Finite Group Databasebeta

← #20286680

Subgroups — #20286680

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811273C211yesmaximal, characteristic, complemented
2#16811267C211yesmaximal, characteristic, complemented
3#16810981C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810940C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810873C211yesmaximal, characteristic, complemented
8#16810850C211yesmaximal, characteristic, complemented
9#16810836C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810735C211yesmaximal, characteristic, complemented
12#16810691C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14C2×#13133138C211yesmaximal, characteristic, complemented
15C53×#447C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719947C2211yescharacteristic
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719912C2211yescharacteristic, complemented
20C22×#88322C2211yescharacteristic
21#13133216C2211yescharacteristic, complemented
22#13133204C2211yescharacteristic
23#13133199—21no—
24#13133198—21no—
25#13133198—21no—
26#13133196C2211yescharacteristic
27#13133194—21no—
28#13133189—21no—
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic, complemented
31#13133186—21no—
32D4×Dic53C2211yescharacteristic
33D4×Dic53C2211yescharacteristic, complemented
34C4×C22⋊D53C2211yescharacteristic
35C2×C4⋊Dic53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37C2×C22⋊Dic53—21no—
38#13133149C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—41no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76#11719947—22no—
77#11719947—41no—
78#11719947—44no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106C2311yescharacteristic
83D8×C53—22no—
84D8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53D411yescharacteristic
97#11719923—41no—
98#11719921D411yescharacteristic
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53C2311yescharacteristic
108C2×#88322—41no—
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111#11719912—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—81no—
118C23×C53C2×D411yescharacteristic
119C22×D53C2×D411yescharacteristic
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—21no—
125D4×C53—42no—
126D4×C53—42no—
127D4×C53#4112yescomplemented
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—22no—
134C22⋊D53—42no—
135C22⋊D53—42no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—81no—
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—42no—
151Dic106—21no—
152Dic106—21no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—84no—
170C22×C53AGL(1,ℤ/8)12yes—
171C22×C53C22×D411yescharacteristic
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—21no—
180Dic53—21no—
181Dic53—21no—
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—42no—
189C2×C53—81no—
190C2×C53—81no—
191C2×C53#54611yescharacteristic
192C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
193C2×C53#57711yescharacteristic
194D53—41no—
195#547—531no—
196#546—531no—
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465—531no—
203#463—531no—
204#449—531no—
205#447D5311yescharacteristic, complemented
206#442—531no—
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—1061no—
213C22×D4C2×D5311yescharacteristic
214C2×SD16—531no—
215C2×D8C2×D5311yescharacteristic
216C2×D8C2×D5311yescharacteristic
217C2×M16—531no—
218C22×C8—531no—
219C4⋊Q8—531no—
220#124—531no—
221#123—1061no—
222#123—1061no—
223#123—1061no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226#121—531no—
227#121—531no—
228#121—1061no—
229#121C2×D5311yescharacteristic
230#120—1061no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C22⋊C4—1061no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8—531no—
241#103—531no—
242#103—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102C2×D5311yescharacteristic
250#102C2×D5311yescharacteristic
251C22⋊C8—531no—
252C22⋊C8C2×D5311yescharacteristic
253#97—531no—
254C24—21no—
255#41—1062no—
256#41—2121no—
257#41—2124no—
258C2×Q8—531no—
259C2×Q8—1061no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4C22×D5311yescharacteristic
269C2×D4C22×D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C22×C4—531no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278SD16—2121no—
279D8—22no—
280D8—41no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—2121no—
285C2×C8C22×D5311yescharacteristic
286C2×C8C22×D5311yescharacteristic
287C4⋊C4—531no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—531no—
298C22⋊C4—531no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C42—531no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—531no—
314C23D4×D5311yescharacteristic
315Q8—1061no—
316Q8—1061no—
317Q8—2122no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—1062no—
326D4—4241no—
327D4—2122no—
328D4—2122no—
329D4#1171994712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4D4×D5311yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—21no—
350C8—21no—
351C8—1061no—
352C8—1061no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—1061no—
366C22—1061no—
367C22—1061no—
368C22#1313324112yes—
369C22C2×D4×D5311yescharacteristic
370C4—21no—
371C4—41no—
372C4—1061no—
373C4—1061no—
374C4—1061no—
375C4—2121no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4C2×#1171994711yescharacteristic
380C4#1313326211yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—2121no—
386C2#1681126011yescharacteristic
387C2C2×#1313324111yescharacteristic
388C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16811273C2semidirect
2. #16811267C2semidirect
3. #16810981C2semidirect
4. #16810954C2semidirect
5. #16810940C2semidirect
6. #16810911C2semidirect
7. #16810873C2semidirect
8. #16810850C2semidirect
9. #16810836C2semidirect
10. #16810765C2semidirect
11. #16810735C2semidirect
12. #16810691C2semidirect
13. #16810621C2semidirect
14. C2×#13133138C2semidirect
15. C53×#447C2semidirect
18. C2×#11719923C22semidirect
19. C2×#11719912C22semidirect
21. #13133216C22semidirect
30. #13133188C22semidirect
33. D4×Dic53C22semidirect
36. C4⋊C4×D53C22semidirect
38. #13133149C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
49. #13133138C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
127. D4×C53#41semidirect
205. #447D53semidirect
210. C53#3306semidirect
329. D4#11719947semidirect