Finite Group Databasebeta

← #20285732

Subgroups — #20285732

Order 6784.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#127×D53C211yesmaximal, characteristic, complemented
2#16811155C211yesmaximal, characteristic, complemented
3#16811011C212yesmaximal, complemented
4#16810986C212yesmaximal, complemented
5#16810824C211yesmaximal, characteristic, complemented
6#16810820C211yesmaximal, characteristic, complemented
7#16810818C211yesmaximal, characteristic, complemented
8#16810775C212yesmaximal, complemented
9#16810755C212yesmaximal, complemented
10#16810679C211yesmaximal, characteristic, complemented
11C53×#459C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15C2×#11719913C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×#11719922C2212yescomplemented
18#13133214C2211yescharacteristic
19#13133213—21no—
20#13133213—21no—
21#13133213C2212yes—
22#13133210—21no—
23#13133208C2211yescharacteristic, complemented
24#13133204C2211yescharacteristic, complemented
25#13133199—21no—
26#13133197C2211yescharacteristic
27#13133188C2211yescharacteristic
28C4×Dic106C2211yescharacteristic, complemented
29C4×C22⋊D53—21no—
30#13133176—21no—
31C42×D53C2211yescharacteristic
32C4⋊#88322C2211yescharacteristic, complemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yes—
37#13133127C2212yescomplemented
38C2×D8×C53C2212yescomplemented
39#127×C53C2211yescharacteristic
40C4×D4×C53C2211yescharacteristic, complemented
41C4⋊C8×C53C2211yescharacteristic, complemented
42#102×C53C2212yescomplemented
43C23×D53—22no—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×D4×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C2×C22⋊D53—21no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53D4×D53—42no—
54D4×D53—42no—
55D4×D53—44no—
56D4×D53—44no—
57C2×D212—21no—
58C2×D212—21no—
59C2×D212C2312yes—
60C2×C4×D53D411yescharacteristic
61C2×C4×D53D411yescharacteristic
62C2×C4×D53C2311yescharacteristic
63C2×Dic106C2311yescharacteristic
64D8×C53—42no—
65C2×C8×C53C2312yes—
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C42×C53C2311yescharacteristic
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719923—42no—
73#11719922—42no—
74#11719921—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53—21no—
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79#11719919—21no—
80C4×Dic53—21no—
81C4×Dic53D411yescharacteristic
82C4×Dic53D411yescharacteristic
83C4×Dic53C2311yescharacteristic
84C2×#88322C2312yes—
85#11719913—42no—
86C23×C53—22no—
87C23×C53—21no—
88C22×D53—22no—
89C22×D53—44no—
90C22×D53—44no—
91C22×D53—44no—
92C22×D53C2×D411yescharacteristic
93D4×C53—22no—
94D4×C53—22no—
95D4×C53—21no—
96D4×C53—21no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—42no—
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—44no—
105C22⋊D53—44no—
106C22⋊D53—44no—
107C22⋊D53—42no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53C2×D411yescharacteristic
111C2×Dic53C2×D411yescharacteristic
112C2×Dic53C2×D411yescharacteristic
113D212—22no—
114D212—22no—
115D212—42no—
116D212—42no—
117C4×D53—22no—
118C4×D53—22no—
119C4×D53—21no—
120C4×D53—21no—
121Dic106—21no—
122Dic106—21no—
123C8×C53—22no—
124#88322—22no—
125C22×C53—42no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53C22×D411yescharacteristic
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—88no—
138C2×D53AGL(1,ℤ/8)12yes—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53AGL(1,ℤ/8)12yes—
142C4×C53H(5,𝔽2)11yescharacteristic
143C4×C53H(5,𝔽2)11yescharacteristic
144Dic53—21no—
145Dic53—21no—
146Dic53—21no—
147Dic53—41no—
148Dic53—41no—
149#3217—531nomaximal, Sylow
150C2×C53—82no—
151C2×C53—81no—
152C2×C53#53511yescharacteristic
153C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
155D53—22no—
156D53—82no—
157C2×#127—531no—
158#529—531no—
159#496—532no—
160#490—532no—
161#460—531no—
162#460—531no—
163#459—531no—
164#459D5311yescharacteristic, complemented
165#450—532no—
166#447—532no—
167#432—531no—
168C53#321711yescharacteristic, complemented, Sylow
169C22×D4—532no—
170C22×D4—1061no—
171C22×D4—1061no—
172C2×SD16—532no—
173C2×SD16—532no—
174C2×D8—532no—
175C2×D8C2×D5312yescomplemented
176#127—532no—
177#127—531no—
178#127—1061no—
179#127—1061no—
180#127C2×D5311yescharacteristic
181#126—1061no—
182#125—531no—
183#124—531no—
184#123—1061no—
185C22⋊Q8—531no—
186#121—531no—
187C4×Q8—531no—
188C4×D4—1061no—
189C4×D4C2×D5311yescharacteristic, complemented
190#117—1061no—
191C2×C42—531no—
192C4⋊C8—531no—
193C4⋊C8C2×D5311yescharacteristic, complemented
194#102—532no—
195#102—532no—
196#102—532no—
197#102C2×D5312yescomplemented
198C22⋊C8—532no—
199#97—532no—
200C24—1062no—
201C2×Q8—531no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—532no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—1061no—
209C2×D4—1061no—
210C2×D4—1061no—
211C2×D4—2122no—
212C2×D4—2122no—
213C2×D4—2124no—
214C2×D4—2124no—
215C2×D4C22×D5312yes—
216C2×D4C22×D5311yescharacteristic
217C22×C4—21no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221SD16—2122no—
222SD16—2122no—
223D8—42no—
224D8—2122no—
225C2×C8—532no—
226C2×C8C22×D5312yes—
227C4⋊C4—531no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4—1061no—
232C4⋊C4C22×D5311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—1061no—
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C22⋊C4—1061no—
238C42—531no—
239C42—531no—
240C42—531no—
241C42—1061no—
242C42C22×D5311yescharacteristic
243C23—21no—
244C23—22no—
245C23—531no—
246C23—1062no—
247C23—2124no—
248C23—2124no—
249C23—2124no—
250Q8—1061no—
251Q8—1061no—
252D4—21no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—1062no—
259D4—1062no—
260D4—2122no—
261D4—2124no—
262D4—2124no—
263D4—2124no—
264D4—2122no—
265D4—2122no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4D4×D5311yescharacteristic
278C2×C4D4×D5311yescharacteristic
279C2×C4C23×D5311yescharacteristic
280C8—22no—
281C8—1062no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—532no—
289C22—1061no—
290C22—1061no—
291C22—4248no—
292C22—2122no—
293C22—2122no—
294C22—2122no—
295C22C2×D4×D5311yescharacteristic
296C4—21no—
297C4—41no—
298C4—1061no—
299C4—1061no—
300C4—1061no—
301C4—2121no—
302C4—2121no—
303C4#1313324112yes—
304C4#1313326211yescharacteristic
305C4#1313326311yescharacteristic
306C2—82no—
307C2—81no—
308C2—1062no—
309C2—4242no—
310C2#1681118911yescharacteristic
311C2AGL(1,ℤ/8)×D5311yescharacteristic
312C2C2×#1313324111yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3217
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #127×D53C2semidirect
2. #16811155C2semidirect
3. #16811011C2semidirect
4. #16810986C2semidirect
5. #16810824C2semidirect
6. #16810820C2semidirect
7. #16810818C2semidirect
8. #16810775C2semidirect
9. #16810755C2semidirect
10. #16810679C2semidirect
11. C53×#459C2semidirect
15. C2×#11719913C22semidirect
16. C2×#11719923C22semidirect
17. C2×#11719922C22semidirect
23. #13133208C22semidirect
24. #13133204C22semidirect
28. C4×Dic106C22semidirect
32. C4⋊#88322C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. #13133127C22semidirect
38. C2×D8×C53C22semidirect
40. C4×D4×C53C22semidirect
41. C4⋊C8×C53C22semidirect
42. #102×C53C22semidirect
164. #459D53semidirect
168. C53#3217semidirect
175. C2×D8C2×D53semidirect
189. C4×D4C2×D53semidirect
193. C4⋊C8C2×D53semidirect
197. #102C2×D53semidirect