Finite Group Databasebeta

← #20284804

Subgroups — #20284804

Order 6784.

488 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811170C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic, complemented
3#16810943C211yesmaximal, characteristic, complemented
4#16810940C211yesmaximal, characteristic, complemented
5#16810912C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810817C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810758C211yesmaximal, characteristic, complemented
10#16810755C211yesmaximal, characteristic, complemented
11#16810736C211yesmaximal, characteristic, complemented
12#16810733C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14#16810678C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×D4×D53C2211yescharacteristic
18C22×C4×D53—21no—
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22#13133216C2211yescharacteristic, complemented
23#13133213C2211yescharacteristic, complemented
24#13133203—21no—
25#13133192C2211yescharacteristic
26#13133190—21no—
27#13133188C2211yescharacteristic
28C4×D212C2211yescharacteristic
29C4×D212C2211yescharacteristic, complemented
30D4×Dic53C2211yescharacteristic
31#13133179C2211yescharacteristic
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34#13133177—21no—
35C4⋊C4×D53C2211yescharacteristic
36C22⋊C4×D53—21no—
37C42×D53C2211yescharacteristic
38#13133149C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic, complemented
52C2×D8×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69D4×D53—22no—
70D4×D53—22no—
71#11719945—44no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—44no—
80C2×C4×D53D411yescharacteristic
81C2×C4×D53D411yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84D8×C53—22no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94#11719922—22no—
95#11719922—22no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53D411yescharacteristic
105C4×Dic53D411yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—22no—
110C23×C53—21no—
111C23×C53—21no—
112C22×D53—24no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53C2×D411yescharacteristic
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53#4112yescomplemented
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—24no—
130C22⋊D53—42no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—21no—
142D212—42no—
143D212#4112yescomplemented
144C4×D53—22no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—22no—
148C4×D53—42no—
149C4×D53—42no—
150C4×D53—42no—
151C4×D53—42no—
152Dic106—42no—
153C8×C53—21no—
154C8×C53—21no—
155#88322—21no—
156#88322—21no—
157C22×C53—22no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×D411yescharacteristic
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—22no—
167C2×D53—48no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53AGL(1,ℤ/8)12yes—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53C2×#4111yescharacteristic
179C4×C53C2×#4111yescharacteristic
180Dic53—21no—
181Dic53—21no—
182Dic53—41no—
183Dic53—41no—
184Dic53AGL(1,ℤ/8)12yes—
185#3109—531nomaximal, Sylow
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#52511yescharacteristic
189C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191D53—22no—
192D53—42no—
193D53—81no—
194#532—531no—
195C2×C4×D4—531no—
196#481—531no—
197#481—531no—
198#476—531no—
199#476—531no—
200#459—531no—
201#459—531no—
202#447—531no—
203#447—531no—
204#442—531no—
205#442—531no—
206#442—531no—
207#442D5311yescharacteristic, complemented
208#432—531no—
209C53#310911yescharacteristic, complemented, Sylow
210C2×#41—1061no—
211C22×D4—531no—
212C23×C4—1061no—
213C2×D8—531no—
214C2×D8—531no—
215C2×D8—531no—
216C2×D8C2×D5311yescharacteristic
217C4⋊Q8—531no—
218#127—531no—
219#124—1061no—
220#121—531no—
221#121—531no—
222#121—1061no—
223C4×D4—531no—
224C4×D4—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C4×D4—1061no—
229C4×D4—1061no—
230C4×D4C2×D5311yescharacteristic
231C4×D4C2×D5311yescharacteristic, complemented
232C2×C4⋊C4—531no—
233C2×C22⋊C4—1061no—
234C2×C42—531no—
235#106—531no—
236#106—531no—
237#106—531no—
238#106C2×D5311yescharacteristic, complemented
239C4⋊C8—531no—
240C4⋊C8—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102C2×D5311yescharacteristic
248#102C2×D5311yescharacteristic, complemented
249C22⋊C8—531no—
250C22⋊C8—531no—
251#97—531no—
252#97C2×D5311yescharacteristic, complemented
253C24—1061no—
254#41—2124no—
255C2×Q8—1061no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1062no—
263C2×D4—1062no—
264C2×D4—1061no—
265C2×D4C22×D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—2124no—
278D8—22no—
279D8—1062no—
280D8—1062no—
281D8—1062no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8C22×D5311yescharacteristic
285C2×C8C22×D5311yescharacteristic
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4C22×D5311yescharacteristic
291C4⋊C4C22×D5311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C42—531no—
301C42—531no—
302C42—531no—
303C42C22×D5311yescharacteristic
304C23—21no—
305C23—21no—
306C23—531no—
307C23—1064no—
308C23—1061no—
309C23—1061no—
310C23—2122no—
311C23—2122no—
312Q8—2122no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—532no—
317D4—1064no—
318D4—1061no—
319D4—1061no—
320D4—1061no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324D4—2122no—
325D4#1171994512yescomplemented
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1062no—
334C2×C4—1062no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4D4×D5311yescharacteristic
345C2×C4D4×D5311yescharacteristic
346C2×C4C23×D5311yescharacteristic
347C8—21no—
348C8—21no—
349C8—1061no—
350C8—1061no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—532no—
358C22—1061no—
359C22—1061no—
360C22—1062no—
361C22—2128no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22C2×D4×D5311yescharacteristic
368C4—21no—
369C4—21no—
370C4—41no—
371C4—41no—
372C4—532no—
373C4—1061no—
374C4—1061no—
375C4—2121no—
376C4—2121no—
377C4#41×D5311yescharacteristic
378C4C2×#1171994511yescharacteristic
379C2—42no—
380C2—81no—
381C2—1062no—
382C2—4241no—
383C2—2122no—
384C2#1681113211yescharacteristic
385C2AGL(1,ℤ/8)×D5311yescharacteristic
386C2AGL(1,ℤ/8)×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3109
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. C4×D4×D53C2semidirect
3. #16810943C2semidirect
4. #16810940C2semidirect
5. #16810912C2semidirect
6. #16810911C2semidirect
7. #16810817C2semidirect
8. #16810816C2semidirect
9. #16810758C2semidirect
10. #16810755C2semidirect
11. #16810736C2semidirect
12. #16810733C2semidirect
13. #16810733C2semidirect
14. #16810678C2semidirect
15. C53×#442C2semidirect
22. #13133216C22semidirect
23. #13133213C22semidirect
29. C4×D212C22semidirect
38. #13133149C22semidirect
39. #13133148C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
44. #13133139C22semidirect
46. #13133138C22semidirect
48. #13133137C22semidirect
51. #13133128C22semidirect
54. C4×D4×C53C22semidirect
55. #106×C53C22semidirect
57. #102×C53C22semidirect
58. #97×C53C22semidirect
121. D4×C53#41semidirect
143. D212#41semidirect
207. #442D53semidirect
209. C53#3109semidirect
231. C4×D4C2×D53semidirect
238. #106C2×D53semidirect
248. #102C2×D53semidirect
252. #97C2×D53semidirect
325. D4#11719945semidirect