Finite Group Databasebeta

← #20286956

Subgroups — #20286956

Order 6784.

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

normal subgroups only

This group has 460 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#16811339C211yesmaximal, characteristic, complemented
2#16811258C211yesmaximal, characteristic, complemented
3#16810945C212yesmaximal, complemented
4#16810910C211yesmaximal, characteristic, complemented
5#16810875C212yesmaximal, complemented
6#16810815C211yesmaximal, characteristic, complemented
7#16810758C212yesmaximal, complemented
8C4×D424C211yesmaximal, characteristic, complemented
9#16810702C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal, complemented
11C53×#486C211yesmaximal, characteristic, complemented
12C2×D4×D53—21no—
13C2×D4×D53C2212yes—
14C2×D424C2211yescharacteristic
15C2×#11719922C2212yescomplemented
16C2×#11719912C2212yescomplemented
17#13133206—22no—
18#13133192—21no—
19#13133192C2212yescomplemented
20#13133188—21no—
21#13133188C2212yes—
22#13133185—22no—
23C4×D212C2211yescharacteristic
24C4×D212C2211yescharacteristic
25C4×D212C2211yescharacteristic, complemented
26#13133175—21no—
27#13133175C2212yescomplemented
28#13133151C2211yescharacteristic, complemented
29#13133148C2212yescomplemented
30C4⋊#88322C2211yescharacteristic, complemented
31#13133139C2211yescharacteristic
32#13133139C2211yescharacteristic, complemented
33#13133138C2212yescomplemented
34#13133137C2212yes—
35#13133129C2212yescomplemented
36C4⋊Q8×C53C2211yescharacteristic, complemented
37#127×C53C2211yescharacteristic
38#102×C53C2212yescomplemented
39#102×C53C2212yescomplemented
40C4×C8×C53C2211yescharacteristic, complemented
41C23×D53—22no—
42C2×Q8×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53C2312yes—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48D4×D53—24no—
49D4×D53—42no—
50D4×D53—44no—
51C2×D212—22no—
52C2×D212D411yescharacteristic
53C2×D212C2311yescharacteristic
54C2×D212C2311yescharacteristic
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×C4×D53C2312yes—
58C2×C8×C53C2311yescharacteristic
59C2×C8×C53C2311yescharacteristic
60C4⋊C4×C53—21no—
61C4⋊C4×C53C2312yes—
62C42×C53C2311yescharacteristic
63C22⋊Dic53—22no—
64#11719922—42no—
65#11719921—22no—
66#11719921—21no—
67#11719921—21no—
68#11719921—22no—
69#11719921D412yes—
70C4⋊Dic53D411yescharacteristic
71C4⋊Dic53C2311yescharacteristic
72C4⋊Dic53C2311yescharacteristic
73C4×Dic53—22no—
74C2×#88322C2312yes—
75D424—22no—
76#11719912—42no—
77C23×C53—22no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—44no—
81C22×D53—44no—
82C22×D53—82no—
83C22×D53C2×D412yes—
84Q8×C53—81no—
85D4×C53—22no—
86D4×C53—22no—
87D4×C53—42no—
88D4×C53—81no—
89C2×C4×C53—22no—
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—44no—
94C22⋊D53—44no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53C2×D412yes—
98D212—22no—
99D212—21no—
100D212—21no—
101D212—21no—
102D212—82no—
103D212#4112yes—
104C4×D53—22no—
105C4×D53—22no—
106C4×D53—42no—
107C4×D53—42no—
108C8×C53—21no—
109C8×C53—21no—
110#88322—22no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53C22×D411yescharacteristic
115C2×D53—22no—
116C2×D53—22no—
117C2×D53—22no—
118C2×D53—22no—
119C2×D53—41no—
120C2×D53—41no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—88no—
125C4×C53—21no—
126C4×C53—42no—
127C4×C53AGL(1,ℤ/8)12yes—
128C4×C53C2×#4111yescharacteristic
129C4×C53H(5,𝔽2)11yescharacteristic
130Dic53—22no—
131Dic53—41no—
132Dic53—41no—
133#3325—531nomaximal, Sylow
134C2×C53—82no—
135C2×C53#54611yescharacteristic
136C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C53#57611yescharacteristic
138D53—42no—
139D53—42no—
140D53—81no—
141#555—531no—
142D42—531no—
143#486D5311yescharacteristic, complemented
144#482—532no—
145#476—531no—
146#469—532no—
147#459—531no—
148#447—532no—
149C4×D8—531no—
150#436—531no—
151#418—532no—
152C53#332511yescharacteristic, complemented, Sylow
153C22×D4—532no—
154C22×D4—1061no—
155C2×D8—532no—
156C2×D8—531no—
157C2×M16—532no—
158C4⋊Q8C2×D5311yescharacteristic, complemented
159#127C2×D5311yescharacteristic
160#124—1062no—
161#121—532no—
162#121—532no—
163#121—1061no—
164#121—1061no—
165#120—1062no—
166C4×D4—531no—
167C4×D4—531no—
168C4×D4—531no—
169#117—532no—
170#117—1061no—
171#107—531no—
172#106—532no—
173C4⋊C8—531no—
174#102—532no—
175#102—531no—
176#102—531no—
177#102—532no—
178#102C2×D5312yescomplemented
179#102C2×D5312yescomplemented
180C22⋊C8—532no—
181C4×C8C2×D5311yescharacteristic, complemented
182C24—1062no—
183C2×Q8—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—531no—
187C2×D4—531no—
188C2×D4—531no—
189C2×D4—1062no—
190C2×D4—1062no—
191C2×D4—1064no—
192C2×D4—1062no—
193C2×D4—2122no—
194C2×D4—2124no—
195C2×D4C22×D5312yes—
196C22×C4—532no—
197C22×C4—1061no—
198C22×C4—1061no—
199D8—1062no—
200D8—2122no—
201M16—2122no—
202C2×C8—532no—
203C2×C8C22×D5311yescharacteristic
204C2×C8C22×D5311yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—531no—
207C4⋊C4—531no—
208C4⋊C4—531no—
209C4⋊C4C22×D5312yes—
210C22⋊C4—532no—
211C22⋊C4—1062no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C22⋊C4—1061no—
215C22⋊C4—1061no—
216C42—1062no—
217C42C22×D5311yescharacteristic
218C23—22no—
219C23—532no—
220C23—1061no—
221C23—1061no—
222C23—4242no—
223C23—2124no—
224C23—2124no—
225Q8—81no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—81no—
230D4—532no—
231D4—1062no—
232D4—1061no—
233D4—1061no—
234D4—1061no—
235D4—4242no—
236D4—2124no—
237D4—2124no—
238C2×C4—22no—
239C2×C4—532no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4D4×D5311yescharacteristic
247C2×C4D4×D5311yescharacteristic
248C2×C4C23×D5311yescharacteristic
249C8—21no—
250C8—21no—
251C8—1062no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—1062no—
256C22—1062no—
257C22—1062no—
258C22—1062no—
259C22—4248no—
260C22—2121no—
261C22—2121no—
262C22—2121no—
263C22—2121no—
264C22—2121no—
265C22C2×D4×D5311yescharacteristic
266C4—21no—
267C4—42no—
268C4—1062no—
269C4—2121no—
270C4—2121no—
271C4#1313324412yes—
272C4#41×D5311yescharacteristic
273C4#1313326311yescharacteristic
274C2—82no—
275C2—4241no—
276C2—2122no—
277C2—2122no—
278C2#1681126611yescharacteristic
279C2C2×#1313324411yescharacteristic
280C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3325
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811339C2semidirect
2. #16811258C2semidirect
3. #16810945C2semidirect
4. #16810910C2semidirect
5. #16810875C2semidirect
6. #16810815C2semidirect
7. #16810758C2semidirect
8. C4×D424C2semidirect
9. #16810702C2semidirect
10. #16810626C2semidirect
11. C53×#486C2semidirect
15. C2×#11719922C22semidirect
16. C2×#11719912C22semidirect
19. #13133192C22semidirect
25. C4×D212C22semidirect
27. #13133175C22semidirect
28. #13133151C22semidirect
29. #13133148C22semidirect
30. C4⋊#88322C22semidirect
32. #13133139C22semidirect
33. #13133138C22semidirect
35. #13133129C22semidirect
36. C4⋊Q8×C53C22semidirect
38. #102×C53C22semidirect
39. #102×C53C22semidirect
40. C4×C8×C53C22semidirect
143. #486D53semidirect
152. C53#3325semidirect
158. C4⋊Q8C2×D53semidirect
178. #102C2×D53semidirect
179. #102C2×D53semidirect
181. C4×C8C2×D53semidirect