Finite Group Databasebeta

← #20273060

Subgroups — #20273060

Order 6784.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#13133199C211yesmaximal, characteristic
2#16810585C211yesmaximal, characteristic, complemented
3C22⋊C8×D53C211yesmaximal, characteristic, complemented
4#16810348C211yesmaximal, characteristic, complemented
5#16810348C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#323C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C411yescharacteristic
9C22×C4×D53C2211yescharacteristic
10C2×#11719912—21no—
11C2×C8×D53—21no—
12#13133199—21no—
13#13133199—21no—
14#13133199—21no—
15#13133199—21no—
16C2×C4⋊Dic53C411yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×C22⋊Dic53C411yescharacteristic
19C2×#11719921C411yescharacteristic
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22#13133129—21no—
23#13133129—21no—
24C2×C22⋊C4×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26C22⋊C8×C53C2211yescharacteristic
27C24×C53C2×C411yescharacteristic
28C23×D53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—42no—
34C22×Dic53C2×C411yescharacteristic
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C411yescharacteristic
37C2×C4×D53—42no—
38C2×C4×D53D411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C8×C53—21no—
43C2×C8×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊Dic53—21no—
47C22⋊Dic53—21no—
48#11719921—21no—
49#11719921—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52#11719919—21no—
53#11719919—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56#11719912—42no—
57C8×D53—42no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53C22×C411yescharacteristic
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—22no—
67C22×D53—41no—
68C22×D53C22⋊C411yescharacteristic
69C22×D53C22⋊C411yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C22⋊D53—42no—
76C22⋊D53—42no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53C22⋊C411yescharacteristic
84C2×Dic53C22⋊C411yescharacteristic
85C4×D53—22no—
86C4×D53—22no—
87C4×D53—41no—
88C4×D53—41no—
89C8×C53—41no—
90C8×C53—41no—
91#88322—41no—
92#88322—41no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—81no—
99C22×C53—81no—
100C22×C53C2×C22⋊C411yescharacteristic
101C22×C53#13111yescharacteristic
102C22×C53#13111yescharacteristic
103C2×D53—21no—
104C2×D53—21no—
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53#9912yes—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—41no—
114Dic53—21no—
115Dic53—21no—
116Dic53—41no—
117Dic53—41no—
118#1468—531nomaximal, Sylow
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—81no—
122C2×C53#40811yescharacteristic
123C2×C53C2×#9911yescharacteristic
124C2×C53#41311yescharacteristic
125D53—22no—
126D53—41no—
127C2×#123—531no—
128#407—531no—
129C2×C22⋊C8—531no—
130#324—531no—
131#324—531no—
132#323—531no—
133#323D5311yescharacteristic, complemented
134C53#146811yescharacteristic, complemented, Sylow
135C22×D4—531no—
136C23×C4—531no—
137C2×M16—1061no—
138C22×C8—1061no—
139#123—1061no—
140#123—1061no—
141#123—1061no—
142#123—1061no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C22⋊C4—531no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4C2×D5311yescharacteristic
148C22⋊C8—531no—
149C22⋊C8—531no—
150C22⋊C8—1061no—
151C22⋊C8—1061no—
152C22⋊C8C2×D5311yescharacteristic
153C22⋊C8C2×D5311yescharacteristic
154C24—531no—
155C24C4×D5311yescharacteristic
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4—2122no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—2122no—
167C22×C4C4×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169M16—2122no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—1061no—
173C2×C8—1061no—
174C2×C8—2122no—
175C4⋊C4—1061no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23—531no—
191C23—531no—
192C23—1062no—
193C23—1061no—
194C23—1061no—
195C23—2121no—
196C23C2×C4×D5311yescharacteristic
197D4—2122no—
198D4—2122no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—531no—
203C2×C4—531no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4D4×D5311yescharacteristic
215C2×C4D4×D5311yescharacteristic
216C8—41no—
217C8—41no—
218C8—2121no—
219C8—2121no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22—532no—
228C22—1061no—
229C22—1061no—
230C22—1061no—
231C22—1061no—
232C22—1061no—
233C22—2122no—
234C22—2122no—
235C22C22⋊C4×D5311yescharacteristic
236C22#1313322311yescharacteristic
237C22#1313322511yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—1061no—
242C4—1061no—
243C4—2121no—
244C4—2121no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—1062no—
249C2—2121no—
250C2#1681058811yescharacteristic
251C2#99×D5311yescharacteristic
252C2#1681060411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53C2×C22⋊C4
hypercenter#323D53
hypocenterC53#1468
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810585C2semidirect
3. C22⋊C8×D53C2semidirect
4. #16810348C2semidirect
5. #16810348C2semidirect
6. #16810347C2semidirect
7. C53×#323C2semidirect
133. #323D53semidirect
134. C53#1468semidirect