Finite Group Databasebeta

← #20286677

Subgroups — #20286677

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#16811274C211yesmaximal, characteristic, complemented
2#16811261C211yesmaximal, characteristic, complemented
3#16810984C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810941C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810855C211yesmaximal, characteristic, complemented
9#16810841C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810732C211yesmaximal, characteristic, complemented
13#16810692C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#417C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216C2211yescharacteristic, complemented
23#13133206C2211yescharacteristic
24#13133202—21no—
25#13133200—21no—
26#13133200—21no—
27#13133195—21no—
28#13133193C2211yescharacteristic
29#13133192C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133191C2211yescharacteristic, complemented
32#13133189—21no—
33#13133185—21no—
34D4×Dic53C2211yescharacteristic
35#13133179C2211yescharacteristic, complemented
36C4×C22⋊D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38C2×#11719921—21no—
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic, complemented
54C2×C4⋊C4×C53C2211yescharacteristic, complemented
55#103×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72C22×Dic53—21no—
73D4×D53—22no—
74D4×D53—41no—
75#11719945—44no—
76C2×D212—21no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83M16×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719924—41no—
92#11719922—22no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110D424—41no—
111C8×D53—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—42no—
124D4×C53#4112yescomplemented
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—22no—
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—81no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—42no—
152C4×D53—42no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—22no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—84no—
174C2×D53AGL(1,ℤ/8)12yes—
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—42no—
188C2×C53—41no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—22no—
193D53—81no—
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#442—531no—
207#435—531no—
208#417D5311yescharacteristic, complemented
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—1061no—
212C2×#41C2×D5311yescharacteristic
213C22×D4—531no—
214C2×SD16—531no—
215C2×D8—531no—
216C2×D8—531no—
217C2×M16C2×D5311yescharacteristic, complemented
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×D4—531no—
233C4×D4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4C2×D5311yescharacteristic, complemented
236C2×C22⋊C4—1061no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8—531no—
241#103—531no—
242#103C2×D5311yescharacteristic, complemented
243#103C2×D5311yescharacteristic, complemented
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⋊C8—531no—
253#97—531no—
254C24—1061no—
255#41—22no—
256#41—41no—
257#41—2124no—
258C2×Q8—1061no—
259C2×Q8C22×D5311yescharacteristic
260C2×D4—21no—
261C2×D4—531no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1062no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—2121no—
270C2×D4C22×D5311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278SD16—2121no—
279D8—1062no—
280D8—2121no—
281M16—41no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—2121no—
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—531no—
297C22⋊C4—531no—
298C22⋊C4—1061no—
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—531no—
309C23—1061no—
310C23—1061no—
311C23—4241no—
312C23—2122no—
313C23—2122no—
314C23D4×D5311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—2122no—
318D4—21no—
319D4—42no—
320D4—1062no—
321D4—1061no—
322D4—1061no—
323D4—1061no—
324D4—1061no—
325D4—4241no—
326D4—2122no—
327D4—2122no—
328D4—2122no—
329D4#1171994512yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—531no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—4241no—
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—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—532no—
360C22—1061no—
361C22—1061no—
362C22—4244no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22C2×D4×D5311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—1061no—
375C4—1061no—
376C4—1061no—
377C4—2121no—
378C4—2121no—
379C4C2×#1171994511yescharacteristic
380C4#1313326211yescharacteristic
381C2—41no—
382C2—42no—
383C2—1062no—
384C2—4241no—
385C2—4241no—
386C2#1681126311yescharacteristic
387C2AGL(1,ℤ/8)×D5311yescharacteristic
388C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#417C2
socleC22×C53C22×D4
hypercenter#417D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #16811274C2semidirect
2. #16811261C2semidirect
3. #16810984C2semidirect
4. #16810952C2semidirect
5. #16810941C2semidirect
6. #16810911C2semidirect
7. #16810876C2semidirect
8. #16810855C2semidirect
9. #16810841C2semidirect
10. #16810768C2semidirect
11. #16810758C2semidirect
12. #16810732C2semidirect
13. #16810692C2semidirect
14. #102×D53C2semidirect
15. C53×#417C2semidirect
18. C2×#11719924C22semidirect
22. #13133216C22semidirect
31. #13133191C22semidirect
35. #13133179C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133148C22semidirect
42. #13133146C22semidirect
43. #13133141C22semidirect
45. #13133139C22semidirect
49. C22⋊#88322C22semidirect
51. #13133127C22semidirect
53. C2×M16×C53C22semidirect
54. C2×C4⋊C4×C53C22semidirect
55. #103×C53C22semidirect
56. #103×C53C22semidirect
124. D4×C53#41semidirect
208. #417D53semidirect
210. C53#3306semidirect
217. C2×M16C2×D53semidirect
235. C2×C4⋊C4C2×D53semidirect
242. #103C2×D53semidirect
243. #103C2×D53semidirect
329. D4#11719945semidirect