Finite Group Databasebeta

← #20283942

Subgroups — #20283942

Order 6784.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#13133243C212yesmaximal, complemented
2#16811134C211yesmaximal, characteristic, complemented
3#16810774C214yesmaximal, complemented
4#16810757C214yesmaximal, complemented
5#16810628C212yesmaximal, complemented
6#16810587C211yesmaximal, characteristic, complemented
7C23×D4×C53C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic, complemented
9#13133243—28no—
10C2×#11719923C2214yescomplemented
11C2×#11719922C2214yescomplemented
12C2×#11719917C2212yescomplemented
13#13133200—21no—
14#13133196C2212yescomplemented
15#13133191C2212yescomplemented
16#13133186—21no—
17C4×C22⋊D53—22no—
18#13133176C2211yescharacteristic, complemented
19#13133138C2218yescomplemented
20C22⋊#88322C2214yescomplemented
21C25×C53—21no—
22C22×D4×C53—28no—
23C22×D4×C53—22no—
24C22×D4×C53—21no—
25C22×D4×C53C2214yes—
26C22×D4×C53C2212yes—
27C23×C4×C53C2211yescharacteristic
28C24×C53—28no—
29C24×C53—24no—
30C24×C53—22no—
31C24×C53—48no—
32C24×C53D411yescharacteristic
33C2×D4×C53—28no—
34C2×D4×C53—216no—
35C2×D4×C53—216no—
36C2×D4×C53—24no—
37C2×D4×C53—22no—
38C2×D4×C53—24no—
39C2×D4×C53D418yescomplemented
40C2×D4×C53C2314yes—
41C22×C4×C53—24no—
42C22×C4×C53D412yes—
43C22×C4×C53D411yescharacteristic
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—22no—
47C2×C22⋊D53—21no—
48#11719945—24no—
49C2×D212C2311yescharacteristic, complemented
50C2×C4×D53—21no—
51C2×Dic106C2311yescharacteristic, complemented
52C22⋊Dic53—21no—
53C22⋊Dic53—22no—
54#11719923—28no—
55#11719922—28no—
56#11719921—22no—
57C4⋊Dic53C2312yescomplemented
58#11719919—22no—
59C4×Dic53—21no—
60#11719917—24no—
61C2×#88322C2314yescomplemented
62C23×C53—22no—
63C23×C53—24no—
64C23×C53—216no—
65C23×C53—28no—
66C23×C53—24no—
67C23×C53—432no—
68C23×C53—48no—
69C23×C53—48no—
70C23×C53—48no—
71C23×C53C2×D411yescharacteristic
72C23×C53C2×D412yes—
73C22×D53—21no—
74D4×C53—216no—
75D4×C53—28no—
76D4×C53—24no—
77D4×C53C2×D418yes—
78C2×C4×C53—22no—
79C2×C4×C53—24no—
80C2×C4×C53—24no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D414yes—
84C2×C4×C53C2411yescharacteristic
85C22⋊D53—42no—
86C22⋊D53—44no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89D212—22no—
90C4×D53—42no—
91Dic106—22no—
92#88322—24no—
93C22×C53—28no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—22no—
97C22×C53—28no—
98C22×C53—416no—
99C22×C53—432no—
100C22×C53—48no—
101C22×C53#12012yes—
102C22×C53AGL(1,ℤ/8)14yescomplemented
103C22×C53C22×D411yescharacteristic
104C2×D53—41no—
105C2×D53—42no—
106C4×C53—22no—
107C4×C53#12012yes—
108C4×C53C22×D412yes—
109Dic53—41no—
110Dic53—42no—
111#3003—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—24no—
114C2×C53—48no—
115C2×C53C2×#12011yescharacteristic
116C2×C53C2×AGL(1,ℤ/8)12yes—
117D53—81no—
118C23×D4D5311yescharacteristic, complemented
119C2×AGL(1,ℤ/8)—532no—
120#525—531no—
121#450—534no—
122#447—534no—
123#418—532no—
124#407—531no—
125C53#300311yescharacteristic, complemented, Sylow
126C25—21no—
127C2×#41—531no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D5314yes—
132C22×D4C2×D5312yes—
133C23×C4C2×D5311yescharacteristic
134AGL(1,ℤ/8)—1068no—
135C2×SD16—534no—
136C2×D8—534no—
137C2×M16—532no—
138#123—1061no—
139C22⋊Q8—532no—
140#121—532no—
141#120—1061no—
142C4×D4—1062no—
143#117—531no—
144#102—538no—
145C22⋊C8—534no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D5311yescharacteristic
151#41—1064no—
152C2×Q8—531no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—531no—
160C2×D4—1062no—
161C2×D4—1061no—
162C2×D4C22⋊D5318yescomplemented
163C2×D4C22×D5314yes—
164C22×C4—24no—
165C22×C4—1061no—
166C22×C4C22⋊D5312yes—
167C22×C4C22⋊D5311yescharacteristic
168C22×C4C22×D5312yes—
169C22×C4C22×D5311yescharacteristic
170SD16—1068no—
171D8—1068no—
172M16—1064no—
173C2×C8—534no—
174C4⋊C4—532no—
175C4⋊C4—1062no—
176C22⋊C4—1061no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C42—1061no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—1061no—
190C23C2×C22⋊D5311yescharacteristic
191C23C2×C22⋊D5312yes—
192Q8—1062no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—1062no—
197D4—2122no—
198D4—2124no—
199D4C2×C22⋊D5318yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4C2×C22⋊D5312yes—
207C2×C4C2×C22⋊D5311yescharacteristic
208C2×C4C2×C22⋊D5314yes—
209C2×C4C23×D5311yescharacteristic
210C8—1064no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—2121no—
220C22—2122no—
221C22#1313318612yes—
222C22#1313324314yescomplemented
223C22C22×C22⋊D5311yescharacteristic
224C4—22no—
225C4—2121no—
226C4—2122no—
227C4#1313318612yes—
228C4C22×C22⋊D5312yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—4241no—
233C2C2×#1313318611yescharacteristic
234C2C2×#1313324312yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC23×D4×C53C2
socleC22×C53C22×D4
hypercenterC23×D4D53
hypocenterC53#3003
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133243C2semidirect
2. #16811134C2semidirect
3. #16810774C2semidirect
4. #16810757C2semidirect
5. #16810628C2semidirect
6. #16810587C2semidirect
7. C23×D4×C53C2semidirect
8. C2×#11719945C22semidirect
10. C2×#11719923C22semidirect
11. C2×#11719922C22semidirect
12. C2×#11719917C22semidirect
14. #13133196C22semidirect
15. #13133191C22semidirect
18. #13133176C22semidirect
19. #13133138C22semidirect
20. C22⋊#88322C22semidirect
39. C2×D4×C53D4semidirect
49. C2×D212C23semidirect
51. C2×Dic106C23semidirect
57. C4⋊Dic53C23semidirect
61. C2×#88322C23semidirect
102. C22×C53AGL(1,ℤ/8)semidirect
118. C23×D4D53semidirect
125. C53#3003semidirect
162. C2×D4C22⋊D53semidirect
222. C22#13133243semidirect