Finite Group Databasebeta

← #20284766

Subgroups — #20284766

Order 6784.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811160C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic, complemented
3#16810957C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810922C211yesmaximal, characteristic, complemented
6#16810921C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810772C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic
11C4×#11719924C211yesmaximal, characteristic, complemented
12C4×D424C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810674C211yesmaximal, characteristic, complemented
15C4×SD16×C53C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C22×C4×D53—21no—
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21#13133212—21no—
22#13133208C2211yescharacteristic, complemented
23#13133206C2211yescharacteristic, complemented
24#13133202—21no—
25#13133193C2211yescharacteristic
26#13133192C2211yescharacteristic
27C4×D212—21no—
28C4×D212C2211yescharacteristic
29C4×D212C2211yescharacteristic, complemented
30D4×Dic53C2211yescharacteristic
31#13133179C2211yescharacteristic
32C4×C22⋊D53—21no—
33#13133177—21no—
34#13133174—21no—
35C4⋊C4×D53C2211yescharacteristic
36C22⋊C4×D53—21no—
37C42×D53C2211yescharacteristic
38#13133151C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic, complemented
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65D4×D53—22no—
66D4×D53—22no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—44no—
75C2×C4×D53D411yescharacteristic
76C2×C4×D53D411yescharacteristic
77C2×C4×D53C2311yescharacteristic
78SD16×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88#11719924—22no—
89#11719922—22no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—22no—
109C23×C53—21no—
110C22×D53—24no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53#4112yes—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—42no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—24no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134D212—21no—
135D212—21no—
136D212—21no—
137D212—42no—
138D212#4112yes—
139C4×D53—22no—
140C4×D53—22no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—22no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—22no—
158C2×D53—48no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53#13512yes—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53AGL(1,ℤ/8)12yes—
170C4×C53C2×#4111yescharacteristic
171C4×C53C2×#4111yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177#3105—531nomaximal, Sylow
178C2×C53—42no—
179C2×C53#52511yescharacteristic
180C2×C53C2×#13511yescharacteristic
181C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
182D53—22no—
183D53—42no—
184D53—81no—
185#529—531no—
186C2×C4×D4—531no—
187#484—531no—
188#481—531no—
189#478—531no—
190#478—531no—
191#463—531no—
192#463—531no—
193#450—531no—
194#447—531no—
195C4×SD16—531no—
196C4×SD16D5311yescharacteristic, complemented
197C4×D8—531no—
198C4×D8—531no—
199#431—531no—
200C53#310511yescharacteristic, complemented, Sylow
201C22×D4—531no—
202C23×C4—1061no—
203C2×SD16—531no—
204C2×SD16C2×D5311yescharacteristic
205C2×D8—531no—
206C2×D8—531no—
207#126—1061no—
208#125—531no—
209#124—531no—
210#123—1061no—
211C22⋊Q8—531no—
212#121—531no—
213C4×Q8C2×D5311yescharacteristic, complemented
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—531no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4—1061no—
221C4×D4C2×D5311yescharacteristic
222#117—1061no—
223C2×C4⋊C4—531no—
224C2×C22⋊C4—1061no—
225C2×C42—531no—
226#107—531no—
227#107—531no—
228#106—531no—
229#106C2×D5311yescharacteristic, complemented
230C4⋊C8—531no—
231C4⋊C8—531no—
232#103—531no—
233#103C2×D5311yescharacteristic, complemented
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102C2×D5311yescharacteristic
240C22⋊C8—531no—
241C22⋊C8—531no—
242C4×C8—531no—
243C4×C8C2×D5311yescharacteristic, complemented
244C24—1061no—
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—531no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1062no—
251C2×D4—1062no—
252C2×D4C22×D5311yescharacteristic
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—2124no—
263SD16—22no—
264SD16—1062no—
265D8—1062no—
266D8—1062no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C4⋊C4C22×D5311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—21no—
290C42—531no—
291C42—531no—
292C42—531no—
293C42C22×D5311yescharacteristic
294C23—21no—
295C23—531no—
296C23—1064no—
297C23—1061no—
298C23—1061no—
299C23—2122no—
300C23—2122no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—532no—
305D4—1064no—
306D4—1061no—
307D4—1061no—
308D4—1061no—
309D4—2122no—
310D4#1171994512yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4D4×D5311yescharacteristic
330C2×C4D4×D5311yescharacteristic
331C2×C4C23×D5311yescharacteristic
332C8—21no—
333C8—21no—
334C8—1061no—
335C8—1061no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—532no—
340C22—1061no—
341C22—1061no—
342C22—1062no—
343C22—2128no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—1061no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4#1313324412yes—
360C4#41×D5311yescharacteristic
361C4C2×#1171994511yescharacteristic
362C2—42no—
363C2—1062no—
364C2—4241no—
365C2—2122no—
366C2#1681113211yescharacteristic
367C2#135×D5311yescharacteristic
368C2C2×#1313324411yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC4×SD16×C53C2
socleC22×C53C22×D4
hypercenterC4×SD16D53
hypocenterC53#3105
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #16811160C2semidirect
2. C4×D4×D53C2semidirect
3. #16810957C2semidirect
4. #16810943C2semidirect
5. #16810922C2semidirect
6. #16810921C2semidirect
7. #16810841C2semidirect
8. #16810840C2semidirect
9. #16810772C2semidirect
11. C4×#11719924C2semidirect
12. C4×D424C2semidirect
13. C4×#11719922C2semidirect
14. #16810674C2semidirect
15. C4×SD16×C53C2semidirect
22. #13133208C22semidirect
23. #13133206C22semidirect
29. C4×D212C22semidirect
38. #13133151C22semidirect
39. #13133150C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133141C22semidirect
45. #13133139C22semidirect
48. #13133137C22semidirect
51. C4×#88322C22semidirect
53. C4×Q8×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
58. C4×C8×C53C22semidirect
196. C4×SD16D53semidirect
200. C53#3105semidirect
213. C4×Q8C2×D53semidirect
229. #106C2×D53semidirect
233. #103C2×D53semidirect
243. C4×C8C2×D53semidirect