Finite Group Databasebeta

← #20286943

Subgroups — #20286943

Order 6784.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811347C211yesmaximal, characteristic, complemented
2D4×D212C211yesmaximal, characteristic
3#16810971C211yesmaximal, characteristic, complemented
4#16810950C212yesmaximal, complemented
5#16810870C212yesmaximal, complemented
6#16810824C211yesmaximal, characteristic, complemented
7#16810775C212yesmaximal, complemented
8C4×#11719923C211yesmaximal, characteristic, complemented
9#16810700C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal, complemented
11C53×#474C211yesmaximal, characteristic, complemented
12C22×D212—21no—
13C2×D4×D53C2212yes—
14C2×#11719913C2212yescomplemented
15C2×#11719923C2211yescharacteristic
16C2×#11719912C2212yescomplemented
17#13133214C2211yescharacteristic
18#13133212—22no—
19#13133206—21no—
20#13133197C2212yescomplemented
21#13133192C2212yes—
22#13133191—21no—
23#13133187—22no—
24Q8×Dic53—21no—
25C4×Dic106C2211yescharacteristic, complemented
26C4×D212C2211yescharacteristic
27#13133175C2212yescomplemented
28#13133148C2211yescharacteristic, complemented
29#13133140C2211yescharacteristic, complemented
30#13133139C2212yescomplemented
31#13133138C2211yescharacteristic
32#13133137C2212yescomplemented
33#13133137C2212yescomplemented
34#13133129C2212yescomplemented
35C4×#88322C2211yescharacteristic, complemented
36C4⋊Q8×C53C2211yescharacteristic, complemented
37C4×D4×C53C2211yescharacteristic
38#107×C53C2212yescomplemented
39C4⋊C8×C53C2211yescharacteristic, complemented
40#102×C53C2212yes—
41C23×D53—22no—
42C2×Q8×C53—21no—
43C2×D4×C53C2311yescharacteristic
44C22×C4×C53—21no—
45C2×C22⋊D53—22no—
46D4×D53—24no—
47D4×D53—42no—
48C2×D212—22no—
49C2×D212—21no—
50C2×D212—21no—
51C2×D212—44no—
52C2×D212D411yescharacteristic
53C2×D212C2312yes—
54C2×C4×D53C2312yes—
55C2×Dic106C2311yescharacteristic
56C2×C8×C53C2312yes—
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2312yes—
59C4⋊C4×C53C2311yescharacteristic
60C22⋊C4×C53—21no—
61C42×C53C2311yescharacteristic
62#11719923—22no—
63#11719921—22no—
64#11719921—22no—
65#11719921—22no—
66#11719921D412yes—
67C4⋊Dic53—22no—
68C4⋊Dic53D411yescharacteristic
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C2×#88322C2311yescharacteristic
74C2×#88322C2311yescharacteristic
75#11719913—42no—
76#11719912—42no—
77C23×C53—21no—
78C22×D53—22no—
79C22×D53—44no—
80C22×D53—44no—
81C22×D53—82no—
82C22×D53C2×D412yes—
83Q8×C53—42no—
84D4×C53—21no—
85D4×C53#4112yes—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2411yescharacteristic
92C22⋊D53—44no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53C2×D412yes—
96D212—22no—
97D212—22no—
98D212—22no—
99D212—44no—
100D212—42no—
101D212—81no—
102C4×D53—22no—
103C4×D53—22no—
104Dic106—21no—
105Dic106—21no—
106C8×C53—22no—
107#88322—21no—
108#88322—21no—
109C22×C53—22no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53C22×D411yescharacteristic
113C2×D53—22no—
114C2×D53—22no—
115C2×D53—22no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—88no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53—42no—
123C4×C53AGL(1,ℤ/8)12yes—
124C4×C53C2×#4111yescharacteristic
125C4×C53H(5,𝔽2)11yescharacteristic
126Dic53—22no—
127Dic53—41no—
128Dic53—41no—
129#3324—531nomaximal, Sylow
130C2×C53—42no—
131C2×C53#54611yescharacteristic
132C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
133C2×C53#57711yescharacteristic
134D53—42no—
135D53—82no—
136#556—531no—
137D42—531no—
138#486—531no—
139#482—532no—
140#474D5311yescharacteristic, complemented
141#468—532no—
142#460—531no—
143#450—532no—
144C4×SD16—531no—
145#436—531no—
146#418—532no—
147C53#332411yescharacteristic, complemented, Sylow
148C22×D4—532no—
149C22×D4—1061no—
150C2×SD16—531no—
151C2×SD16—532no—
152C2×M16—532no—
153C4⋊Q8C2×D5311yescharacteristic, complemented
154#127—531no—
155#126—1062no—
156#124—1061no—
157C22⋊Q8—532no—
158#121—532no—
159#121—1061no—
160#120—1062no—
161C4×Q8—531no—
162C4×Q8—1061no—
163C4×D4—531no—
164C4×D4C2×D5311yescharacteristic
165#117—532no—
166#107C2×D5312yescomplemented
167#106—531no—
168C4⋊C8C2×D5311yescharacteristic, complemented
169#103—531no—
170#102—531no—
171#102—532no—
172#102—532no—
173#102—532no—
174#102C2×D5312yes—
175C22⋊C8—532no—
176C4×C8—531no—
177C24—1062no—
178C2×Q8—21no—
179C2×Q8—531no—
180C2×D4—531no—
181C2×D4—532no—
182C2×D4—1062no—
183C2×D4—1062no—
184C2×D4—1064no—
185C2×D4—1061no—
186C2×D4—1061no—
187C2×D4—2124no—
188C2×D4—2122no—
189C2×D4C22×D5311yescharacteristic
190C22×C4—21no—
191C22×C4—532no—
192SD16—1062no—
193SD16—2122no—
194M16—2122no—
195C2×C8—531no—
196C2×C8—531no—
197C2×C8C22×D5312yes—
198C4⋊C4—21no—
199C4⋊C4—531no—
200C4⋊C4—531no—
201C4⋊C4—1062no—
202C4⋊C4—1061no—
203C4⋊C4C22×D5312yes—
204C4⋊C4C22×D5311yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—532no—
207C22⋊C4—1062no—
208C22⋊C4—1062no—
209C22⋊C4—1062no—
210C42—1062no—
211C42—1061no—
212C42C22×D5311yescharacteristic
213C23—21no—
214C23—532no—
215C23—1062no—
216C23—4242no—
217C23—2124no—
218C23—2124no—
219Q8—42no—
220Q8—1061no—
221Q8—1061no—
222D4—21no—
223D4—1062no—
224D4—1062no—
225D4—1062no—
226D4—4241no—
227D4—2124no—
228D4—2124no—
229D4—2122no—
230D4#1171994912yes—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—532no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4D4×D5311yescharacteristic
240C2×C4D4×D5311yescharacteristic
241C2×C4C23×D5311yescharacteristic
242C8—22no—
243C8—1061no—
244C8—1061no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—1062no—
249C22—1062no—
250C22—1062no—
251C22—4248no—
252C22—2122no—
253C22—2122no—
254C22—2122no—
255C22C2×D4×D5311yescharacteristic
256C4—21no—
257C4—41no—
258C4—42no—
259C4—1062no—
260C4—2121no—
261C4—2121no—
262C4#1313324112yes—
263C4C2×#1171994911yescharacteristic
264C4#1313326311yescharacteristic
265C2—42no—
266C2—4242no—
267C2—2122no—
268C2#1681126411yescharacteristic
269C2C2×#1313324111yescharacteristic
270C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3324
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16811347C2semidirect
3. #16810971C2semidirect
4. #16810950C2semidirect
5. #16810870C2semidirect
6. #16810824C2semidirect
7. #16810775C2semidirect
8. C4×#11719923C2semidirect
9. #16810700C2semidirect
10. #16810626C2semidirect
11. C53×#474C2semidirect
14. C2×#11719913C22semidirect
16. C2×#11719912C22semidirect
20. #13133197C22semidirect
25. C4×Dic106C22semidirect
27. #13133175C22semidirect
28. #13133148C22semidirect
29. #13133140C22semidirect
30. #13133139C22semidirect
32. #13133137C22semidirect
33. #13133137C22semidirect
34. #13133129C22semidirect
35. C4×#88322C22semidirect
36. C4⋊Q8×C53C22semidirect
38. #107×C53C22semidirect
39. C4⋊C8×C53C22semidirect
140. #474D53semidirect
147. C53#3324semidirect
153. C4⋊Q8C2×D53semidirect
166. #107C2×D53semidirect
168. C4⋊C8C2×D53semidirect