Finite Group Databasebeta

← #20286536

Subgroups — #20286536

Order 6784.

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

normal subgroups only

This group has 462 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#16811334C211yesmaximal, characteristic, complemented
2D4×D212C211yesmaximal, characteristic
3#16810971C211yesmaximal, characteristic, complemented
4#16810943C212yesmaximal, complemented
5#16810853C212yesmaximal, complemented
6#16810824C211yesmaximal, characteristic, complemented
7#16810775C212yesmaximal, complemented
8C4×#11719923C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic, complemented
10#102×D53C212yesmaximal, complemented
11C53×#476C211yesmaximal, characteristic, complemented
12C22×D212—21no—
13C2×D4×D53C2212yes—
14C2×#11719913C2212yescomplemented
15C2×#11719923C2211yescharacteristic
16C2×C8×D53C2212yescomplemented
17#13133214C2211yescharacteristic
18#13133208—21no—
19#13133197C2212yescomplemented
20#13133193—22no—
21#13133192C2212yes—
22#13133191—21no—
23#13133187—22no—
24C4×Dic106C2211yescharacteristic, complemented
25C4×D212C2211yescharacteristic
26#13133179—21no—
27C4⋊C4×D53C2212yescomplemented
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#106×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×D53—22no—
55C2×C4×D53C2312yes—
56C2×Dic106C2311yescharacteristic
57C2×C8×C53C2312yes—
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2312yes—
60C4⋊C4×C53C2311yescharacteristic
61C22⋊C4×C53—21no—
62C42×C53C2311yescharacteristic
63#11719923—22no—
64#11719921—22no—
65#11719921—22no—
66#11719921D412yes—
67C4⋊Dic53—22no—
68C4⋊Dic53D411yescharacteristic
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—22no—
72C4×Dic53—21no—
73C2×#88322C2311yescharacteristic
74C2×#88322C2311yescharacteristic
75#11719913—42no—
76C8×D53—42no—
77C23×C53—21no—
78C22×D53—22no—
79C22×D53—44no—
80C22×D53—44no—
81C22×D53—82no—
82C22×D53C2×D412yes—
83Q8×C53—81no—
84D4×C53—21no—
85D4×C53#4112yes—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
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—44no—
99D212—42no—
100D212—81no—
101D212SD1614yescomplemented
102C4×D53—22no—
103C4×D53—22no—
104C4×D53—82no—
105Dic106—21no—
106Dic106—21no—
107C8×C53—22no—
108#88322—21no—
109#88322—21no—
110C22×C53—22no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53C22×D411yescharacteristic
114C2×D53—22no—
115C2×D53—22no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—88no—
120C2×D53C2×SD1614yes—
121C4×C53—21no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53C2×SD1612yes—
125C4×C53C2×#4111yescharacteristic
126C4×C53H(5,𝔽2)11yescharacteristic
127Dic53—22no—
128Dic53—41no—
129Dic53—41no—
130#3295—531nomaximal, Sylow
131C2×C53—42no—
132C2×C53#54611yescharacteristic
133C2×C53C22×SD1611yescharacteristic
134C2×C53#57611yescharacteristic
135D53—24no—
136D53—82no—
137#554—531no—
138D42—531no—
139#486—531no—
140#481—532no—
141#476D5311yescharacteristic, complemented
142#465—532no—
143#460—531no—
144#450—532no—
145C4×SD16—531no—
146C8×D4—531no—
147C2×#102—532no—
148C53#329511yescharacteristic, complemented, Sylow
149C22×D4—532no—
150C22×D4—1061no—
151C2×SD16—531no—
152C2×SD16—532no—
153C22×C8—532no—
154C4⋊Q8C2×D5311yescharacteristic, complemented
155#127—531no—
156#125—1061no—
157C22⋊Q8—532no—
158C22⋊Q8—1062no—
159#121—532no—
160#121—1061no—
161#120—1062no—
162C4×Q8—531no—
163C4×D4—531no—
164C4×D4—1061no—
165C4×D4C2×D5311yescharacteristic
166C2×C4⋊C4—532no—
167#106—531no—
168#106C2×D5312yescomplemented
169C4⋊C8C2×D5311yescharacteristic, complemented
170#103—531no—
171#102—531no—
172#102—532no—
173#102—532no—
174#102—532no—
175#102C2×D5312yes—
176C22⋊C8—532no—
177C4×C8—531no—
178C24—1062no—
179C2×Q8—21no—
180C2×Q8—531no—
181C2×D4—531no—
182C2×D4—532no—
183C2×D4—1062no—
184C2×D4—1062no—
185C2×D4—1064no—
186C2×D4—1061no—
187C2×D4—1061no—
188C2×D4—2124no—
189C2×D4—2122no—
190C2×D4C22×D5311yescharacteristic
191C22×C4—21no—
192C22×C4—532no—
193C22×C4—1062no—
194SD16—1062no—
195SD16—2122no—
196C2×C8—531no—
197C2×C8—531no—
198C2×C8—2122no—
199C2×C8C22×D5312yes—
200C4⋊C4—21no—
201C4⋊C4—531no—
202C4⋊C4—531no—
203C4⋊C4—1062no—
204C4⋊C4—1062no—
205C4⋊C4—1061no—
206C4⋊C4C22×D5312yes—
207C4⋊C4C22×D5311yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—532no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C42—1061no—
213C42C22×D5311yescharacteristic
214C23—21no—
215C23—532no—
216C23—1062no—
217C23—4242no—
218C23—2124no—
219C23—2124no—
220Q8—81no—
221Q8—1061no—
222Q8—1061no—
223D4—21no—
224D4—534no—
225D4—1062no—
226D4—1062no—
227D4—4241no—
228D4—2124no—
229D4—2124no—
230D4—2122no—
231D4#1171994912yes—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—532no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—4242no—
241C2×C4D4×D5311yescharacteristic
242C2×C4D4×D5311yescharacteristic
243C2×C4C23×D5311yescharacteristic
244C8—22no—
245C8—1061no—
246C8—1061no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—534no—
251C22—1062no—
252C22—1062no—
253C22—4248no—
254C22—2122no—
255C22—2122no—
256C22—2122no—
257C22C2×D4×D5311yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—1062no—
262C4—2121no—
263C4—2121no—
264C4SD16×D5312yes—
265C4C2×#1171994911yescharacteristic
266C4#1313326311yescharacteristic
267C2—42no—
268C2—1064no—
269C2—4242no—
270C2#1681126411yescharacteristic
271C2C2×SD16×D5311yescharacteristic
272C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#476C2
socleC22×C53C22×D4
hypercenter#476D53
hypocenterC53#3295
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
3. #16810971C2semidirect
4. #16810943C2semidirect
5. #16810853C2semidirect
6. #16810824C2semidirect
7. #16810775C2semidirect
8. C4×#11719923C2semidirect
9. #16810688C2semidirect
10. #102×D53C2semidirect
11. C53×#476C2semidirect
14. C2×#11719913C22semidirect
16. C2×C8×D53C22semidirect
19. #13133197C22semidirect
24. C4×Dic106C22semidirect
27. C4⋊C4×D53C22semidirect
28. #13133148C22semidirect
29. #13133140C22semidirect
30. #13133139C22semidirect
32. #13133137C22semidirect
33. #13133137C22semidirect
34. #13133129C22semidirect
35. C4×#88322C22semidirect
36. C4⋊Q8×C53C22semidirect
38. #106×C53C22semidirect
39. C4⋊C8×C53C22semidirect
101. D212SD16semidirect
141. #476D53semidirect
148. C53#3295semidirect
154. C4⋊Q8C2×D53semidirect
168. #106C2×D53semidirect
169. C4⋊C8C2×D53semidirect