Finite Group Databasebeta

← #20284853

Subgroups — #20284853

Order 6784.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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
2#16810953C211yesmaximal, characteristic, complemented
3#16810936C211yesmaximal, characteristic, complemented
4#16810923C212yesmaximal, complemented
5#16810838C212yesmaximal, complemented
6#16810774C211yesmaximal, characteristic, complemented
7#16810757C211yesmaximal, characteristic
8#16810733C212yesmaximal, complemented
9#16810723C212yesmaximal, complemented
10#16810680C211yesmaximal, characteristic, complemented
11C2×C4×D4×C53C211yesmaximal, characteristic, complemented
12C2×#11719923C2212yes—
13C2×#11719922C2212yes—
14#13133211—21no—
15#13133209C2211yescharacteristic, complemented
16#13133205C2211yescharacteristic, complemented
17#13133200—21no—
18#13133196C2211yescharacteristic
19#13133191C2211yescharacteristic
20C4×Dic106C2211yescharacteristic, complemented
21C4×D212C2211yescharacteristic, complemented
22C4×C22⋊D53—21no—
23#13133174—21no—
24#13133150C2212yescomplemented
25#13133148C2212yescomplemented
26C4⋊#88322C2212yescomplemented
27#13133140C2212yescomplemented
28#13133138C2212yes—
29#13133138C2212yes—
30#13133137C2212yescomplemented
31C22⋊#88322C2211yescharacteristic
32C22⋊#88322C2211yescharacteristic
33#13133128C2212yescomplemented
34C22×D4×C53C2211yescharacteristic
35C23×C4×C53—21no—
36C4×D4×C53—21no—
37C4×D4×C53—21no—
38C4×D4×C53C2212yes—
39C4×D4×C53C2212yes—
40C2×C4⋊C4×C53C2211yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C42×C53C2211yescharacteristic
43C24×C53—21no—
44C2×D4×C53—24no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—44no—
51C22×C4×C53D411yescharacteristic
52C22×C4×C53D411yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C2×C22⋊D53—21no—
55C2×D212C2311yescharacteristic
56C2×C4×D53—21no—
57C2×Dic106C2311yescharacteristic
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2312yes—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—22no—
62C42×C53D411yescharacteristic
63C42×C53D411yescharacteristic
64C42×C53C2311yescharacteristic
65C42×C53C2311yescharacteristic
66C22⋊Dic53—21no—
67#11719923—24no—
68#11719922—24no—
69#11719921—21no—
70#11719921—21no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53C2311yescharacteristic
74C4⋊Dic53C2311yescharacteristic
75#11719919—21no—
76#11719919—21no—
77#11719919—21no—
78#11719919—21no—
79C4×Dic53—21no—
80C2×#88322C2312yes—
81C2×#88322C2312yes—
82C23×C53—24no—
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53C2×D411yescharacteristic
87C22×D53—21no—
88D4×C53—24no—
89D4×C53—22no—
90D4×C53#4114yes—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—44no—
96C2×C4×C53—44no—
97C2×C4×C53—42no—
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—42no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108D212—22no—
109C4×D53—42no—
110Dic106—22no—
111#88322—22no—
112#88322—22no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—24no—
116C22×C53—48no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53AGL(1,ℤ/8)12yes—
120C22×C53C22×D411yescharacteristic
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×C53—42no—
130C4×C53C2×#4112yes—
131Dic53—41no—
132Dic53—41no—
133Dic53—41no—
134#3114—531nomaximal, Sylow
135C2×C53—22no—
136C2×C53—44no—
137C2×C53#52511yescharacteristic
138C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
139C2×C53#57511yescharacteristic
140D53—81no—
141#529—531no—
142C2×C4×D4D5311yescharacteristic, complemented
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#432—531no—
152C53#311411yescharacteristic, complemented, Sylow
153C22×D4C2×D5311yescharacteristic
154C23×C4—21no—
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—21no—
165C4×D4—21no—
166C4×D4—531no—
167C4×D4—1061no—
168C4×D4C2×D5312yes—
169C4×D4C2×D5312yes—
170#117—1061no—
171C2×C4⋊C4C2×D5311yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D5311yescharacteristic
174#107—532no—
175#106—532no—
176C4⋊C8—532no—
177#103—532no—
178#102—532no—
179#102—532no—
180#102—532no—
181C22⋊C8—531no—
182C22⋊C8—531no—
183#97—532no—
184C24—21no—
185C2×Q8—531no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—531no—
189C2×D4—1061no—
190C2×D4C22×D5312yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1061no—
196C22×C4C22⋊D5311yescharacteristic
197C22×C4C22⋊D5311yescharacteristic
198C22×C4C22×D5311yescharacteristic
199SD16—1064no—
200D8—1064no—
201C2×C8—532no—
202C2×C8—532no—
203C4⋊C4—21no—
204C4⋊C4—531no—
205C4⋊C4—531no—
206C4⋊C4—1061no—
207C4⋊C4—1061no—
208C4⋊C4—1061no—
209C4⋊C4—1061no—
210C4⋊C4C22×D5312yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—1061no—
214C22⋊C4—1061no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1061no—
218C42—1061no—
219C42C22⋊D5311yescharacteristic
220C42C22⋊D5311yescharacteristic
221C42C22×D5311yescharacteristic
222C42C22×D5311yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1061no—
228C23C2×C22⋊D5311yescharacteristic
229Q8—1062no—
230D4—24no—
231D4—22no—
232D4—1062no—
233D4—2122no—
234D4#1171994514yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—2122no—
246C2×C4C2×C22⋊D5311yescharacteristic
247C2×C4C2×C22⋊D5311yescharacteristic
248C2×C4C2×C22⋊D5311yescharacteristic
249C2×C4C2×C22⋊D5311yescharacteristic
250C2×C4C2×C22⋊D5311yescharacteristic
251C2×C4C23×D5311yescharacteristic
252C8—1062no—
253C8—1062no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—2121no—
261C22—2121no—
262C22—2121no—
263C22#1313324312yes—
264C22C22×C22⋊D5311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—2121no—
272C4—2121no—
273C4—2121no—
274C4C2×#1171994512yes—
275C2—22no—
276C2—44no—
277C2—4241no—
278C2#1681113411yescharacteristic
279C2C2×#1313324311yescharacteristic
280C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C4×D4×C53C2
socleC22×C53C22×D4
hypercenterC2×C4×D4D53
hypocenterC53#3114
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811157C2semidirect
2. #16810953C2semidirect
3. #16810936C2semidirect
4. #16810923C2semidirect
5. #16810838C2semidirect
6. #16810774C2semidirect
8. #16810733C2semidirect
9. #16810723C2semidirect
10. #16810680C2semidirect
11. C2×C4×D4×C53C2semidirect
15. #13133209C22semidirect
16. #13133205C22semidirect
20. C4×Dic106C22semidirect
21. C4×D212C22semidirect
24. #13133150C22semidirect
25. #13133148C22semidirect
26. C4⋊#88322C22semidirect
27. #13133140C22semidirect
30. #13133137C22semidirect
33. #13133128C22semidirect
142. C2×C4×D4D53semidirect
152. C53#3114semidirect