Finite Group Databasebeta

← #20286559

Subgroups — #20286559

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811269C211yesmaximal, characteristic, complemented
2#16810980C211yesmaximal, characteristic, complemented
3#16810958C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810857C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810838C211yesmaximal, characteristic, complemented
9#16810766C211yesmaximal, characteristic, complemented
10#16810756C211yesmaximal, characteristic, complemented
11C4×#11719922C211yesmaximal, characteristic, complemented
12D4×#88322C211yesmaximal, characteristic, complemented
13#16810617C211yesmaximal, characteristic, complemented
14C2×#13133137C211yesmaximal, characteristic, complemented
15C53×#553C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C22×#88322C2211yescharacteristic
22C22×#88322C2211yescharacteristic, complemented
23#13133205C2211yescharacteristic
24#13133202—21no—
25#13133196C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133190—21no—
28#13133185—21no—
29C4×D212C2211yescharacteristic
30C22⋊C4×D53—21no—
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133140C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic, complemented
39#13133137C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48#126×C53—21no—
49#125×C53C2211yescharacteristic, complemented
50#123×C53—21no—
51#121×C53—21no—
52#121×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic, complemented
54C4×D4×C53—21no—
55C4×D4×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic, complemented
57#117×C53C2211yescharacteristic, complemented
58C2×C4⋊C4×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53D411yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70#11719945—22no—
71#11719945—41no—
72C2×D212—22no—
73C2×D212—21no—
74C2×D212—41no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×Dic106C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53D411yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53D411yescharacteristic
92C22⋊C4×C53D411yescharacteristic
93C42×C53—21no—
94C42×C53C2311yescharacteristic
95C22⋊Dic53—21no—
96#11719923—41no—
97#11719922—22no—
98#11719922—41no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53C2311yescharacteristic
104#11719919—21no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—81no—
119D4×C53—22no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123D4×C53—81no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—81no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—42no—
146D212#4112yes—
147C4×D53—42no—
148C4×D53—81no—
149Dic106—21no—
150Dic106—21no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53#13512yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—22no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C2×#4111yescharacteristic
180C4×C53H(5,𝔽2)11yescharacteristic
181Dic53—41no—
182Dic53—41no—
183#3297—531nomaximal, Sylow
184C2×C53—22no—
185C2×C53—41no—
186C2×C53—81no—
187C2×C53#54611yescharacteristic
188C2×C53C2×#13511yescharacteristic
189C2×C53#57611yescharacteristic
190D53—42no—
191D53—81no—
192#553D5311yescharacteristic, complemented
193#546—531no—
194#488—531no—
195#484—531no—
196#482—531no—
197#478—531no—
198#466—531no—
199#465—531no—
200#463—531no—
201#449—531no—
202#447—531no—
203C4×D8—531no—
204C8×D4—531no—
205#416—531no—
206C2×#102—531no—
207C53#329711yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4—531no—
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C22×C8—531no—
214C22×C8—531no—
215#126—21no—
216#125C2×D5311yescharacteristic, complemented
217#124—531no—
218#123—21no—
219#123—1061no—
220C22⋊Q8—531no—
221#121—21no—
222#121—531no—
223#121—1061no—
224#121C2×D5311yescharacteristic
225#121C2×D5311yescharacteristic, complemented
226#120—1061no—
227C4×D4—21no—
228C4×D4—531no—
229C4×D4C2×D5311yescharacteristic
230C4×D4C2×D5311yescharacteristic, complemented
231#117C2×D5311yescharacteristic, complemented
232C2×C4⋊C4C2×D5311yescharacteristic
233C2×C22⋊C4—1061no—
234#107—531no—
235#107—531no—
236#106—531no—
237C4⋊C8—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248C22⋊C8—531no—
249C22⋊C8—531no—
250C4×C8—531no—
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8—531no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1062no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—2121no—
264C2×D4C22⋊D5311yescharacteristic
265C2×D4C22×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—2121no—
280C2×C8—2121no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4C22⋊D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4C22⋊D5311yescharacteristic
300C22⋊C4C22⋊D5311yescharacteristic
301C42—21no—
302C42C22×D5311yescharacteristic
303C23—21no—
304C23—1061no—
305C23—1061no—
306C23—4241no—
307C23—2122no—
308C23—2122no—
309C23C2×C22⋊D5311yescharacteristic
310C23C2×C22⋊D5311yescharacteristic
311Q8—1061no—
312Q8—1061no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—532no—
319D4—1061no—
320D4—1061no—
321D4—1061no—
322D4—4241no—
323D4—2122no—
324D4—2122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4C2×C22⋊D5311yescharacteristic
339C2×C4C2×C22⋊D5311yescharacteristic
340C2×C4C2×C22⋊D5311yescharacteristic
341C2×C4C2×C22⋊D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—1061no—
344C8—1061no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—1062no—
356C22—4244no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22#1313323712yes—
363C22C22×C22⋊D5311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—2121no—
372C4—2121no—
373C4#41×D5311yescharacteristic
374C4#1313326311yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—4241no—
379C2—2122no—
380C2#1681126111yescharacteristic
381C2C2×#1313323711yescharacteristic
382C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#553C2
socleC22×C53C22×D4
hypercenter#553D53
hypocenterC53#3297
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16811269C2semidirect
2. #16810980C2semidirect
3. #16810958C2semidirect
4. #16810947C2semidirect
5. #16810920C2semidirect
6. #16810857C2semidirect
7. #16810850C2semidirect
8. #16810838C2semidirect
9. #16810766C2semidirect
10. #16810756C2semidirect
11. C4×#11719922C2semidirect
12. D4×#88322C2semidirect
13. #16810617C2semidirect
14. C2×#13133137C2semidirect
15. C53×#553C2semidirect
18. C2×#11719923C22semidirect
22. C22×#88322C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133140C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
38. #13133138C22semidirect
46. C22⋊#88322C22semidirect
47. C4×#88322C22semidirect
49. #125×C53C22semidirect
53. #121×C53C22semidirect
56. C4×D4×C53C22semidirect
57. #117×C53C22semidirect
192. #553D53semidirect
207. C53#3297semidirect
216. #125C2×D53semidirect
225. #121C2×D53semidirect
230. C4×D4C2×D53semidirect
231. #117C2×D53semidirect