Finite Group Databasebeta

← #20284657

Subgroups — #20284657

Order 6784.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#13133192C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic, complemented
3#16810950C212yesmaximal, complemented
4#16810949C212yesmaximal, complemented
5#16810943C212yesmaximal, complemented
6#16810938C212yesmaximal, complemented
7#16810663C211yesmaximal, characteristic, complemented
8#16810629C211yesmaximal, characteristic, complemented
9#16810627C211yesmaximal, characteristic, complemented
10#16810586C211yesmaximal, characteristic, complemented
11C53×#428C211yesmaximal, characteristic, complemented
12C22×D212—21no—
13C22×D212C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C2×#11719917C2211yescharacteristic
16#13133192—21no—
17#13133192—21no—
18#13133192C2212yes—
19#13133192C2212yes—
20#13133176C2211yescharacteristic
21#13133175C2211yescharacteristic, complemented
22#13133175C2211yescharacteristic, complemented
23#13133174—21no—
24#13133173—21no—
25C4⋊C4×D53C2211yescharacteristic, complemented
26C4⋊C4×D53C2211yescharacteristic, complemented
27C2×#11719921—21no—
28C22⋊C4×D53—21no—
29C42×D53—21no—
30#13133150C2212yescomplemented
31#13133148C2212yescomplemented
32#13133139C2212yescomplemented
33#13133139C2212yescomplemented
34#13133137C2212yescomplemented
35#13133137C2212yescomplemented
36#13133129C2212yescomplemented
37#13133129C2212yescomplemented
38C2×M16×C53C2211yescharacteristic
39#117×C53C2211yescharacteristic
40C2×C4⋊C4×C53C2211yescharacteristic
41#107×C53C2212yescomplemented
42#106×C53C2212yescomplemented
43C23×D53—21no—
44C23×D53D411yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C22×Dic53D411yescharacteristic
48C2×D212—22no—
49C2×D212—22no—
50C2×D212—22no—
51C2×D212—22no—
52C2×D212—21no—
53C2×D212—44no—
54C2×D212C2312yes—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—21no—
58C2×C4×D53—21no—
59C2×C4×D53D411yescharacteristic
60C2×C4×D53D411yescharacteristic
61C2×C4×D53C2311yescharacteristic
62C2×C4×D53C2311yescharacteristic
63M16×C53—22no—
64C2×C8×C53C2312yes—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2311yescharacteristic
67C4⋊C4×C53C2312yes—
68C4⋊C4×C53C2311yescharacteristic
69C22⋊C4×C53—21no—
70C42×C53—21no—
71C22⋊Dic53—21no—
72#11719921—21no—
73#11719921—21no—
74#11719921—22no—
75#11719921—22no—
76C4⋊Dic53C2311yescharacteristic
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79#11719919—21no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83#11719917—22no—
84C2×#88322C2312yes—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—41no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53C2×D411yescharacteristic
94C22×D53C2×D411yescharacteristic
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53#4112yes—
101C2×C4×C53#4112yes—
102C2×C4×C53C2411yescharacteristic
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108D212—22no—
109D212—22no—
110D212—44no—
111D212—44no—
112D212—42no—
113C4×D53—22no—
114C4×D53—22no—
115C4×D53—21no—
116C4×D53—21no—
117C4×D53—21no—
118C4×D53—21no—
119C4×D53—42no—
120C4×D53—42no—
121C8×C53—22no—
122#88322—22no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53#12312yes—
126C22×C53C22×D411yescharacteristic
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—44no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—84no—
137C2×D53AGL(1,ℤ/8)12yes—
138C4×C53—41no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53#12312yes—
142C4×C53C2×#4111yescharacteristic
143C4×C53C2×#4111yescharacteristic
144Dic53—21no—
145Dic53—21no—
146Dic53—41no—
147Dic53—41no—
148#3094—531nomaximal, Sylow
149C2×C53—22no—
150C2×C53C2×#12311yescharacteristic
151C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C53#57511yescharacteristic
153D53—22no—
154D53—41no—
155D53—82no—
156C2×#121—531no—
157C2×#117—531no—
158#482—532no—
159#482—532no—
160#481—532no—
161#480—532no—
162#428—531no—
163#428D5311yescharacteristic, complemented
164#418—531no—
165#418—531no—
166#407—531no—
167C53#309411yescharacteristic, complemented, Sylow
168C22×D4—531no—
169C22×D4—1061no—
170C23×C4—531no—
171C2×M16—531no—
172C2×M16C2×D5311yescharacteristic
173#121—532no—
174#121—532no—
175#121—1061no—
176#121—1061no—
177#117—531no—
178#117—531no—
179#117—531no—
180#117—1061no—
181#117—1061no—
182#117C2×D5311yescharacteristic
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4C2×D5311yescharacteristic
186C2×C22⋊C4—1061no—
187C2×C22⋊C4—1061no—
188C2×C42—1061no—
189#107—532no—
190#107C2×D5312yescomplemented
191#106—532no—
192#106C2×D5312yescomplemented
193#102—532no—
194#102—532no—
195#102—532no—
196#102—532no—
197C22⋊C8—532no—
198C22⋊C8—532no—
199C24—531no—
200C24—1061no—
201C2×D4—532no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1061no—
206C2×D4—1062no—
207C2×D4—2124no—
208C22×C4—21no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—1062no—
217C22×C4—1062no—
218C22×C4C22×D5311yescharacteristic
219M16—22no—
220M16—1062no—
221C2×C8—532no—
222C2×C8C22×D5312yes—
223C4⋊C4—21no—
224C4⋊C4—531no—
225C4⋊C4—531no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4C22×D5311yescharacteristic
229C4⋊C4C22×D5311yescharacteristic
230C4⋊C4C22×D5312yes—
231C22⋊C4—21no—
232C22⋊C4—1062no—
233C22⋊C4—1062no—
234C22⋊C4—1061no—
235C22⋊C4—1061no—
236C22⋊C4—1061no—
237C42—21no—
238C42—1061no—
239C42—1061no—
240C42—1061no—
241C23—531no—
242C23—531no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—2121no—
247C23—2122no—
248C23—2122no—
249C23—2122no—
250C23D4×D5311yescharacteristic
251D4—1062no—
252D4—1062no—
253D4—2124no—
254D4—2124no—
255D4—2122no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—531no—
261C2×C4—531no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1171994512yes—
274C2×C4D4×D5311yescharacteristic
275C2×C4#1171994912yes—
276C2×C4C23×D5311yescharacteristic
277C8—22no—
278C8—1062no—
279C22—21no—
280C22—21no—
281C22—532no—
282C22—1061no—
283C22—1061no—
284C22—1061no—
285C22—1061no—
286C22—1061no—
287C22—4244no—
288C22—2124no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22#1313320212yes—
293C22C2×D4×D5311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—1061no—
298C4—1061no—
299C4—2121no—
300C4—2121no—
301C4#1313320212yes—
302C4C2×#1171994511yescharacteristic
303C4C2×#1171994911yescharacteristic
304C2—22no—
305C2—1062no—
306C2—4242no—
307C2—2121no—
308C2C2×#1313320211yescharacteristic
309C2AGL(1,ℤ/8)×D5311yescharacteristic
310C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#428C2
socleC22×C53C22×D4
hypercenter#428D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D53C2semidirect
3. #16810950C2semidirect
4. #16810949C2semidirect
5. #16810943C2semidirect
6. #16810938C2semidirect
7. #16810663C2semidirect
8. #16810629C2semidirect
9. #16810627C2semidirect
10. #16810586C2semidirect
11. C53×#428C2semidirect
21. #13133175C22semidirect
22. #13133175C22semidirect
25. C4⋊C4×D53C22semidirect
26. C4⋊C4×D53C22semidirect
30. #13133150C22semidirect
31. #13133148C22semidirect
32. #13133139C22semidirect
33. #13133139C22semidirect
34. #13133137C22semidirect
35. #13133137C22semidirect
36. #13133129C22semidirect
37. #13133129C22semidirect
41. #107×C53C22semidirect
42. #106×C53C22semidirect
163. #428D53semidirect
167. C53#3094semidirect
190. #107C2×D53semidirect
192. #106C2×D53semidirect