Finite Group Databasebeta

← #20285363

Subgroups — #20285363

Order 6784.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811224C212yesmaximal, complemented
2#16811023C211yesmaximal, characteristic, complemented
3#16810998C211yesmaximal, characteristic, complemented
4#16810986C212yesmaximal, complemented
5#16810877C211yesmaximal, characteristic, complemented
6#16810873C211yesmaximal, characteristic, complemented
7#16810858C212yesmaximal, complemented
8#16810736C211yesmaximal, characteristic, complemented
9C4×#11719922C212yesmaximal, complemented
10#16810580C211yesmaximal, characteristic, complemented
11C53×#442C211yesmaximal, characteristic, complemented
12C2×D424C2211yescharacteristic
13C2×#11719922C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×C8×D53C2211yescharacteristic
16#13133213C2212yes—
17#13133208C2212yes—
18#13133200—22no—
19#13133199—22no—
20#13133192C2212yes—
21#13133191—22no—
22#13133190—22no—
23#13133188C2212yes—
24C4×D212C2212yes—
25#13133174C2211yescharacteristic
26#13133150C2212yes—
27#13133149C2211yescharacteristic
28#13133139C2212yes—
29#13133138C2212yes—
30#13133137C2212yes—
31#13133127C2211yescharacteristic
32C4×#88322C2211yescharacteristic
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—
44C2×D212—22no—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×C4×D53C2311yescharacteristic
48D8×C53—22no—
49C2×C8×C53C2311yescharacteristic
50C2×C8×C53C2311yescharacteristic
51C4⋊C4×C53C2312yes—
52C22⋊C4×C53—22no—
53C42×C53C2311yescharacteristic
54C22⋊Dic53—22no—
55#11719922—24no—
56#11719921—22no—
57#11719921—22no—
58#11719921D412yes—
59C4⋊Dic53—22no—
60C4⋊Dic53C2312yes—
61#11719919—22no—
62#11719919D412yes—
63C4×Dic53C2311yescharacteristic
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66D424—22no—
67#11719912—22no—
68C8×D53—22no—
69C23×C53—22no—
70C22×D53—22no—
71C22×D53C2×D411yescharacteristic
72D4×C53—22no—
73D4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—82no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—82no—
79C22⋊D53—82no—
80C22⋊D53—82no—
81C2×Dic53—22no—
82C2×Dic53C2×D411yescharacteristic
83C2×Dic53C2×D412yes—
84D212—22no—
85D212—22no—
86D212—82no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—82no—
90C8×C53—21no—
91C8×C53#4112yes—
92#88322—21no—
93#88322#4112yes—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53C22×D411yescharacteristic
98C2×D53—21no—
99C2×D53—22no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C4×C53—22no—
104C4×C53—42no—
105C4×C53C2×#4111yescharacteristic
106C4×C53C2×#4111yescharacteristic
107Dic53—21no—
108Dic53—22no—
109Dic53—42no—
110#3176—531nomaximal, Sylow
111C2×C53—82no—
112C2×C53#53211yescharacteristic
113C2×C53#57612yes—
114D53—41no—
115D53—82no—
116#540—532no—
117#499—531no—
118#493—531no—
119#490—532no—
120#469—531no—
121#469—531no—
122#466—532no—
123#442—531no—
124#442D5311yescharacteristic, complemented
125C4×D8—532no—
126#405—531no—
127C53#317611yescharacteristic, complemented, Sylow
128C2×D8—532no—
129C2×D8—531no—
130C2×D8C2×D5311yescharacteristic
131C2×M16—531no—
132C22×C8—531no—
133#127—532no—
134#125—532no—
135#123—1062no—
136#123—1062no—
137#121—532no—
138#121—532no—
139#121—1062no—
140#121—1062no—
141C4×D4—532no—
142C4×D4C2×D5312yes—
143#117—531no—
144#107—532no—
145#106—531no—
146#106C2×D5311yescharacteristic
147#102—532no—
148#102—532no—
149#102—532no—
150#102C2×D5312yes—
151#97—531no—
152#97C2×D5311yescharacteristic
153C4×C8—531no—
154C4×C8—531no—
155C2×D4—532no—
156C2×D4—1062no—
157C2×D4—1062no—
158C2×D4—1062no—
159C2×D4—1062no—
160C2×D4C22×D5312yes—
161C22×C4—22no—
162C22×C4—531no—
163C22×C4—1062no—
164D8—22no—
165D8—1064no—
166D8—1062no—
167M16—1062no—
168C2×C8—531no—
169C2×C8—531no—
170C2×C8—1062no—
171C2×C8C22×D5311yescharacteristic
172C2×C8C22×D5311yescharacteristic
173C4⋊C4—532no—
174C4⋊C4—532no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4C22×D5312yes—
178C22⋊C4—22no—
179C22⋊C4—532no—
180C22⋊C4—1062no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C42—531no—
184C42C22×D5311yescharacteristic
185C23—22no—
186C23—531no—
187C23—1062no—
188D4—22no—
189D4—22no—
190D4—1062no—
191D4—1062no—
192D4—4242no—
193D4—4242no—
194D4—4242no—
195D4—4242no—
196C2×C4—22no—
197C2×C4—82no—
198C2×C4—532no—
199C2×C4—531no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1062no—
203C2×C4—4242no—
204C2×C4D4×D5312yes—
205C2×C4C23×D5311yescharacteristic
206C8—21no—
207C8—532no—
208C8—1061no—
209C8#1171994512yes—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—1061no—
214C22—1062no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22C2×D4×D5311yescharacteristic
219C4—22no—
220C4—42no—
221C4—1061no—
222C4—1062no—
223C4—2122no—
224C4#41×D5311yescharacteristic
225C4C2×#1171994511yescharacteristic
226C2—82no—
227C2—4242no—
228C2—2121no—
229C2#1681117011yescharacteristic
230C2#1681142712yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3176
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. #16811224C2semidirect
2. #16811023C2semidirect
3. #16810998C2semidirect
4. #16810986C2semidirect
5. #16810877C2semidirect
6. #16810873C2semidirect
7. #16810858C2semidirect
8. #16810736C2semidirect
9. C4×#11719922C2semidirect
10. #16810580C2semidirect
11. C53×#442C2semidirect
124. #442D53semidirect
127. C53#3176semidirect