Finite Group Databasebeta

← #20286526

Subgroups — #20286526

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811274C211yesmaximal, characteristic, complemented
2#16811257C211yesmaximal, characteristic, complemented
3#16810968C211yesmaximal, characteristic, complemented
4#16810938C211yesmaximal, characteristic, complemented
5#16810902C211yesmaximal, characteristic, complemented
6#16810860C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810817C211yesmaximal, characteristic, complemented
9#16810759C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic, complemented
11#16810705C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13C2×#13133137C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#480C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C22×D212C2211yescharacteristic
18C2×D4×D53—21no—
19C2×D4×D53C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×D424C2211yescharacteristic, complemented
22C2×#11719922C2211yescharacteristic, complemented
23C22×#88322C2211yescharacteristic, complemented
24C2×C8×D53C2211yescharacteristic, complemented
25#13133216C2211yescharacteristic, complemented
26#13133213C2211yescharacteristic
27#13133202—21no—
28#13133200—21no—
29#13133195—21no—
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic, complemented
32#13133191C2211yescharacteristic, complemented
33#13133190—21no—
34#13133187—21no—
35#13133185—21no—
36#13133179C2211yescharacteristic
37#13133179C2211yescharacteristic, complemented
38C4×C22⋊D53C2211yescharacteristic
39C4⋊C4×D53C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic, complemented
44#13133138C2211yescharacteristic, complemented
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#107×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×C53—21no—
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70D4×D53—22no—
71D4×D53—41no—
72D4×D53—44no—
73#11719945—44no—
74C2×D212—22no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—41no—
79C2×D212C2311yescharacteristic
80C2×D212C2311yescharacteristic
81C2×D212C2311yescharacteristic
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×Dic106—21no—
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊Dic53D411yescharacteristic
95#11719922—41no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919D411yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—22no—
111D424—41no—
112C8×D53—41no—
113C23×C53—21no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53C2×D411yescharacteristic
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2411yescharacteristic
136C22⋊D53—42no—
137C22⋊D53—42no—
138C22⋊D53—42no—
139C22⋊D53—81no—
140C22⋊D53D814yescomplemented
141C2×Dic53—21no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212—21no—
150D212—42no—
151D212—42no—
152D212—42no—
153D212#4112yescomplemented
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159Dic106—42no—
160C8×C53—21no—
161C8×C53—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53C2×D812yes—
170C22×C53C22×D411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—22no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—84no—
183C2×D53—84no—
184C2×D53C2×D812yes—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53C2×#4111yescharacteristic
190C4×C53H(5,𝔽2)11yescharacteristic
191Dic53—21no—
192Dic53—21no—
193Dic53—41no—
194Dic53C2×D812yes—
195#3294—531nomaximal, Sylow
196C2×C53—22no—
197C2×C53—81no—
198C2×C53#54611yescharacteristic
199C2×C53C22×D811yescharacteristic
200C2×C53#57711yescharacteristic
201D53—22no—
202D53—42no—
203D53—81no—
204D53—81no—
205#547—531no—
206D42—531no—
207#486—531no—
208#480—531no—
209#480D5311yescharacteristic, complemented
210#474—531no—
211#466—531no—
212#466—531no—
213#459—531no—
214#447—531no—
215#447—531no—
216C4×D8—531no—
217C8×D4—531no—
218C2×#102—531no—
219C2×#102—531no—
220C53#329411yescharacteristic, complemented, Sylow
221C2×#41—1061no—
222C22×D4—531no—
223C22×D4—531no—
224C22×D4—1061no—
225C2×D8—531no—
226C2×D8—531no—
227C2×D8—531no—
228C22×C8—531no—
229C22×C8—531no—
230C4⋊Q8—531no—
231#127—531no—
232#123—1061no—
233#123—1061no—
234C22⋊Q8—1061no—
235#121—531no—
236#121—531no—
237#121—531no—
238#121—1061no—
239#121C2×D5311yescharacteristic
240#120—1061no—
241#120—1061no—
242C4×D4—531no—
243C4×D4—531no—
244C4×D4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4C2×D5311yescharacteristic, complemented
247#107—531no—
248#107C2×D5311yescharacteristic, complemented
249#107C2×D5311yescharacteristic, complemented
250C4⋊C8—531no—
251#102—531no—
252#102—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259#102C2×D5311yescharacteristic, complemented
260#102C2×D5311yescharacteristic, complemented
261C22⋊C8—531no—
262C22⋊C8C2×D5311yescharacteristic, complemented
263C4×C8—531no—
264C24—1061no—
265C24—1061no—
266#41—2124no—
267C2×Q8—1061no—
268C2×D4—21no—
269C2×D4—531no—
270C2×D4—531no—
271C2×D4—531no—
272C2×D4—531no—
273C2×D4—1062no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1062no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2124no—
285C2×D4C22×D5311yescharacteristic
286C22×C4—21no—
287C22×C4—531no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4C22×D5311yescharacteristic
292D8—1062no—
293D8—2121no—
294D8—2121no—
295C2×C8—531no—
296C2×C8—531no—
297C2×C8—2121no—
298C2×C8—2121no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4C22×D5311yescharacteristic
307C4⋊C4C22×D5311yescharacteristic
308C4⋊C4C22×D5311yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—531no—
311C22⋊C4—531no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—531no—
318C23—21no—
319C23—531no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326C23—2122no—
327C23—2122no—
328C23—2122no—
329C23D4×D5311yescharacteristic
330Q8—2122no—
331D4—21no—
332D4—21no—
333D4—42no—
334D4—534no—
335D4—532no—
336D4—1061no—
337D4—1061no—
338D4—1061no—
339D4—1061no—
340D4—1061no—
341D4—4241no—
342D4—2122no—
343D4—2122no—
344D4—2122no—
345D4—2122no—
346D4—2122no—
347D4—2122no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—531no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4D4×D5311yescharacteristic
364C2×C4C23×D5311yescharacteristic
365C8—21no—
366C8—21no—
367C8—1061no—
368C8—1061no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—532no—
375C22—1061no—
376C22—1061no—
377C22—1062no—
378C22—4244no—
379C22—4244no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22—2121no—
388C22D8×D5312yes—
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—532no—
395C4—1061no—
396C4—1061no—
397C4—2121no—
398C4#41×D5311yescharacteristic
399C4#1313326311yescharacteristic
400C2—22no—
401C2—81no—
402C2—1062no—
403C2—4241no—
404C2—4241no—
405C2—2122no—
406C2#1681126511yescharacteristic
407C2C2×D8×D5311yescharacteristic
408C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3294
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #16811274C2semidirect
2. #16811257C2semidirect
3. #16810968C2semidirect
4. #16810938C2semidirect
5. #16810902C2semidirect
6. #16810860C2semidirect
7. #16810857C2semidirect
8. #16810817C2semidirect
9. #16810759C2semidirect
10. #16810758C2semidirect
11. #16810705C2semidirect
12. #16810686C2semidirect
13. C2×#13133137C2semidirect
14. #102×D53C2semidirect
15. C53×#480C2semidirect
21. C2×D424C22semidirect
22. C2×#11719922C22semidirect
23. C22×#88322C22semidirect
24. C2×C8×D53C22semidirect
25. #13133216C22semidirect
31. #13133192C22semidirect
32. #13133191C22semidirect
37. #13133179C22semidirect
39. C4⋊C4×D53C22semidirect
40. #13133150C22semidirect
41. #13133146C22semidirect
43. #13133139C22semidirect
44. #13133138C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. #13133137C22semidirect
50. #13133129C22semidirect
51. C8×Dic53C22semidirect
53. C2×C4⋊C4×C53C22semidirect
54. #107×C53C22semidirect
55. #107×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
140. C22⋊D53D8semidirect
153. D212#41semidirect
209. #480D53semidirect
220. C53#3294semidirect
246. C2×C4⋊C4C2×D53semidirect
248. #107C2×D53semidirect
249. #107C2×D53semidirect
259. #102C2×D53semidirect
260. #102C2×D53semidirect
262. C22⋊C8C2×D53semidirect