Finite Group Databasebeta

← #20273270

Subgroups — #20273270

Order 6784.

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

normal subgroups only

This group has 284 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
1#16811136C211yesmaximal, characteristic
2C22⋊C8×D53C211yesmaximal, characteristic, complemented
3#16810485C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810358C211yesmaximal, characteristic, complemented
6#16810356C211yesmaximal, characteristic, complemented
7C53×#328C211yesmaximal, characteristic, complemented
8C2×#11719949C411yescharacteristic, complemented
9C22×C4×D53C2211yescharacteristic
10C2×C8×D53—21no—
11#13133202—21no—
12#13133193C411yescharacteristic, complemented
13#13133192C411yescharacteristic, complemented
14#13133191C2211yescharacteristic
15#13133187—21no—
16#13133179—21no—
17#13133177—21no—
18#13133175C411yescharacteristic, complemented
19C2×#11719921—21no—
20C22⋊#88322C2211yescharacteristic
21#13133129—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×Q8×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34#11719949—42no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212C2×C411yescharacteristic
38C2×C4×D53—21no—
39C2×C4×D53—42no—
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53C2×C411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×C8×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53C2×C411yescharacteristic
47C22⋊C4×C53—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52#11719921—42no—
53C4⋊Dic53C2×C411yescharacteristic
54#11719919—21no—
55C4×Dic53—21no—
56C2×#88322—21no—
57C8×D53—42no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—22no—
62C22×D53—21no—
63C22×D53—41no—
64C22×D53C22⋊C411yescharacteristic
65C22×D53C22⋊C411yescharacteristic
66Q8×C53—41no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C22×C411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C22⋊D53—41no—
76C22⋊D53—41no—
77C2×Dic53—21no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C22⋊C411yescharacteristic
84D212—41no—
85D212—42no—
86D212—42no—
87C4×D53—22no—
88C4×D53—22no—
89C4×D53—41no—
90C4×D53—41no—
91C4×D53—42no—
92C8×C53—41no—
93#88322—41no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53C2×C22⋊C411yescharacteristic
97C22×C53#13511yescharacteristic
98C22×C53#13511yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53#9912yes—
110C4×C53—21no—
111C4×C53—21no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53—81no—
115Dic53—21no—
116Dic53—21no—
117Dic53—41no—
118Dic53—81no—
119#1516—531nomaximal, Sylow
120C2×C53—21no—
121C2×C53—21no—
122C2×C53C2×#9911yescharacteristic
123C2×C53#41611yescharacteristic
124C2×C53#42111yescharacteristic
125D53—22no—
126D53—41no—
127D53—81no—
128#525—531no—
129C2×C22⋊C8—531no—
130#381—531no—
131#328—531no—
132#328D5311yescharacteristic, complemented
133#327—531no—
134#327—531no—
135C53#151611yescharacteristic, complemented, Sylow
136C2×#41—531no—
137C23×C4—531no—
138C22×C8—1061no—
139#123—1061no—
140C22⋊Q8—531no—
141C22⋊Q8C2×D5311yescharacteristic
142#121—531no—
143#121—531no—
144#120—1061no—
145C4×D4—1061no—
146C4×D4—1061no—
147#117—531no—
148C2×C22⋊C4—1061no—
149C22⋊C8—531no—
150C22⋊C8—1061no—
151C22⋊C8C2×D5311yescharacteristic
152#95—531no—
153#95—1061no—
154#95C2×D5311yescharacteristic
155C24—531no—
156#41—2122no—
157C2×Q8C4×D5311yescharacteristic
158C2×D4—531no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—1061no—
162C22×C4—21no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—1061no—
169C22×C4—1061no—
170C22×C4—2122no—
171C22×C4C22×D5311yescharacteristic
172C2×C8—21no—
173C2×C8—1061no—
174C2×C8—2122no—
175C4⋊C4—21no—
176C4⋊C4—531no—
177C4⋊C4—1061no—
178C4⋊C4C4×D5311yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—2122no—
185C42—1061no—
186C23—531no—
187C23—531no—
188C23—1062no—
189C23—1061no—
190C23—1061no—
191C23—1061no—
192C23—2121no—
193C23D4×D5311yescharacteristic
194Q8—41no—
195D4—2121no—
196D4—2122no—
197D4—2122no—
198D4—2121no—
199D4—2121no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—531no—
207C2×C4—531no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1061no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4—2122no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2121no—
217C2×C4—2121no—
218C2×C4C2×C4×D5311yescharacteristic
219C2×C4D4×D5311yescharacteristic
220C8—41no—
221C8—2121no—
222C22—21no—
223C22—21no—
224C22—532no—
225C22—1061no—
226C22—1061no—
227C22—1061no—
228C22—1061no—
229C22—1061no—
230C22—2122no—
231C22—2122no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22C22⋊C4×D5311yescharacteristic
236C22#1313323811yescharacteristic
237C22#1313324011yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—81no—
243C4—1061no—
244C4—1061no—
245C4—4241no—
246C4—2121no—
247C2—21no—
248C2—21no—
249C2—1062no—
250C2—4241no—
251C2—2121no—
252C2#99×D5311yescharacteristic
253C2#1681061811yescharacteristic
254C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53C2×C22⋊C4
hypercenter#328D53
hypocenterC53#1516
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×D53C2semidirect
3. #16810485C2semidirect
4. #16810360C2semidirect
5. #16810358C2semidirect
6. #16810356C2semidirect
7. C53×#328C2semidirect
8. C2×#11719949C4semidirect
12. #13133193C4semidirect
13. #13133192C4semidirect
18. #13133175C4semidirect
132. #328D53semidirect
135. C53#1516semidirect