Finite Group Databasebeta

← #20285450

Subgroups — #20285450

Order 6784.

446 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 119 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#16811170C212yesmaximal, complemented
2#16811020C211yesmaximal, characteristic, complemented
3#16810998C211yesmaximal, characteristic, complemented
4#16810968C212yesmaximal, complemented
5#16810877C211yesmaximal, characteristic, complemented
6#16810875C212yesmaximal, complemented
7#16810873C211yesmaximal, characteristic, complemented
8#16810736C211yesmaximal, characteristic, complemented
9#16810733C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic, complemented
11C53×#442C211yesmaximal, characteristic, complemented
12C2×#11719945—22no—
13C2×D424C2211yescharacteristic
14C2×#11719922C2212yes—
15C2×#11719912C2211yescharacteristic
16C2×#11719912C2211yescharacteristic
17#13133216C2212yes—
18#13133213C2212yes—
19#13133203—22no—
20#13133192C2212yes—
21#13133190—22no—
22#13133188C2212yes—
23C4×D212C2212yes—
24C4×C22⋊D53—22no—
25C42×D53C2211yescharacteristic
26#13133149C2211yescharacteristic
27#13133148C2212yes—
28#13133139C2212yes—
29#13133138C2212yes—
30#13133137C2212yes—
31#13133128C2211yescharacteristic
32C8×Dic53C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34C2×D8×C53C2211yescharacteristic
35C4×D4×C53C2212yes—
36#106×C53C2211yescharacteristic
37#102×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×D4×C53C2312yes—
40C22×C4×C53—22no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53—22no—
43C2×C22⋊D53—22no—
44#11719945—48no—
45C2×D212—22no—
46C2×D212C2312yes—
47C2×C4×D53—22no—
48C2×C4×D53D412yes—
49C2×C4×D53C2311yescharacteristic
50C2×Dic106—22no—
51D8×C53—22no—
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—22no—
56C42×C53C2311yescharacteristic
57C22⋊Dic53—22no—
58#11719922—24no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63C4×Dic53D412yes—
64C4×Dic53C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66C2×#88322C2311yescharacteristic
67D424—22no—
68#11719912—22no—
69#11719912—22no—
70C23×C53—22no—
71C22×D53—22no—
72C22×D53C2×D411yescharacteristic
73D4×C53—22no—
74D4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—44no—
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2411yescharacteristic
79C22⋊D53—44no—
80C22⋊D53—44no—
81C22⋊D53—44no—
82C2×Dic53—22no—
83C2×Dic53C2×D411yescharacteristic
84C2×Dic53C2×D412yes—
85D212—22no—
86D212—22no—
87D212—44no—
88C4×D53—21no—
89C4×D53—21no—
90C4×D53—24no—
91C4×D53—44no—
92Dic106—44no—
93C8×C53—21no—
94C8×C53#4112yescomplemented
95#88322—21no—
96#88322#4112yescomplemented
97C22×C53—42no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53C22×D411yescharacteristic
101C2×D53—21no—
102C2×D53—22no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C4×C53—22no—
107C4×C53—42no—
108C4×C53C2×#4111yescharacteristic
109C4×C53C2×#4111yescharacteristic
110Dic53—21no—
111Dic53—42no—
112Dic53AGL(1,ℤ/8)14yes—
113#3184—531nomaximal, Sylow
114C2×C53—82no—
115C2×C53#53211yescharacteristic
116C2×C53C2×AGL(1,ℤ/8)12yes—
117D53—41no—
118D53—82no—
119#532—532no—
120#498—531no—
121#493—531no—
122#486—532no—
123#469—531no—
124#469—531no—
125#469—532no—
126#442—532no—
127#442—531no—
128#442D5311yescharacteristic, complemented
129C4×M16—531no—
130C53#318411yescharacteristic, complemented, Sylow
131C2×#41—1062no—
132C2×D8—532no—
133C2×D8—531no—
134C2×D8C2×D5311yescharacteristic
135C2×M16—531no—
136C2×M16—531no—
137C4⋊Q8—532no—
138#127—532no—
139#124—1062no—
140#121—532no—
141#121—532no—
142#121—1062no—
143C4×D4—532no—
144C4×D4—1062no—
145C4×D4C2×D5312yes—
146C2×C42—531no—
147#106—531no—
148#106—532no—
149#106C2×D5311yescharacteristic
150#102—532no—
151#102—532no—
152#102—532no—
153#102C2×D5312yes—
154#97—531no—
155#97C2×D5311yescharacteristic
156C4×C8—531no—
157C4×C8—531no—
158#41—2128no—
159C2×Q8—1062no—
160C2×D4—532no—
161C2×D4—1062no—
162C2×D4—1062no—
163C2×D4—1062no—
164C2×D4—1062no—
165C2×D4C22×D5312yes—
166C22×C4—22no—
167C22×C4—532no—
168C22×C4—531no—
169C22×C4—1062no—
170D8—22no—
171D8—1064no—
172D8—1062no—
173M16—1062no—
174M16—1062no—
175C2×C8—531no—
176C2×C8—531no—
177C2×C8C22×D5311yescharacteristic
178C2×C8C22×D5311yescharacteristic
179C4⋊C4—532no—
180C4⋊C4—1062no—
181C4⋊C4C22×D5312yes—
182C22⋊C4—22no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—532no—
187C42—531no—
188C42C22×D5311yescharacteristic
189C23—22no—
190C23—531no—
191C23—1062no—
192Q8—2124no—
193D4—22no—
194D4—22no—
195D4—1062no—
196D4—1062no—
197D4—2124no—
198D4—2124no—
199D4—2124no—
200D4—2124no—
201C2×C4—22no—
202C2×C4—44no—
203C2×C4—532no—
204C2×C4—531no—
205C2×C4—1064no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1062no—
209C2×C4—2124no—
210C2×C4D4×D5312yes—
211C2×C4C23×D5311yescharacteristic
212C8—21no—
213C8—532no—
214C8—1061no—
215C8#1171994512yescomplemented
216C22—42no—
217C22—42no—
218C22—42no—
219C22—1062no—
220C22—1061no—
221C22—2122no—
222C22—2122no—
223C22—2122no—
224C22C2×D4×D5311yescharacteristic
225C4—22no—
226C4—42no—
227C4—534no—
228C4—1061no—
229C4—2122no—
230C4#41×D5311yescharacteristic
231C4C2×#1171994511yescharacteristic
232C2—82no—
233C2—4242no—
234C2—2121no—
235C2#1681117011yescharacteristic
236C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3184
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. #16811020C2semidirect
3. #16810998C2semidirect
4. #16810968C2semidirect
5. #16810877C2semidirect
6. #16810875C2semidirect
7. #16810873C2semidirect
8. #16810736C2semidirect
9. #16810733C2semidirect
10. #16810575C2semidirect
11. C53×#442C2semidirect
94. C8×C53#41semidirect
96. #88322#41semidirect
128. #442D53semidirect
130. C53#3184semidirect
215. C8#11719945semidirect