Finite Group Databasebeta

← #20285364

Subgroups — #20285364

Order 6784.

386 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811229C211yesmaximal, characteristic, complemented
2#16811227C211yesmaximal, characteristic, complemented
3#16811022C211yesmaximal, characteristic, complemented
4#16810999C211yesmaximal, characteristic, complemented
5#16810988C211yesmaximal, characteristic, complemented
6#16810875C212yesmaximal, complemented
7#16810858C212yesmaximal, complemented
8#16810732C212yesmaximal, complemented
9#16810703C212yesmaximal, complemented
10#16810580C211yesmaximal, characteristic, complemented
11C53×#490C211yesmaximal, characteristic, complemented
12C2×#11719922C2212yes—
13C2×#11719922C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×C8×D53C2211yescharacteristic
16#13133214C2211yescharacteristic
17#13133209C2211yescharacteristic
18#13133202—22no—
19#13133198—22no—
20#13133192—22no—
21#13133192C2212yes—
22#13133189—22no—
23#13133188C2212yes—
24D4×Dic53C2212yes—
25#13133179C2212yes—
26#13133174C2211yescharacteristic
27#13133150C2212yes—
28#13133148C2212yes—
29#13133139C2212yes—
30#13133139C2212yes—
31#13133127C2211yescharacteristic
32C4×#88322C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34#127×C53C2211yescharacteristic
35#125×C53C2211yescharacteristic
36#102×C53C2212yes—
37#102×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×D4×C53—22no—
40C2×D4×C53C2312yes—
41C2×C22⋊D53—22no—
42C22×Dic53—22no—
43C2×D212—22no—
44C2×D212—22no—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×C4×D53C2311yescharacteristic
48C2×C8×C53C2311yescharacteristic
49C2×C8×C53C2311yescharacteristic
50C4⋊C4×C53—22no—
51C4⋊C4×C53C2312yes—
52C42×C53C2311yescharacteristic
53C22⋊Dic53—22no—
54C22⋊Dic53—22no—
55#11719922—24no—
56#11719922—24no—
57#11719921—22no—
58#11719921—22no—
59#11719921D412yes—
60C4⋊Dic53C2312yes—
61#11719919—22no—
62#11719919D412yes—
63C4×Dic53C2311yescharacteristic
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66#11719912—22no—
67C8×D53—22no—
68C23×C53—22no—
69C22×D53—22no—
70C22×D53C2×D411yescharacteristic
71D4×C53—22no—
72D4×C53—22no—
73D4×C53—82no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2411yescharacteristic
77C22⋊D53—82no—
78C2×Dic53—22no—
79C2×Dic53—82no—
80C2×Dic53C2×D412yes—
81C2×Dic53C2×D411yescharacteristic
82D212—22no—
83D212—22no—
84D212—82no—
85D212—82no—
86C4×D53—21no—
87C4×D53—21no—
88C4×D53—82no—
89C8×C53—21no—
90C8×C53—21no—
91#88322#4112yes—
92#88322#4112yes—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53C22×D411yescharacteristic
97C2×D53—21no—
98C2×D53—22no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53C2×#4111yescharacteristic
105C4×C53C2×#4111yescharacteristic
106Dic53—21no—
107Dic53—22no—
108Dic53—42no—
109#3176—531nomaximal, Sylow
110C2×C53—82no—
111C2×C53#53211yescharacteristic
112C2×C53#57612yes—
113D53—41no—
114D53—82no—
115#540—531no—
116#540—531no—
117#499—531no—
118#493—531no—
119#490—531no—
120#490D5311yescharacteristic, complemented
121#469—532no—
122#466—532no—
123#442—532no—
124C4×D8—532no—
125#405—531no—
126C53#317611yescharacteristic, complemented, Sylow
127C2×D8—532no—
128C2×D8—532no—
129C2×M16—531no—
130C22×C8—531no—
131#127—531no—
132#127C2×D5311yescharacteristic
133#125—531no—
134#125C2×D5311yescharacteristic
135#123—1062no—
136#123—1062no—
137#121—532no—
138#121—532no—
139#121—1062no—
140#121—1062no—
141C4×D4—532no—
142C4×D4—532no—
143#117—531no—
144#107—532no—
145#106—532no—
146#102—532no—
147#102—532no—
148#102C2×D5312yes—
149#102C2×D5312yes—
150#97—531no—
151#97C2×D5311yescharacteristic
152C4×C8—531no—
153C4×C8—531no—
154C2×D4—22no—
155C2×D4—532no—
156C2×D4—1062no—
157C2×D4—1062no—
158C2×D4—1062no—
159C2×D4C22×D5312yes—
160C22×C4—531no—
161C22×C4—1062no—
162C22×C4—1062no—
163D8—1064no—
164D8—1064no—
165M16—1062no—
166C2×C8—531no—
167C2×C8—531no—
168C2×C8—1062no—
169C2×C8C22×D5311yescharacteristic
170C2×C8C22×D5311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—532no—
173C4⋊C4—532no—
174C4⋊C4—1062no—
175C4⋊C4C22×D5312yes—
176C22⋊C4—532no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C22⋊C4—1062no—
180C22⋊C4—1062no—
181C42—531no—
182C42C22×D5311yescharacteristic
183C23—22no—
184C23—531no—
185C23—1062no—
186D4—22no—
187D4—22no—
188D4—82no—
189D4—1062no—
190D4—1062no—
191D4—4242no—
192D4—4242no—
193D4—4242no—
194C2×C4—22no—
195C2×C4—532no—
196C2×C4—531no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1062no—
200C2×C4—4242no—
201C2×C4—4242no—
202C2×C4D4×D5312yes—
203C2×C4C23×D5311yescharacteristic
204C8—21no—
205C8—21no—
206C8—532no—
207C8—532no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—1062no—
212C22—1061no—
213C22—2122no—
214C22—2122no—
215C22—2122no—
216C22C2×D4×D5311yescharacteristic
217C4—22no—
218C4—42no—
219C4—1062no—
220C4—1061no—
221C4—2122no—
222C4#41×D5311yescharacteristic
223C4#41×D5311yescharacteristic
224C2—82no—
225C2—4242no—
226C2—2121no—
227C2#1681117211yescharacteristic
228C2#1681142712yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3176
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811229C2semidirect
2. #16811227C2semidirect
3. #16811022C2semidirect
4. #16810999C2semidirect
5. #16810988C2semidirect
6. #16810875C2semidirect
7. #16810858C2semidirect
8. #16810732C2semidirect
9. #16810703C2semidirect
10. #16810580C2semidirect
11. C53×#490C2semidirect
120. #490D53semidirect
126. C53#3176semidirect