Finite Group Databasebeta

← #20285736

Subgroups — #20285736

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
1C2×#13133213C211yesmaximal, characteristic, complemented
2#16811159C211yesmaximal, characteristic, complemented
3#16811009C212yesmaximal, complemented
4#16810986C212yesmaximal, complemented
5#16810821C212yesmaximal, complemented
6#16810817C212yesmaximal, complemented
7#16810773C211yesmaximal, characteristic, complemented
8#16810759C211yesmaximal, characteristic, complemented
9#16810757C211yesmaximal, characteristic, complemented
10#16810677C211yesmaximal, characteristic, complemented
11C53×#450C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53—21no—
14C22×C22⋊D53—21no—
15C2×#11719924C2212yescomplemented
16C2×D424C2212yescomplemented
17C2×#11719922C2212yescomplemented
18#13133213—21no—
19#13133213—21no—
20#13133213C2212yes—
21#13133213C2212yes—
22#13133212—21no—
23#13133208C2211yescharacteristic, complemented
24#13133206C2211yescharacteristic, complemented
25#13133201—21no—
26#13133191C2211yescharacteristic
27Q8×Dic53C2211yescharacteristic, complemented
28#13133179C2211yescharacteristic, complemented
29#13133177—21no—
30#13133175—21no—
31C2×C4×Dic53C2211yescharacteristic
32#13133146C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133127C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×SD16×C53C2212yescomplemented
40C22⋊Q8×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53—21no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—21no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—44no—
56C2×C22⋊D53—44no—
57C22×Dic53D411yescharacteristic
58C22×Dic53D411yescharacteristic
59C2×D212—21no—
60C2×D212—42no—
61C2×D212C2312yes—
62C2×D212C2311yescharacteristic
63C2×C4×D53—21no—
64SD16×C53—42no—
65C2×C8×C53C2312yes—
66C4⋊C4×C53—21no—
67C4⋊C4×C53C2311yescharacteristic
68C22⋊C4×C53—21no—
69#11719924—42no—
70#11719922—42no—
71#11719921—21no—
72#11719921—21no—
73#11719921—21no—
74#11719921—21no—
75C4⋊Dic53—21no—
76C4⋊Dic53—21no—
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79C4×Dic53—21no—
80C4×Dic53D411yescharacteristic
81C4×Dic53D411yescharacteristic
82C4×Dic53C2311yescharacteristic
83C4×Dic53C2311yescharacteristic
84C2×#88322C2312yes—
85D424—42no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×D411yescharacteristic
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96Q8×C53—22no—
97D4×C53—22no—
98D4×C53—22no—
99D4×C53—42no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2411yescharacteristic
105C22⋊D53—44no—
106C22⋊D53—44no—
107C22⋊D53—44no—
108C22⋊D53—42no—
109C22⋊D53—44no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—21no—
113C2×Dic53C2×D411yescharacteristic
114C2×Dic53C2×D411yescharacteristic
115C2×Dic53C2×D411yescharacteristic
116C2×Dic53C2×D411yescharacteristic
117D212—22no—
118D212—22no—
119D212—22no—
120D212—42no—
121C4×D53—42no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—84no—
130C22×C53AGL(1,ℤ/8)12yes—
131C22×C53C22×D411yescharacteristic
132C2×D53—42no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—41no—
138C2×D53—84no—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53H(5,𝔽2)12yes—
143Dic53—21no—
144Dic53—21no—
145Dic53—21no—
146Dic53—41no—
147Dic53AGL(1,ℤ/8)12yes—
148#3217—531nomaximal, Sylow
149C2×C53—22no—
150C2×C53—82no—
151C2×C53#53511yescharacteristic
152C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
154D53—81no—
155D53—82no—
156C2×#127—531no—
157#529—531no—
158#496—532no—
159#490—532no—
160#460—532no—
161#459—532no—
162#450—531no—
163#450D5311yescharacteristic, complemented
164#447—531no—
165#447—531no—
166#432—531no—
167C53#321711yescharacteristic, complemented, Sylow
168C22×D4—531no—
169C22×D4—1061no—
170C22×D4—1061no—
171C22×D4C2×D5311yescharacteristic
172C2×SD16—532no—
173C2×SD16C2×D5312yescomplemented
174C2×D8—532no—
175C2×D8—532no—
176#127—532no—
177#127—532no—
178#127—1061no—
179#127—1061no—
180#126—1061no—
181#125—531no—
182#124—531no—
183#123—1061no—
184C22⋊Q8C2×D5311yescharacteristic
185#121—531no—
186C4×Q8—531no—
187C4×D4—531no—
188C4×D4—1061no—
189#117—1061no—
190C2×C42—531no—
191C4⋊C8—532no—
192#102—532no—
193#102—532no—
194#102—532no—
195#102C2×D5312yescomplemented
196C22⋊C8—531no—
197C22⋊C8C2×D5311yescharacteristic
198#97—532no—
199C24—21no—
200C24—1061no—
201C2×Q8C22×D5311yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—532no—
205C2×D4—531no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1061no—
211C2×D4—1061no—
212C2×D4—2122no—
213C2×D4—2124no—
214C2×D4—2124no—
215C2×D4C22×D5312yes—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4C22×D5311yescharacteristic
220SD16—42no—
221SD16—2122no—
222D8—2122no—
223D8—2122no—
224C2×C8—532no—
225C2×C8C22×D5312yes—
226C4⋊C4—21no—
227C4⋊C4—531no—
228C4⋊C4—1061no—
229C4⋊C4—1061no—
230C4⋊C4—1061no—
231C4⋊C4C22×D5311yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—1061no—
234C22⋊C4—1061no—
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C42—531no—
238C42—531no—
239C42—531no—
240C42—531no—
241C42—1061no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—1061no—
247C23—1062no—
248C23—2122no—
249C23—2122no—
250C23—2122no—
251C23D4×D5311yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—1062no—
257D4—1062no—
258D4—1062no—
259D4—2122no—
260D4—2124no—
261D4—2124no—
262D4—2124no—
263D4—2124no—
264D4—2122no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1061no—
275C2×C4—2122no—
276C2×C4D4×D5311yescharacteristic
277C2×C4C23×D5311yescharacteristic
278C8—22no—
279C8—1062no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—4244no—
287C22—2121no—
288C22—2122no—
289C22—2122no—
290C22—2121no—
291C22—2122no—
292C22—2121no—
293C22#1313324412yes—
294C22C2×D4×D5311yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—532no—
299C4—1061no—
300C4—1061no—
301C4—1061no—
302C4—2121no—
303C4#1313326212yes—
304C2—22no—
305C2—82no—
306C2—4242no—
307C2—4241no—
308C2#1681118811yescharacteristic
309C2AGL(1,ℤ/8)×D5311yescharacteristic
310C2C2×#1313324411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3217
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. #16811159C2semidirect
3. #16811009C2semidirect
4. #16810986C2semidirect
5. #16810821C2semidirect
6. #16810817C2semidirect
7. #16810773C2semidirect
8. #16810759C2semidirect
9. #16810757C2semidirect
10. #16810677C2semidirect
11. C53×#450C2semidirect
15. C2×#11719924C22semidirect
16. C2×D424C22semidirect
17. C2×#11719922C22semidirect
23. #13133208C22semidirect
24. #13133206C22semidirect
27. Q8×Dic53C22semidirect
28. #13133179C22semidirect
32. #13133146C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133137C22semidirect
37. #13133127C22semidirect
39. C2×SD16×C53C22semidirect
41. #102×C53C22semidirect
163. #450D53semidirect
167. C53#3217semidirect
173. C2×SD16C2×D53semidirect
195. #102C2×D53semidirect