Finite Group Databasebeta

← #20285087

Subgroups — #20285087

Order 6784.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#16811176C211yesmaximal, characteristic, complemented
2C2×#13133213C211yesmaximal, characteristic
3#16810986C214yesmaximal, complemented
4#16810968C214yesmaximal, complemented
5#16810629C211yesmaximal, characteristic, complemented
6#16810628C211yesmaximal, characteristic, complemented
7#16810627C211yesmaximal, characteristic, complemented
8M16×Dic53C211yesmaximal, characteristic, complemented
9C53×#418C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53—22no—
12C2×#11719917C2211yescharacteristic
13#13133216C2212yescomplemented
14#13133213—22no—
15#13133213C2212yes—
16#13133213C2212yes—
17#13133208C2212yescomplemented
18#13133194—22no—
19C4×Dic106—22no—
20#13133176C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22#13133139C2214yescomplemented
23#13133138C2214yescomplemented
24#13133137C2214yescomplemented
25#13133127C2212yescomplemented
26C8×Dic53C2212yescomplemented
27C22×D4×C53C2211yescharacteristic
28C2×M16×C53C2211yescharacteristic
29#117×C53C2211yescharacteristic
30#102×C53C2214yescomplemented
31C24×C53—21no—
32C23×D53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—22no—
35C2×D4×C53—22no—
36C2×D4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—24no—
39C2×C22⋊D53—24no—
40C2×C22⋊D53—48no—
41C22×Dic53D412yes—
42C2×D212—22no—
43C2×D212—22no—
44C2×D212—21no—
45C2×D212C2312yes—
46C2×Dic106—22no—
47M16×C53—22no—
48C2×C8×C53C2312yes—
49C4⋊C4×C53C2312yes—
50C22⋊C4×C53—21no—
51C42×C53—21no—
52C22⋊Dic53—21no—
53C4⋊Dic53—22no—
54C4⋊Dic53C2312yes—
55#11719919—22no—
56#11719919—22no—
57C4×Dic53—21no—
58C4×Dic53D412yes—
59C4×Dic53C2311yescharacteristic
60C4×Dic53C2311yescharacteristic
61#11719917—22no—
62C2×#88322C2312yes—
63C23×C53—22no—
64C23×C53—44no—
65C23×C53—42no—
66C23×C53C2×D411yescharacteristic
67C22×D53—22no—
68C22×D53—44no—
69C22×D53—42no—
70D4×C53—22no—
71D4×C53—22no—
72D4×C53—42no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53#4112yes—
76C2×C4×C53#4112yes—
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—48no—
79C22⋊D53—48no—
80C2×Dic53—24no—
81C2×Dic53—22no—
82C2×Dic53C2×D412yes—
83C2×Dic53C2×D412yes—
84D212—22no—
85D212—22no—
86D212—42no—
87Dic106—44no—
88C8×C53—22no—
89#88322—22no—
90C22×C53—22no—
91C22×C53—42no—
92C22×C53—44no—
93C22×C53—84no—
94C22×C53#12412yes—
95C22×C53C22×D411yescharacteristic
96C2×D53—42no—
97C2×D53—44no—
98C2×D53—84no—
99C4×C53—42no—
100C4×C53#12412yes—
101C4×C53C2×#4111yescharacteristic
102C4×C53C2×#4111yescharacteristic
103Dic53—22no—
104Dic53—42no—
105Dic53AGL(1,ℤ/8)14yes—
106#3138—531nomaximal, Sylow
107C2×C53—22no—
108C2×C53—82no—
109C2×C53C2×#12411yescharacteristic
110C2×C53C2×AGL(1,ℤ/8)12yes—
111D53—82no—
112#533—531no—
113C2×#127—531no—
114#490—534no—
115#486—534no—
116#418—531no—
117#418—531no—
118#418—531no—
119#418D5311yescharacteristic, complemented
120C4×M16—531no—
121C53#313811yescharacteristic, complemented, Sylow
122C22×D4—531no—
123C22×D4—1062no—
124C22×D4C2×D5311yescharacteristic
125C2×M16—531no—
126C2×M16C2×D5311yescharacteristic
127C4⋊Q8—532no—
128#127—532no—
129#127—532no—
130#127—1062no—
131#125—532no—
132C22⋊Q8—1062no—
133C4×Q8—1062no—
134#117—531no—
135#117C2×D5311yescharacteristic
136C2×C42—531no—
137#102—534no—
138#102—534no—
139#102—534no—
140#102C2×D5314yescomplemented
141#97—532no—
142C4×C8—532no—
143C24—21no—
144C24—1061no—
145C2×Q8—1062no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—532no—
150C2×D4—1064no—
151C2×D4—1064no—
152C2×D4—1062no—
153C2×D4—1061no—
154C2×D4—1062no—
155C2×D4—2128no—
156C2×D4C22×D5312yes—
157C22×C4—532no—
158C22×C4C22×D5311yescharacteristic
159M16—22no—
160M16—1062no—
161C2×C8—532no—
162C2×C8C22×D5312yes—
163C4⋊C4—532no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4C22×D5312yes—
168C22⋊C4—21no—
169C22⋊C4—1061no—
170C42—21no—
171C42—532no—
172C42—531no—
173C42—531no—
174C42—1061no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—1062no—
179C23—2124no—
180C23—2122no—
181C23D4×D5311yescharacteristic
182Q8—2124no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—1062no—
187D4—1062no—
188D4—2128no—
189D4—2128no—
190D4—2122no—
191C2×C4—22no—
192C2×C4—532no—
193C2×C4—532no—
194C2×C4—1064no—
195C2×C4—1062no—
196C2×C4#1171994512yes—
197C2×C4D4×D5311yescharacteristic
198C2×C4#1171994712yes—
199C2×C4C23×D5311yescharacteristic
200C8—22no—
201C8—1062no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—4244no—
207C22—2122no—
208C22—2124no—
209C22#1313320312yes—
210C22C2×D4×D5311yescharacteristic
211C4—42no—
212C4—534no—
213C4—1062no—
214C4—2122no—
215C4#1313320312yes—
216C4C2×#1171994711yescharacteristic
217C4C2×#1171994511yescharacteristic
218C2—22no—
219C2—82no—
220C2—4242no—
221C2C2×#1313320311yescharacteristic
222C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#418C2
socleC22×C53C22×D4
hypercenter#418D53
hypocenterC53#3138
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #16811176C2semidirect
3. #16810986C2semidirect
4. #16810968C2semidirect
5. #16810629C2semidirect
6. #16810628C2semidirect
7. #16810627C2semidirect
8. M16×Dic53C2semidirect
9. C53×#418C2semidirect
13. #13133216C22semidirect
17. #13133208C22semidirect
22. #13133139C22semidirect
23. #13133138C22semidirect
24. #13133137C22semidirect
25. #13133127C22semidirect
26. C8×Dic53C22semidirect
30. #102×C53C22semidirect
119. #418D53semidirect
121. C53#3138semidirect
140. #102C2×D53semidirect