Finite Group Databasebeta

← #20286557

Subgroups — #20286557

Order 6784.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811314C211yesmaximal, characteristic, complemented
2#16811266C211yesmaximal, characteristic
3#16810977C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810945C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810861C211yesmaximal, characteristic, complemented
8#16810849C211yesmaximal, characteristic, complemented
9#16810784C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11C4×#11719924C211yesmaximal, characteristic, complemented
12#16810688C211yesmaximal, characteristic, complemented
13#16810618C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic, complemented
21#13133211—21no—
22#13133209C2211yescharacteristic, complemented
23#13133207—21no—
24#13133202—21no—
25#13133199—21no—
26#13133193C2211yescharacteristic
27#13133193C2211yescharacteristic
28#13133193C2211yescharacteristic, complemented
29#13133190—21no—
30#13133188C2211yescharacteristic
31#13133187—21no—
32C4×D212C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133175C2211yescharacteristic, complemented
35#13133174—21no—
36C4⋊C4×D53C2211yescharacteristic
37C22⋊C4×D53—21no—
38#13133151C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133141C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51C4×#88322C2211yescharacteristic, complemented
52C2×Q16×C53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65#11719949—22no—
66#11719949—41no—
67D4×D53—22no—
68D4×D53—41no—
69C2×D212—21no—
70C2×D212D411yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×C4×D53C2311yescharacteristic
77Q16×C53—41no—
78SD16×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88#11719924—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95#11719921D411yescharacteristic
96C4⋊Dic53D411yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C8×D53—41no—
110C8×D53—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53—81no—
116C22×D53C2×D411yescharacteristic
117C22×D53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—42no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137D212—22no—
138D212—21no—
139D212—42no—
140D212#4112yes—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—42no—
146C4×D53—42no—
147C4×D53—81no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—22no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—84no—
165C2×D53#13512yes—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3296—531nomaximal, Sylow
179C2×C53—81no—
180C2×C53#54611yescharacteristic
181C2×C53C2×#13511yescharacteristic
182C2×C53#57711yescharacteristic
183D53—22no—
184D53—41no—
185D53—42no—
186#552—531no—
187#546—531no—
188#488—531no—
189#483—531no—
190#482—531no—
191#478—531no—
192#467—531no—
193#465—531no—
194#464D5311yescharacteristic, complemented
195#452—531no—
196#450—531no—
197C4×SD16—531no—
198C8×D4—531no—
199#416—531no—
200C2×#102—531no—
201C53#329611yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C22×D4—531no—
204C2×Q16C2×D5311yescharacteristic, complemented
205C2×SD16—531no—
206C2×SD16C2×D5311yescharacteristic
207C22×C8—531no—
208C22×C8—531no—
209#126—1061no—
210#125—531no—
211#125—1061no—
212#124C2×D5311yescharacteristic
213#123—1061no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—531no—
218#121—531no—
219#121—1061no—
220#120—1061no—
221C4×Q8C2×D5311yescharacteristic, complemented
222C4×D4—531no—
223C4×D4—531no—
224#117—531no—
225#117—1061no—
226C2×C4⋊C4—531no—
227C2×C22⋊C4—1061no—
228#107—531no—
229#106—531no—
230#106—531no—
231C4⋊C8C2×D5311yescharacteristic, complemented
232#103—531no—
233#103—531no—
234#103—531no—
235#103C2×D5311yescharacteristic, complemented
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102C2×D5311yescharacteristic
242C22⋊C8—531no—
243C22⋊C8—531no—
244C4×C8—531no—
245C24—1061no—
246#41—1062no—
247#41—2121no—
248C2×Q8C22×D5311yescharacteristic
249C2×Q8C22×D5311yescharacteristic
250C2×D4—531no—
251C2×D4—531no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1062no—
255C2×D4—1061no—
256C2×D4—2121no—
257C2×D4C22×D5311yescharacteristic
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263Q16—41no—
264SD16—41no—
265SD16—1062no—
266C2×C8—531no—
267C2×C8—531no—
268C2×C8—2121no—
269C2×C8—2121no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4C22×D5311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—531no—
287C22⋊C4—531no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—21no—
295C42—1061no—
296C42C22×D5311yescharacteristic
297C23—21no—
298C23—531no—
299C23—531no—
300C23—1061no—
301C23—4241no—
302C23—2122no—
303C23—2122no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—532no—
311D4—1062no—
312D4—1061no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—4241no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4D4×D5311yescharacteristic
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—532no—
342C22—1061no—
343C22—1061no—
344C22—1061no—
345C22—1062no—
346C22—1061no—
347C22—1061no—
348C22—4244no—
349C22—2121no—
350C22—2121no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—1061no—
358C4—1061no—
359C4—2121no—
360C4—2121no—
361C4—2121no—
362C4#41×D5311yescharacteristic
363C4#1313326211yescharacteristic
364C2—81no—
365C2—1062no—
366C2—2121no—
367C2—2122no—
368C2#1681127011yescharacteristic
369C2#135×D5311yescharacteristic
370C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3296
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #16811314C2semidirect
3. #16810977C2semidirect
4. #16810952C2semidirect
5. #16810945C2semidirect
6. #16810924C2semidirect
7. #16810861C2semidirect
8. #16810849C2semidirect
9. #16810784C2semidirect
10. #16810772C2semidirect
11. C4×#11719924C2semidirect
12. #16810688C2semidirect
13. #16810618C2semidirect
14. #102×D53C2semidirect
15. C53×#464C2semidirect
20. C2×C8×D53C22semidirect
22. #13133209C22semidirect
28. #13133193C22semidirect
34. #13133175C22semidirect
38. #13133151C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133141C22semidirect
42. #13133141C22semidirect
43. #13133141C22semidirect
50. #13133129C22semidirect
51. C4×#88322C22semidirect
52. C2×Q16×C53C22semidirect
55. C4×Q8×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
194. #464D53semidirect
201. C53#3296semidirect
204. C2×Q16C2×D53semidirect
221. C4×Q8C2×D53semidirect
231. C4⋊C8C2×D53semidirect
235. #103C2×D53semidirect