Finite Group Databasebeta

← #20273062

Subgroups — #20273062

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×#13133202C211yesmaximal, characteristic
2#16810585C211yesmaximal, characteristic, complemented
3C22⋊C8×D53C211yesmaximal, characteristic, complemented
4#16810349C211yesmaximal, characteristic, complemented
5#16810346C211yesmaximal, characteristic, complemented
6#16810346C211yesmaximal, characteristic, complemented
7C53×#324C211yesmaximal, characteristic, complemented
8C22×D212C411yescharacteristic
9C22×C4×D53C2211yescharacteristic
10C2×#11719912—21no—
11C2×C8×D53—21no—
12#13133202—21no—
13#13133202—21no—
14#13133202—21no—
15#13133202—21no—
16C2×#11719919C411yescharacteristic
17C2×#11719921C411yescharacteristic
18C2×#11719921C411yescharacteristic
19C2×#11719921C2211yescharacteristic
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22#13133129—21no—
23#13133129—21no—
24C2×C4⋊C4×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26C22⋊C8×C53C2211yescharacteristic
27C23×D53C2×C411yescharacteristic
28C23×D53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53C2×C411yescharacteristic
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—42no—
37C2×C4×D53—42no—
38C2×C4×D53D411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C8×C53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46#11719921—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52#11719919—21no—
53#11719919—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56#11719912—42no—
57C8×D53—42no—
58C23×C53C22×C411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—22no—
62C22×D53—21no—
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—41no—
66C22×D53—41no—
67C22×D53—41no—
68C22×D53C22⋊C411yescharacteristic
69C22×D53C22⋊C411yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×Dic53—21no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83D212—42no—
84D212—42no—
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×C53C2×C22⋊C411yescharacteristic
96C22×C53#13111yescharacteristic
97C22×C53#13111yescharacteristic
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—41no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—81no—
109C2×D53—81no—
110C2×D53#9912yes—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115Dic53—21no—
116Dic53—21no—
117Dic53—41no—
118#1468—531nomaximal, Sylow
119C2×C53—21no—
120C2×C53—21no—
121C2×C53#40811yescharacteristic
122C2×C53C2×#9911yescharacteristic
123C2×C53#41311yescharacteristic
124D53—22no—
125D53—41no—
126D53—81no—
127C2×#123—531no—
128#407—531no—
129C2×C22⋊C8—531no—
130#324—531no—
131#324D5311yescharacteristic, complemented
132#323—531no—
133#323—531no—
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⋊C4C2×D5311yescharacteristic
145C2×C22⋊C4—531no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4—531no—
148C22⋊C8—531no—
149C22⋊C8—531no—
150C22⋊C8—1061no—
151C22⋊C8—1061no—
152C22⋊C8C2×D5311yescharacteristic
153C22⋊C8C2×D5311yescharacteristic
154C24—531no—
155C24—531no—
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—2122no—
166C22×C4C4×D5311yescharacteristic
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—21no—
176C4⋊C4—21no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C23—531no—
186C23—531no—
187C23—1062no—
188C23—1061no—
189C23—1061no—
190C23—1061no—
191C23—1061no—
192C23—1061no—
193C23—2121no—
194C23—2121no—
195C23—2121no—
196C23C2×C4×D5311yescharacteristic
197D4—2122no—
198D4—2122no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—531no—
206C2×C4—531no—
207C2×C4—1062no—
208C2×C4—1062no—
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—532no—
223C22—1061no—
224C22—1061no—
225C22—1061no—
226C22—1061no—
227C22—1061no—
228C22—4241no—
229C22—4241no—
230C22—2122no—
231C22—2122no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22C22⋊C4×D5311yescharacteristic
236C22#1313322311yescharacteristic
237C22#1313322511yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—1061no—
243C4—1061no—
244C4—2121no—
245C2—21no—
246C2—21no—
247C2—1062no—
248C2—4241no—
249C2—2121no—
250C2#1681058811yescharacteristic
251C2#99×D5311yescharacteristic
252C2#1681060611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53C2×C22⋊C4
hypercenter#324D53
hypocenterC53#1468
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #16810585C2semidirect
3. C22⋊C8×D53C2semidirect
4. #16810349C2semidirect
5. #16810346C2semidirect
6. #16810346C2semidirect
7. C53×#324C2semidirect
131. #324D53semidirect
134. C53#1468semidirect