Finite Group Databasebeta

← #20285534

Subgroups — #20285534

Order 6784.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811136C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3#16810949C212yesmaximal, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810877C212yesmaximal, complemented
6#16810872C212yesmaximal, complemented
7#16810773C211yesmaximal, characteristic, complemented
8#16810771C212yesmaximal, complemented
9#16810759C211yesmaximal, characteristic, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#483C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic, complemented
13C22×D212—21no—
14C22×D212C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16C2×#11719924C2212yescomplemented
17C2×D424C2212yescomplemented
18C2×#11719912C2212yescomplemented
19#13133202—21no—
20#13133193C2211yescharacteristic, complemented
21#13133192—21no—
22#13133192—21no—
23#13133192C2212yes—
24#13133192C2212yes—
25#13133192C2211yescharacteristic, complemented
26#13133191C2211yescharacteristic
27#13133187—21no—
28#13133179—21no—
29#13133177—21no—
30#13133175C2211yescharacteristic, complemented
31C2×#11719921—21no—
32#13133150C2212yescomplemented
33#13133139C2212yescomplemented
34#13133137C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yescomplemented
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#106×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53—21no—
44C23×D53D411yescharacteristic
45C2×Q8×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—21no—
49C22×Dic53D411yescharacteristic
50#11719949—42no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212—21no—
56C2×D212—42no—
57C2×D212—44no—
58C2×D212C2311yescharacteristic
59C2×D212C2312yes—
60C2×C4×D53—21no—
61C2×C4×D53—42no—
62C2×C4×D53D411yescharacteristic
63C2×C4×D53D411yescharacteristic
64C2×C4×D53C2311yescharacteristic
65C2×C4×D53C2311yescharacteristic
66C2×C8×C53C2312yes—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—21no—
69C4⋊C4×C53C2312yes—
70C4⋊C4×C53C2311yescharacteristic
71C22⋊C4×C53—21no—
72#11719924—42no—
73#11719921—21no—
74#11719921—21no—
75#11719921—21no—
76#11719921—22no—
77#11719921—22no—
78#11719921—21no—
79C4⋊Dic53C2311yescharacteristic
80#11719919—21no—
81C4×Dic53—21no—
82C2×#88322C2312yes—
83D424—42no—
84#11719912—42no—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—41no—
90C22×D53—41no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53—42no—
94C22×D53C2×D411yescharacteristic
95C22×D53C2×D411yescharacteristic
96Q8×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—42no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53C2×D411yescharacteristic
108C2×Dic53C2×D411yescharacteristic
109D212—22no—
110D212—22no—
111D212—22no—
112D212—42no—
113D212—42no—
114D212—44no—
115D212—44no—
116D212—42no—
117C4×D53—22no—
118C4×D53—22no—
119C4×D53—41no—
120C4×D53—41no—
121C4×D53—42no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53AGL(1,ℤ/8)12yes—
127C22×C53C22×D411yescharacteristic
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—44no—
135C2×D53—41no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—42no—
141C2×D53—84no—
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53H(5,𝔽2)12yes—
147Dic53—21no—
148Dic53—21no—
149Dic53—41no—
150#3194—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53#53411yescharacteristic
153C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C53#57511yescharacteristic
155D53—41no—
156D53—41no—
157D53—81no—
158D53—82no—
159#525—531no—
160C2×#121—531no—
161#483D5311yescharacteristic, complemented
162#482—532no—
163#480—531no—
164#469—532no—
165#468—532no—
166#450—531no—
167#449—532no—
168#447—531no—
169#407—531no—
170C53#319411yescharacteristic, complemented, Sylow
171C2×#41—531no—
172C22×D4—531no—
173C22×D4—1061no—
174C23×C4—531no—
175C2×SD16—532no—
176C2×D8—532no—
177C2×M16—532no—
178#123—1061no—
179C22⋊Q8—531no—
180C22⋊Q8C2×D5311yescharacteristic
181#121—531no—
182#121—532no—
183#121—532no—
184#121—531no—
185#121—1061no—
186#121—1061no—
187#120—1061no—
188C4×D4—1061no—
189C4×D4—1061no—
190#117—531no—
191C2×C4⋊C4C2×D5311yescharacteristic
192C2×C22⋊C4—1061no—
193#107—532no—
194#106C2×D5312yescomplemented
195#103C2×D5312yescomplemented
196#102—532no—
197#102—532no—
198#102—532no—
199C22⋊C8—531no—
200C22⋊C8—532no—
201C22⋊C8C2×D5311yescharacteristic
202C24—531no—
203C24—1061no—
204#41—2122no—
205C2×Q8C22×D5311yescharacteristic
206C2×D4—531no—
207C2×D4—532no—
208C2×D4—1061no—
209C2×D4—1061no—
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—1061no—
213C2×D4—1061no—
214C2×D4—2122no—
215C2×D4—2124no—
216C22×C4—21no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—1061no—
223C22×C4—2122no—
224C22×C4C22×D5311yescharacteristic
225SD16—2122no—
226D8—2122no—
227M16—2122no—
228C2×C8—532no—
229C2×C8C22×D5312yes—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—531no—
233C4⋊C4—1061no—
234C4⋊C4C22×D5311yescharacteristic
235C4⋊C4C22×D5312yes—
236C22⋊C4—21no—
237C22⋊C4—1062no—
238C22⋊C4—1062no—
239C22⋊C4—1061no—
240C22⋊C4—1061no—
241C22⋊C4—1061no—
242C22⋊C4—1061no—
243C42—1061no—
244C23—531no—
245C23—531no—
246C23—1062no—
247C23—1061no—
248C23—1062no—
249C23—2121no—
250C23—2121no—
251C23—2122no—
252C23—2122no—
253C23—2122no—
254C23D4×D5311yescharacteristic
255Q8—22no—
256D4—1062no—
257D4—1062no—
258D4—1062no—
259D4—2122no—
260D4—2122no—
261D4—2124no—
262D4—2124no—
263D4—2122no—
264D4—2122no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—531no—
271C2×C4—531no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1061no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2122no—
278C2×C4—2121no—
279C2×C4D4×D5311yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—22no—
282C8—1062no—
283C22—21no—
284C22—21no—
285C22—1061no—
286C22—1061no—
287C22—1061no—
288C22—1061no—
289C22—1061no—
290C22—1061no—
291C22—4244no—
292C22—2124no—
293C22—2122no—
294C22—2121no—
295C22—2121no—
296C22—2122no—
297C22—2121no—
298C22—2122no—
299C22#1313324412yes—
300C22C2×D4×D5311yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—1061no—
306C4—1061no—
307C4—2121no—
308C4#1313326312yes—
309C2—22no—
310C2—4242no—
311C2—4241no—
312C2—2121no—
313C2—2121no—
314C2#1681118411yescharacteristic
315C2C2×#1313324411yescharacteristic
316C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#483C2
socleC22×C53C22×D4
hypercenter#483D53
hypocenterC53#3194
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #16811136C2semidirect
2. C2×#13133192C2semidirect
3. #16810949C2semidirect
4. #16810936C2semidirect
5. #16810877C2semidirect
6. #16810872C2semidirect
7. #16810773C2semidirect
8. #16810771C2semidirect
9. #16810759C2semidirect
10. #16810585C2semidirect
11. C53×#483C2semidirect
12. C2×#11719949C22semidirect
16. C2×#11719924C22semidirect
17. C2×D424C22semidirect
18. C2×#11719912C22semidirect
20. #13133193C22semidirect
25. #13133192C22semidirect
30. #13133175C22semidirect
32. #13133150C22semidirect
33. #13133139C22semidirect
34. #13133137C22semidirect
35. #13133137C22semidirect
37. #13133129C22semidirect
40. #106×C53C22semidirect
41. #103×C53C22semidirect
161. #483D53semidirect
170. C53#3194semidirect
194. #106C2×D53semidirect
195. #103C2×D53semidirect