Finite Group Databasebeta

← #20285649

Subgroups — #20285649

Order 6784.

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

normal subgroups only

This group has 438 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#16811174C211yesmaximal, characteristic, complemented
2#16811171C211yesmaximal, characteristic, complemented
3#16810999C211yesmaximal, characteristic, complemented
4#16810996C211yesmaximal, characteristic, complemented
5#16810975C211yesmaximal, characteristic, complemented
6#16810837C214yesmaximal, complemented
7#16810769C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×#11719949—21no—
11C2×#11719947C2212yes—
12C2×#11719913C2214yes—
13C2×#11719922C2214yes—
14#13133217C2211yescharacteristic
15#13133214C2211yescharacteristic
16#13133206—21no—
17#13133204C2212yes—
18#13133192—21no—
19#13133192C2212yes—
20#13133188—21no—
21#13133188—21no—
22D4×Dic53—21no—
23D4×Dic53—21no—
24#13133179—21no—
25#13133179C2212yes—
26C42×D53C2211yescharacteristic
27#13133146C2212yes—
28#13133140C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32#127×C53C2211yescharacteristic
33#102×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C22×Dic53—22no—
41#11719949—44no—
42#11719947—44no—
43C2×D212—22no—
44C2×D212—21no—
45C2×D212—21no—
46C2×D212C2312yes—
47C2×C4×D53—22no—
48C2×C4×D53D411yescharacteristic
49C2×C4×D53D411yescharacteristic
50C2×C4×D53C2311yescharacteristic
51C2×Dic106C2312yes—
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C42×C53C2311yescharacteristic
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58#11719922—44no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—21no—
62C4⋊Dic53—21no—
63C4×Dic53D411yescharacteristic
64C4×Dic53D411yescharacteristic
65C4×Dic53C2311yescharacteristic
66C2×#88322C2312yes—
67#11719913—44no—
68C23×C53—22no—
69C22×D53—22no—
70C22×D53C2×D411yescharacteristic
71Q8×C53—42no—
72D4×C53—24no—
73D4×C53—42no—
74D4×C53—42no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2411yescharacteristic
79C22⋊D53—44no—
80C2×Dic53—22no—
81C2×Dic53—44no—
82C2×Dic53C2×D411yescharacteristic
83C2×Dic53C2×D411yescharacteristic
84C2×Dic53C2×D411yescharacteristic
85D212—24no—
86D212—44no—
87D212—42no—
88D212—42no—
89C4×D53—22no—
90C4×D53—22no—
91C4×D53—22no—
92C4×D53—44no—
93Dic106—24no—
94C8×C53—22no—
95#88322—22no—
96C22×C53—42no—
97C22×C53—42no—
98C22×C53—42no—
99C22×C53C22×D411yescharacteristic
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C4×C53—42no—
107C4×C53#13512yes—
108C4×C53AGL(1,ℤ/8)12yes—
109C4×C53H(5,𝔽2)12yes—
110Dic53—21no—
111Dic53—21no—
112Dic53—21no—
113Dic53—42no—
114#3207—531nomaximal, Sylow
115C2×C53—82no—
116C2×C53#53511yescharacteristic
117C2×C53C2×#13511yescharacteristic
118C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
119D53—41no—
120D53—82no—
121#532—531no—
122#532—531no—
123#493—531no—
124#492—531no—
125#487—531no—
126#486D5311yescharacteristic, complemented
127#463—534no—
128#449—534no—
129#431—531no—
130C53#320711yescharacteristic, complemented, Sylow
131C2×#41—532no—
132C2×#41—1061no—
133C2×SD16—534no—
134C2×D8—534no—
135C4⋊Q8—531no—
136C4⋊Q8C2×D5311yescharacteristic
137#127—531no—
138#127C2×D5311yescharacteristic
139#124—532no—
140#124—1061no—
141#121—532no—
142#121—1061no—
143#121—1061no—
144#121—1061no—
145C4×D4—532no—
146C4×D4—1061no—
147C4×D4—1061no—
148C4×D4—1061no—
149C2×C42—531no—
150C4⋊C8—532no—
151#103—534no—
152#102C2×D5314yes—
153C22⋊C8—532no—
154C4×C8—531no—
155C4×C8C2×D5311yescharacteristic
156#41—2124no—
157#41—2124no—
158C2×Q8—21no—
159C2×Q8—532no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—532no—
163C2×D4—1062no—
164C2×D4—1062no—
165C2×D4—1061no—
166C2×D4—1061no—
167C2×D4C22×D5312yes—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—1062no—
172C22×C4—1062no—
173SD16—2124no—
174D8—2124no—
175C2×C8—532no—
176C2×C8C22×D5312yes—
177C4⋊C4—21no—
178C4⋊C4—1061no—
179C4⋊C4—1061no—
180C4⋊C4C22×D5312yes—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C42—531no—
186C42—531no—
187C42—531no—
188C42C22×D5311yescharacteristic
189C23—22no—
190C23—531no—
191C23—1062no—
192Q8—42no—
193Q8—1064no—
194D4—24no—
195D4—42no—
196D4—42no—
197D4—1064no—
198D4—2124no—
199D4—2124no—
200D4—2122no—
201D4—2122no—
202C2×C4—22no—
203C2×C4—531no—
204C2×C4—531no—
205C2×C4—531no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—2124no—
211C2×C4—2124no—
212C2×C4D4×D5311yescharacteristic
213C2×C4D4×D5311yescharacteristic
214C2×C4C23×D5311yescharacteristic
215C8—22no—
216C8—1062no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—1061no—
221C22—1061no—
222C22—1061no—
223C22—2122no—
224C22—2122no—
225C22—2122no—
226C22C2×D4×D5311yescharacteristic
227C4—42no—
228C4—1061no—
229C4—1061no—
230C4—1061no—
231C4—2122no—
232C4#1313323812yes—
233C4#1313324112yes—
234C4#1313326312yes—
235C2—82no—
236C2—4242no—
237C2—2121no—
238C2#1681119111yescharacteristic
239C2C2×#1313323811yescharacteristic
240C2C2×#1313324111yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3207
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811174C2semidirect
2. #16811171C2semidirect
3. #16810999C2semidirect
4. #16810996C2semidirect
5. #16810975C2semidirect
6. #16810837C2semidirect
7. #16810769C2semidirect
8. #16810674C2semidirect
9. C53×#486C2semidirect
126. #486D53semidirect
130. C53#3207semidirect