Finite Group Databasebeta

← #20286682

Subgroups — #20286682

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#16811264C211yesmaximal, characteristic, complemented
3#16810984C211yesmaximal, characteristic, complemented
4#16810951C211yesmaximal, characteristic, complemented
5#16810943C211yesmaximal, characteristic, complemented
6#16810912C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810767C211yesmaximal, characteristic, complemented
10#16810756C211yesmaximal, characteristic, complemented
11#16810732C211yesmaximal, characteristic, complemented
12#16810694C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14C2×#13133139C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947—21no—
18C22×D212C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719912C2211yescharacteristic, complemented
22#13133216C2211yescharacteristic, complemented
23#13133206C2211yescharacteristic
24#13133202—21no—
25#13133199—21no—
26#13133198—21no—
27#13133194—21no—
28#13133192—21no—
29#13133192C2211yescharacteristic
30#13133188C2211yescharacteristic, complemented
31#13133187—21no—
32D4×Dic53C2211yescharacteristic, complemented
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35C2×C4⋊Dic53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37C22⋊C4×D53—21no—
38#13133148C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C22×C8×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68#11719949—22no—
69#11719949—41no—
70#11719947—44no—
71C2×D212—22no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—41no—
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83SD16×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53D411yescharacteristic
95#11719922—22no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921D411yescharacteristic
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919D411yescharacteristic
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111#11719912—41no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—81no—
119C22×D53C2×D411yescharacteristic
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—22no—
133C22⋊D53—42no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—21no—
142D212—21no—
143D212—21no—
144D212—42no—
145D212—42no—
146D212—81no—
147D212#4112yescomplemented
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—42no—
151C4×D53—42no—
152C4×D53—81no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53AGL(1,ℤ/8)12yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—22no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—84no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C2×#4111yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—21no—
182Dic53—21no—
183Dic53—21no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—81no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—41no—
193D53—42no—
194D53—81no—
195#547—531no—
196#546—531no—
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465D5311yescharacteristic, complemented
203#463—531no—
204#449—531no—
205#447—531no—
206#442—531no—
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—1061no—
213C22×D4—531no—
214C2×SD16C2×D5311yescharacteristic, complemented
215C2×D8—531no—
216C2×D8—531no—
217C2×M16—531no—
218C22×C8C2×D5311yescharacteristic
219C4⋊Q8—531no—
220#124—531no—
221#123—1061no—
222#123—1061no—
223#123—1061no—
224C22⋊Q8—1061no—
225C22⋊Q8C2×D5311yescharacteristic
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#106C2×D5311yescharacteristic, complemented
240C4⋊C8—531no—
241#103—531no—
242#103—531no—
243#103C2×D5311yescharacteristic, complemented
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102C2×D5311yescharacteristic, complemented
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97—531no—
254C24—1061no—
255#41—1062no—
256#41—2121no—
257#41—2124no—
258C2×Q8—1061no—
259C2×Q8C22×D5311yescharacteristic
260C2×D4—21no—
261C2×D4—531no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—1062no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
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—41no—
279D8—1062no—
280D8—2121no—
281M16—2121no—
282C2×C8—41no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8C22×D5311yescharacteristic
286C2×C8C22×D5311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4C22×D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—531no—
299C22⋊C4—531no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C42—531no—
307C23—21no—
308C23—531no—
309C23—1061no—
310C23—1061no—
311C23—4241no—
312C23—2122no—
313C23—2122no—
314C23D4×D5311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—2122no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—532no—
322D4—1062no—
323D4—1061no—
324D4—1061no—
325D4—1061no—
326D4—4241no—
327D4—2122no—
328D4—2122no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—531no—
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—41no—
356C22—41no—
357C22—41no—
358C22—1061no—
359C22—1061no—
360C22—1061no—
361C22—1062no—
362C22—4244no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22#1313324412yes—
369C22C2×D4×D5311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—1061no—
375C4—1061no—
376C4—1061no—
377C4—2121no—
378C4—2121no—
379C4#41×D5311yescharacteristic
380C4#1313326211yescharacteristic
381C2—22no—
382C2—81no—
383C2—4241no—
384C2—2121no—
385C2—2122no—
386C2#1681126211yescharacteristic
387C2C2×#1313324411yescharacteristic
388C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811273C2semidirect
2. #16811264C2semidirect
3. #16810984C2semidirect
4. #16810951C2semidirect
5. #16810943C2semidirect
6. #16810912C2semidirect
7. #16810875C2semidirect
8. #16810840C2semidirect
9. #16810767C2semidirect
10. #16810756C2semidirect
11. #16810732C2semidirect
12. #16810694C2semidirect
13. #16810623C2semidirect
14. C2×#13133139C2semidirect
15. C53×#465C2semidirect
21. C2×#11719912C22semidirect
22. #13133216C22semidirect
30. #13133188C22semidirect
32. D4×Dic53C22semidirect
36. C4⋊C4×D53C22semidirect
38. #13133148C22semidirect
39. #13133148C22semidirect
40. #13133146C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
52. C2×SD16×C53C22semidirect
56. #106×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
147. D212#41semidirect
202. #465D53semidirect
210. C53#3306semidirect
214. C2×SD16C2×D53semidirect
239. #106C2×D53semidirect
243. #103C2×D53semidirect
250. #102C2×D53semidirect