no—
281C2×D4—531no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4C22×D5311yescharacteristic
296C22×C4—531no—
297C22×C4—531no—
298C22×C4C22×D5311yescharacteristic
299D8—2121no—
300D8—2121no—
301D8—2121no—
302D8—2121no—
303M16—2121no—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—2121no—
307C2×C8C22×D5311yescharacteristic
308C2×C8C22×D5311yescharacteristic
309C4⋊C4—531no—
310C4⋊C4C22×D5311yescharacteristic
311C4⋊C4C22×D5311yescharacteristic
312C4⋊C4C22×D5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C42—21no—
324C42—1061no—
325C23—21no—
326C23—531no—
327C23—531no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—4241no—
332C23—4241no—
333C23—4241no—
334C23—4241no—
335C23—2122no—
336C23—2122no—
337C23D4×D5311yescharacteristic
338Q8—2121no—
339D4—21no—
340D4—21no—
341D4—81no—
342D4—1061no—
343D4—1061no—
344D4—1061no—
345D4—1061no—
346D4—1061no—
347D4—1061no—
348D4—4241no—
349D4—4241no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—2121no—
356D4—2122no—
357D4—2121no—
358D4—2121no—
359D4—2122no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—531no—
366C2×C4—531no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4D4×D5311yescharacteristic
373C2×C4C23×D5311yescharacteristic
374C8—21no—
375C8—21no—
376C8—1061no—
377C8—1061no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—1061no—
385C22—1061no—
386C22—1061no—
387C22—1061no—
388C22—1061no—
389C22—1061no—
390C22—4244no—
391C22—4244no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22—2121no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22C2×D4×D5311yescharacteristic
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—1061no—
407C4—1061no—
408C4—2121no—
409C4#1313326311yescharacteristic
410C4#1313326311yescharacteristic
411C2—41no—
412C2—81no—
413C2—4241no—
414C2—4241no—
415C2—4241no—
416C2—2121no—
417C2—2121no—
418C2#1681118411yescharacteristic
419C2#1681142511yescharacteristic
420C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811192C2semidirect
2. #16811189C2semidirect
3. #16810950C2semidirect
4. #16810949C2semidirect
5. #16810947C2semidirect
6. #16810877C2semidirect
7. #16810875C2semidirect
8. #16810860C2semidirect
9. #16810858C2semidirect
10. #16810759C2semidirect
11. #16810758C2semidirect
12. #16810758C2semidirect
13. #16810756C2semidirect
14. #16810588C2semidirect
15. C53×#482C2semidirect
17. C22×D212C22semidirect
18. C2×D4×D53C22semidirect
23. C2×#11719912C22semidirect
24. C2×C8×D53C22semidirect
31. #13133192C22semidirect
32. #13133192C22semidirect
33. #13133192C22semidirect
35. #13133188C22semidirect
40. #13133175C22semidirect
41. #13133150C22semidirect
42. #13133148C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133137C22semidirect
46. #13133137C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
53. #117×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
216. #482D53semidirect
226. C53#3199semidirect
253. #117C2×D53semidirect
255. #107C2×D53semidirect
257. #106C2×D53semidirect
264. #102C2×D53semidirect
265. #102C2×D53semidirect
269. C22⋊C8C2×D53semidirect