Finite Group Databasebeta

← #20284653

Subgroups — #20284653

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×#13133188C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic, complemented
3#16810946C212yesmaximal, complemented
4#16810945C212yesmaximal, complemented
5#16810940C212yesmaximal, complemented
6#16810935C212yesmaximal, complemented
7#16810666C211yesmaximal, characteristic, complemented
8#16810665C211yesmaximal, characteristic, complemented
9#16810628C211yesmaximal, characteristic, complemented
10#16810586C211yesmaximal, characteristic, complemented
11C53×#418C211yesmaximal, characteristic, complemented
12C22×C22⋊D53—21no—
13C22×C4×D53C2211yescharacteristic
14C2×#11719917C2211yescharacteristic
15#13133188—21no—
16#13133188—21no—
17#13133188C2212yes—
18#13133188C2212yes—
19#13133176C2211yescharacteristic
20#13133175C2211yescharacteristic, complemented
21#13133175C2211yescharacteristic, complemented
22#13133174—21no—
23#13133173—21no—
24C2×C4⋊Dic53C2211yescharacteristic
25C4⋊C4×D53C2211yescharacteristic, complemented
26C4⋊C4×D53C2211yescharacteristic, complemented
27C2×C22⋊Dic53—21no—
28C22⋊C4×D53—21no—
29C42×D53—21no—
30#13133151C2212yescomplemented
31#13133150C2212yescomplemented
32#13133149C2212yescomplemented
33#13133148C2212yescomplemented
34#13133138C2212yescomplemented
35#13133138C2212yescomplemented
36#13133129C2212yescomplemented
37#13133129C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×M16×C53C2211yescharacteristic
40#117×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42#102×C53C2212yescomplemented
43C24×C53—21no—
44C23×D53D411yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—44no—
53C22×Dic53—21no—
54C22×Dic53D411yescharacteristic
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×C53C2311yescharacteristic
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C42×C53—21no—
69C22⋊Dic53—21no—
70C22⋊Dic53—22no—
71C22⋊Dic53—22no—
72#11719921—21no—
73#11719921—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2311yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77C4⋊Dic53C2312yes—
78#11719919—21no—
79#11719919—21no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83#11719917—22no—
84C2×#88322C2312yes—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53C2×D411yescharacteristic
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—41no—
93C22×D53C2×D411yescharacteristic
94C22×D53C2×D411yescharacteristic
95D4×C53—22no—
96D4×C53—22no—
97D4×C53—42no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53#4112yes—
102C2×C4×C53#4112yes—
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—44no—
105C22⋊D53—44no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—42no—
111C2×Dic53C2×D411yescharacteristic
112C2×Dic53C2×D411yescharacteristic
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—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—84no—
129C22×C53#12312yes—
130C22×C53C22×D411yescharacteristic
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—21no—
135C2×D53—21no—
136C2×D53—44no—
137C2×D53AGL(1,ℤ/8)12yes—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53#12312yes—
141C4×C53C2×#4111yescharacteristic
142C4×C53C2×#4111yescharacteristic
143Dic53—21no—
144Dic53—21no—
145Dic53—41no—
146Dic53—42no—
147Dic53—41no—
148#3094—531nomaximal, Sylow
149C2×C53—22no—
150C2×C53—82no—
151C2×C53C2×#12311yescharacteristic
152C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C53#57511yescharacteristic
154D53—22no—
155D53—41no—
156C2×#121—531no—
157C2×#117—531no—
158#482—532no—
159#482—532no—
160#481—532no—
161#480—532no—
162#428—531no—
163#428—531no—
164#418—531no—
165#418D5311yescharacteristic, complemented
166#407—531no—
167C53#309411yescharacteristic, complemented, Sylow
168C22×D4—1061no—
169C22×D4C2×D5311yescharacteristic
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⋊C4—531no—
186C2×C22⋊C4—1061no—
187C2×C22⋊C4—1061no—
188C2×C42—1061no—
189#107—532no—
190#107—532no—
191#106—532no—
192#106—532no—
193#102—532no—
194#102—532no—
195#102C2×D5312yescomplemented
196#102C2×D5312yescomplemented
197C22⋊C8—532no—
198C22⋊C8—532no—
199C24—21no—
200C24—531no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—2124no—
207C2×D4C22×D5312yes—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—1061no—
218C22×C4C22×D5311yescharacteristic
219M16—22no—
220M16—1062no—
221C2×C8—532no—
222C2×C8C22×D5312yes—
223C4⋊C4—531no—
224C4⋊C4—531no—
225C4⋊C4—532no—
226C4⋊C4—1061no—
227C4⋊C4—1061no—
228C4⋊C4—1061no—
229C4⋊C4C22×D5311yescharacteristic
230C4⋊C4C22×D5311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C22⋊C4—1062no—
235C22⋊C4—1062no—
236C22⋊C4—1061no—
237C42—21no—
238C42—1061no—
239C42—1061no—
240C42—1061no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—531no—
246C23—531no—
247C23—1062no—
248C23—1062no—
249C23—2121no—
250C23D4×D5311yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—2124no—
255D4—2124no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1062no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1171994512yes—
274C2×C4D4×D5311yescharacteristic
275C2×C4#1171994712yes—
276C2×C4C23×D5311yescharacteristic
277C8—22no—
278C8—1062no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—532no—
286C22—1061no—
287C22—1061no—
288C22—1061no—
289C22—1061no—
290C22—1061no—
291C22—2124no—
292C22#1313319912yes—
293C22C2×D4×D5311yescharacteristic
294C4—41no—
295C4—41no—
296C4—1061no—
297C4—1061no—
298C4—2121no—
299C4—2122no—
300C4—2121no—
301C4#1313319912yes—
302C4C2×#1171994711yescharacteristic
303C4C2×#1171994511yescharacteristic
304C2—22no—
305C2—82no—
306C2—1062no—
307C2—2121no—
308C2C2×#1313319911yescharacteristic
309C2AGL(1,ℤ/8)×D5311yescharacteristic
310C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#418C2
socleC22×C53C22×D4
hypercenter#418D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D53C2semidirect
3. #16810946C2semidirect
4. #16810945C2semidirect
5. #16810940C2semidirect
6. #16810935C2semidirect
7. #16810666C2semidirect
8. #16810665C2semidirect
9. #16810628C2semidirect
10. #16810586C2semidirect
11. C53×#418C2semidirect
20. #13133175C22semidirect
21. #13133175C22semidirect
25. C4⋊C4×D53C22semidirect
26. C4⋊C4×D53C22semidirect
30. #13133151C22semidirect
31. #13133150C22semidirect
32. #13133149C22semidirect
33. #13133148C22semidirect
34. #13133138C22semidirect
35. #13133138C22semidirect
36. #13133129C22semidirect
37. #13133129C22semidirect
41. #102×C53C22semidirect
42. #102×C53C22semidirect
165. #418D53semidirect
167. C53#3094semidirect
195. #102C2×D53semidirect
196. #102C2×D53semidirect