Finite Group Databasebeta

← #20284855

Subgroups — #20284855

Order 6784.

450 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811157C211yesmaximal, characteristic, complemented
2C2×C4×D212C211yesmaximal, characteristic, complemented
3#16810954C211yesmaximal, characteristic, complemented
4#16810937C211yesmaximal, characteristic, complemented
5#16810924C212yesmaximal, complemented
6#16810839C212yesmaximal, complemented
7#16810776C211yesmaximal, characteristic, complemented
8#16810759C211yesmaximal, characteristic
9#16810736C212yesmaximal, complemented
10#16810725C212yesmaximal, complemented
11C53×#432C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53—21no—
14C2×#11719913C2212yes—
15C2×D424C2212yes—
16#13133211—21no—
17#13133209C2211yescharacteristic, complemented
18#13133205C2211yescharacteristic, complemented
19#13133200—21no—
20#13133196C2211yescharacteristic
21#13133191C2211yescharacteristic
22C4×Dic106C2211yescharacteristic, complemented
23C4×D212—21no—
24C4×D212—21no—
25C4×D212C2212yes—
26C4×D212C2212yes—
27C4×D212C2211yescharacteristic, complemented
28C4×C22⋊D53—21no—
29#13133174—21no—
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719921—21no—
32#13133151C2212yescomplemented
33#13133149C2212yescomplemented
34#13133142C2212yescomplemented
35#13133139C2212yes—
36#13133139C2212yes—
37#13133139C2212yescomplemented
38C2×C42×C53C2211yescharacteristic
39C4⋊C8×C53C2212yescomplemented
40C22⋊C8×C53C2211yescharacteristic
41C22⋊C8×C53C2211yescharacteristic
42#97×C53C2212yescomplemented
43C23×D53—21no—
44C22×C4×C53D411yescharacteristic
45C22×C4×C53D411yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49C2×D212—21no—
50C2×D212—24no—
51C2×D212C2311yescharacteristic
52C2×D212C2312yes—
53C2×C4×D53—21no—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—44no—
57C2×Dic106C2311yescharacteristic
58C2×C8×C53C2312yes—
59C2×C8×C53C2312yes—
60C42×C53D411yescharacteristic
61C42×C53D411yescharacteristic
62C42×C53C2311yescharacteristic
63C42×C53C2311yescharacteristic
64C22⋊Dic53—21no—
65#11719921—22no—
66#11719921—22no—
67#11719921—21no—
68#11719921—21no—
69#11719921—21no—
70#11719921—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53C2311yescharacteristic
73C4⋊Dic53C2311yescharacteristic
74C4⋊Dic53C2312yes—
75#11719919—21no—
76#11719919—21no—
77#11719919—21no—
78#11719919—21no—
79C4×Dic53—21no—
80D424—24no—
81#11719913—24no—
82C23×C53C2×D411yescharacteristic
83C22×D53—24no—
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—42no—
87C22×D53—42no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—42no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—42no—
103D212—24no—
104D212—22no—
105D212—22no—
106D212#4114yes—
107C4×D53—42no—
108C4×D53—44no—
109C4×D53—44no—
110Dic106—22no—
111C8×C53—22no—
112C8×C53—22no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53AGL(1,ℤ/8)12yes—
116C22×C53C22×D411yescharacteristic
117C2×D53—24no—
118C2×D53—48no—
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53C2×#4112yes—
130Dic53—41no—
131Dic53—42no—
132Dic53—41no—
133Dic53—41no—
134#3114—531nomaximal, Sylow
135C2×C53—22no—
136C2×C53#52511yescharacteristic
137C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C53#57511yescharacteristic
139D53—44no—
140D53—81no—
141#529—531no—
142C2×C4×D4—531no—
143#483—531no—
144#480—531no—
145#478—532no—
146#463—532no—
147#450—531no—
148#447—531no—
149#442—532no—
150#440—532no—
151#432D5311yescharacteristic, complemented
152C53#311411yescharacteristic, complemented, Sylow
153C22×D4—531no—
154C23×C4—1061no—
155C2×SD16—532no—
156C2×D8—532no—
157#126—1061no—
158#125—531no—
159#124—531no—
160#123—1061no—
161C22⋊Q8—531no—
162#121—531no—
163C4×Q8—531no—
164C4×D4—532no—
165C4×D4—532no—
166C4×D4—531no—
167C4×D4—1061no—
168C4×D4—1061no—
169C4×D4—1061no—
170#117—1061no—
171C2×C4⋊C4—531no—
172C2×C22⋊C4—1061no—
173C2×C42C2×D5311yescharacteristic
174#107—532no—
175#106—532no—
176C4⋊C8C2×D5312yescomplemented
177#103—532no—
178#102—532no—
179#102—532no—
180#102—532no—
181C22⋊C8C2×D5311yescharacteristic
182C22⋊C8C2×D5311yescharacteristic
183#97C2×D5312yescomplemented
184C24—1061no—
185C2×Q8—531no—
186C2×D4—532no—
187C2×D4—531no—
188C2×D4—1064no—
189C2×D4—1061no—
190C2×D4—1061no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4—1062no—
194C22×C4—1062no—
195C22×C4—2124no—
196C22×C4D21211yescharacteristic
197C22×C4D21211yescharacteristic
198C22×C4C22×D5311yescharacteristic
199SD16—1064no—
200D8—1064no—
201C2×C8C22×D5312yes—
202C2×C8C22×D5312yes—
203C4⋊C4—531no—
204C4⋊C4—531no—
205C4⋊C4—532no—
206C4⋊C4—1061no—
207C4⋊C4—1061no—
208C4⋊C4—1061no—
209C4⋊C4—1061no—
210C4⋊C4—1061no—
211C22⋊C4—1061no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C22⋊C4—1061no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1061no—
218C42—1061no—
219C42D21211yescharacteristic
220C42D21211yescharacteristic
221C42C22×D5311yescharacteristic
222C42C22×D5311yescharacteristic
223C23—1064no—
224C23—1062no—
225C23—1061no—
226C23—2122no—
227C23—2122no—
228C23C2×D21211yescharacteristic
229Q8—1062no—
230D4—534no—
231D4—1064no—
232D4—1062no—
233D4—1062no—
234D4—2122no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—2122no—
243C2×C4—2124no—
244C2×C4—2124no—
245C2×C4—2122no—
246C2×C4C2×D21211yescharacteristic
247C2×C4C2×D21211yescharacteristic
248C2×C4C2×D21211yescharacteristic
249C2×C4C2×D21211yescharacteristic
250C2×C4C2×D21211yescharacteristic
251C2×C4C23×D5311yescharacteristic
252C8—22no—
253C8—22no—
254C22—21no—
255C22—21no—
256C22—1064no—
257C22—2128no—
258C22—2121no—
259C22—2121no—
260C22—2121no—
261C22—2122no—
262C22—2122no—
263C22#1313324512yes—
264C22C22×D21211yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—2121no—
271C4—2122no—
272C4—2121no—
273C4—2121no—
274C4#41×D5312yes—
275C2—22no—
276C2—4241no—
277C2—2124no—
278C2#1681113511yescharacteristic
279C2C2×#1313324511yescharacteristic
280C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3114
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #16811157C2semidirect
2. C2×C4×D212C2semidirect
3. #16810954C2semidirect
4. #16810937C2semidirect
5. #16810924C2semidirect
6. #16810839C2semidirect
7. #16810776C2semidirect
9. #16810736C2semidirect
10. #16810725C2semidirect
11. C53×#432C2semidirect
17. #13133209C22semidirect
18. #13133205C22semidirect
22. C4×Dic106C22semidirect
27. C4×D212C22semidirect
32. #13133151C22semidirect
33. #13133149C22semidirect
34. #13133142C22semidirect
37. #13133139C22semidirect
39. C4⋊C8×C53C22semidirect
42. #97×C53C22semidirect
151. #432D53semidirect
152. C53#3114semidirect
176. C4⋊C8C2×D53semidirect
183. #97C2×D53semidirect