Finite Group Databasebeta

← #20284631

Subgroups — #20284631

Order 6784.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811136C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic, complemented
3#16810963C212yesmaximal, complemented
4#16810951C211yesmaximal, characteristic, complemented
5#16810950C212yesmaximal, complemented
6#16810941C211yesmaximal, characteristic, complemented
7#16810937C211yesmaximal, characteristic, complemented
8#16810660C212yesmaximal, complemented
9#16810618C212yesmaximal, complemented
10C22⋊C8×D53C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C2×C8×D53C2212yes—
15#13133202—21no—
16#13133193C2211yescharacteristic
17#13133192C2211yescharacteristic
18#13133191C2211yescharacteristic
19#13133187—21no—
20#13133179—21no—
21#13133177—21no—
22#13133175—21no—
23#13133175—21no—
24#13133175C2212yes—
25#13133175C2212yes—
26#13133175C2211yescharacteristic
27C2×C4⋊Dic53C2211yescharacteristic
28C2×#11719921—21no—
29C2×C4×Dic53—21no—
30#13133151C2212yescomplemented
31#13133148C2212yescomplemented
32#13133148C2212yescomplemented
33#13133141C2212yescomplemented
34#13133139C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×Q8×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49C22×Dic53D411yescharacteristic
50#11719949—24no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212C2311yescharacteristic
54C2×C4×D53—24no—
55C2×C4×D53—21no—
56C2×C4×D53D411yescharacteristic
57C2×C4×D53D411yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×C4×D53C2311yescharacteristic
60C2×C8×C53C2312yes—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—21no—
63C4⋊C4×C53C2312yes—
64C4⋊C4×C53C2311yescharacteristic
65C22⋊C4×C53—21no—
66#11719921—22no—
67#11719921—22no—
68#11719921—21no—
69#11719921—21no—
70#11719921—21no—
71#11719921—21no—
72C4⋊Dic53—21no—
73C4⋊Dic53C2312yes—
74C4⋊Dic53C2311yescharacteristic
75#11719919—21no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—21no—
79C2×#88322C2312yes—
80C8×D53—24no—
81C23×C53C2×D411yescharacteristic
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53C2×D411yescharacteristic
87C22×D53C2×D411yescharacteristic
88Q8×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—42no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—42no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—42no—
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103D212—22no—
104D212—42no—
105D212—42no—
106C4×D53—22no—
107C4×D53—22no—
108C4×D53—22no—
109C4×D53—42no—
110C4×D53#4114yes—
111C8×C53—22no—
112#88322—22no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53#13512yes—
116C22×C53C22×D411yescharacteristic
117C2×D53—21no—
118C2×D53—21no—
119C2×D53—21no—
120C2×D53—21no—
121C2×D53—21no—
122C2×D53—44no—
123C2×D53—41no—
124C2×D53—41no—
125C2×D53—41no—
126C2×D53#12312yes—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53—42no—
131C4×C53C2×#4112yes—
132Dic53—21no—
133Dic53—42no—
134Dic53—41no—
135Dic53#12312yes—
136#3091—531nomaximal, Sylow
137C2×C53—22no—
138C2×C53C2×#12311yescharacteristic
139C2×C53C2×#13511yescharacteristic
140C2×C53#57511yescharacteristic
141D53—22no—
142D53—41no—
143D53—81no—
144#525—531no—
145C2×#117—531no—
146#485—532no—
147#484D5311yescharacteristic, complemented
148#483—531no—
149#482—532no—
150#481—531no—
151#480—531no—
152#427—532no—
153#416—532no—
154C2×C22⋊C8—531no—
155C53#309111yescharacteristic, complemented, Sylow
156C2×#41—531no—
157C23×C4—531no—
158C22×C8—532no—
159#123—1061no—
160C22⋊Q8—531no—
161C22⋊Q8C2×D5311yescharacteristic
162#121—531no—
163#121—531no—
164#120—1061no—
165C4×D4—1061no—
166C4×D4—1061no—
167#117—532no—
168#117—532no—
169#117—531no—
170#117—1061no—
171#117—1061no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4C2×D5311yescharacteristic
174C2×C22⋊C4—1061no—
175C2×C42—1061no—
176#107—532no—
177#107C2×D5312yescomplemented
178#106—532no—
179#106—532no—
180#103—532no—
181#103C2×D5312yescomplemented
182#102—532no—
183#102—532no—
184C22⋊C8—531no—
185C22⋊C8—532no—
186C22⋊C8C2×D5311yescharacteristic
187C24—531no—
188#41—1064no—
189C2×Q8C22×D5311yescharacteristic
190C2×D4—531no—
191C2×D4—1061no—
192C2×D4—1061no—
193C2×D4—1061no—
194C22×C4—21no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—1064no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4C22×D5311yescharacteristic
204C2×C8—532no—
205C2×C8—1064no—
206C2×C8C22×D5312yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—532no—
210C4⋊C4—531no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C4⋊C4C22×D5312yes—
214C4⋊C4C22×D5311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1061no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C22⋊C4—1061no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C42—1062no—
223C42—1062no—
224C42—1061no—
225C23—531no—
226C23—531no—
227C23—1062no—
228C23—1062no—
229C23—1061no—
230C23—2121no—
231C23D4×D5311yescharacteristic
232Q8—22no—
233D4—1062no—
234D4—2122no—
235D4—2122no—
236D4—2122no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—531no—
243C2×C4—534no—
244C2×C4—531no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4D4×D5311yescharacteristic
254C2×C4C23×D5311yescharacteristic
255C8—22no—
256C8—1062no—
257C22—21no—
258C22—21no—
259C22—532no—
260C22—1061no—
261C22—1061no—
262C22—1061no—
263C22—1061no—
264C22—1061no—
265C22—2124no—
266C22—2121no—
267C22—2121no—
268C22—2121no—
269C22#1313324012yes—
270C22C2×D4×D5311yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—532no—
276C4—1061no—
277C4—2122no—
278C4—2121no—
279C4#41×D5312yes—
280C2—22no—
281C2—1062no—
282C2—4241no—
283C2—2121no—
284C2#123×D5311yescharacteristic
285C2C2×#1313324011yescharacteristic
286C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3091
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#13133175C2semidirect
3. #16810963C2semidirect
4. #16810951C2semidirect
5. #16810950C2semidirect
6. #16810941C2semidirect
7. #16810937C2semidirect
8. #16810660C2semidirect
9. #16810618C2semidirect
10. C22⋊C8×D53C2semidirect
11. C53×#484C2semidirect
30. #13133151C22semidirect
31. #13133148C22semidirect
32. #13133148C22semidirect
33. #13133141C22semidirect
34. #13133139C22semidirect
35. #13133137C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
147. #484D53semidirect
155. C53#3091semidirect
177. #107C2×D53semidirect
181. #103C2×D53semidirect