Finite Group Databasebeta

← #20286681

Subgroups — #20286681

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#16811278C211yesmaximal, characteristic, complemented
2#16811268C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810943C211yesmaximal, characteristic, complemented
6#16810913C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810851C211yesmaximal, characteristic, complemented
9#16810838C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810755C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719947C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719923C2211yescharacteristic, complemented
20C2×#11719922C2211yescharacteristic
21C2×#11719912C2211yescharacteristic, complemented
22C2×C8×D53C2211yescharacteristic
23#13133217C2211yescharacteristic, complemented
24#13133205C2211yescharacteristic
25#13133201—21no—
26#13133199—21no—
27#13133199—21no—
28#13133197C2211yescharacteristic
29#13133196—21no—
30#13133192C2211yescharacteristic
31#13133191—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic, complemented
34#13133187—21no—
35C4×D212C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic, complemented
38C2×#11719921—21no—
39#13133148C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41C4⋊#88322C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic, 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—
69#11719947—22no—
70#11719947—41no—
71D4×D53—22no—
72D4×D53—41no—
73#11719945—44no—
74C2×D212—21no—
75C2×D212D411yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106C2311yescharacteristic
83D8×C53—22no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719923—41no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921D411yescharacteristic
100#11719921D411yescharacteristic
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53D411yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118C22×D53C2×D411yescharacteristic
119C22×D53C2×D411yescharacteristic
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53#4112yescomplemented
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—42no—
132C22⋊D53—42no—
133C22⋊D53—42no—
134C22⋊D53—42no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—42no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—22no—
142D212—21no—
143D212—21no—
144D212—81no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—42no—
150C4×D53—42no—
151Dic106—21no—
152Dic106—21no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—84no—
174C2×D53AGL(1,ℤ/8)12yes—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C2×#4111yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—21no—
182Dic53—21no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—42no—
188C2×C53—81no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—22no—
193D53—41no—
194D53—81no—
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#447—531no—
206#442D5311yescharacteristic, complemented
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—1061no—
213C22×D4—531no—
214C2×SD16—531no—
215C2×D8—531no—
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—531no—
229#121—1061no—
230#120—1061no—
231C4×D4—531no—
232C4×D4C2×D5311yescharacteristic
233C4×D4C2×D5311yescharacteristic, complemented
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#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102C2×D5311yescharacteristic
250#102C2×D5311yescharacteristic, complemented
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97C2×D5311yescharacteristic, complemented
254C24—1061no—
255#41—1062no—
256#41—2124no—
257#41—2121no—
258C2×Q8—531no—
259C2×Q8—1061no—
260C2×D4—531no—
261C2×D4—531no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1062no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C2×D4C22×D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278SD16—2121no—
279D8—22no—
280D8—2121no—
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—1061no—
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—
306C42C22×D5311yescharacteristic
307C23—21no—
308C23—21no—
309C23—531no—
310C23—531no—
311C23—1061no—
312C23—4241no—
313C23—2122no—
314C23—2122no—
315Q8—1061no—
316Q8—1061no—
317Q8—2122no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—1062no—
322D4—1061no—
323D4—1061no—
324D4—4241no—
325D4—2122no—
326D4—2122no—
327D4—2122no—
328D4—2122no—
329D4#1171994512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—81no—
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—1061no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4D4×D5311yescharacteristic
347C2×C4D4×D5311yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—21no—
350C8—21no—
351C8—1061no—
352C8—1061no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—532no—
360C22—1061no—
361C22—1061no—
362C22—1061no—
363C22—1061no—
364C22—1061no—
365C22—4244no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22C2×D4×D5311yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—1061no—
375C4—1061no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4C2×#1171994511yescharacteristic
380C4#1313326311yescharacteristic
381C2—42no—
382C2—81no—
383C2—1062no—
384C2—4241no—
385C2—2121no—
386C2#1681126911yescharacteristic
387C2AGL(1,ℤ/8)×D5311yescharacteristic
388C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3306
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. #16811278C2semidirect
2. #16811268C2semidirect
3. #16810983C2semidirect
4. #16810955C2semidirect
5. #16810943C2semidirect
6. #16810913C2semidirect
7. #16810875C2semidirect
8. #16810851C2semidirect
9. #16810838C2semidirect
10. #16810765C2semidirect
11. #16810755C2semidirect
12. #16810696C2semidirect
13. #16810621C2semidirect
14. #102×D53C2semidirect
15. C53×#442C2semidirect
19. C2×#11719923C22semidirect
21. C2×#11719912C22semidirect
23. #13133217C22semidirect
33. #13133188C22semidirect
37. C4⋊C4×D53C22semidirect
39. #13133148C22semidirect
40. #13133148C22semidirect
41. C4⋊#88322C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133140C22semidirect
51. #13133129C22semidirect
54. C4×D4×C53C22semidirect
55. #106×C53C22semidirect
57. #102×C53C22semidirect
58. #97×C53C22semidirect
123. D4×C53#41semidirect
206. #442D53semidirect
210. C53#3306semidirect
233. C4×D4C2×D53semidirect
239. #106C2×D53semidirect
250. #102C2×D53semidirect
253. #97C2×D53semidirect
329. D4#11719945semidirect