Finite Group Databasebeta

← #20273273

Subgroups — #20273273

Order 6784.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#16811168C211yesmaximal, characteristic
2#16810673C211yesmaximal, characteristic, complemented
3#16810487C211yesmaximal, characteristic, complemented
4#16810358C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810356C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9#13133216C411yescharacteristic, complemented
10#13133213C411yescharacteristic, complemented
11#13133203—21no—
12#13133191C2211yescharacteristic
13#13133189—21no—
14D4×Dic53C411yescharacteristic, complemented
15#13133179C411yescharacteristic, complemented
16#13133177—21no—
17C2×#11719919—21no—
18C2×C4×Dic53C2211yescharacteristic
19#13133146—21no—
20C22⋊#88322C2211yescharacteristic
21C8×Dic53—21no—
22#13133124—21no—
23#13133124C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37C22×Dic53D411yescharacteristic
38#11719947—42no—
39#11719947—42no—
40C2×D212C2×C411yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53C2×C411yescharacteristic
45C22⋊C4×C53—21no—
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53C2×C411yescharacteristic
50#11719919—21no—
51#11719919—42no—
52C4×Dic53C2×C411yescharacteristic
53C4×Dic53C2×C411yescharacteristic
54C4×Dic53D411yescharacteristic
55C4×Dic53D411yescharacteristic
56C2×#88322—21no—
57C23×C53—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C22⋊D53—42no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C22⋊D53—42no—
74C2×Dic53—21no—
75C2×Dic53—41no—
76C2×Dic53—42no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53C22⋊C411yescharacteristic
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C22⋊C411yescharacteristic
84C2×Dic53D812yes—
85C2×Dic53SD1612yes—
86D212—41no—
87C4×D53—42no—
88Dic106—42no—
89C8×C53—41no—
90#88322—41no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53C2×C22⋊C411yescharacteristic
97C22×C53C2×D811yescharacteristic
98C22×C53C2×SD1611yescharacteristic
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—41no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—81no—
106Dic53—21no—
107Dic53—21no—
108Dic53—41no—
109Dic53—81no—
110Dic53#10212yes—
111Dic53#10212yes—
112#1517—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—81no—
116C2×C53#41011yescharacteristic
117C2×C53C2×#10211yescharacteristic
118C2×C53#42111yescharacteristic
119D53—81no—
120#532—531no—
121#431—531no—
122#382—531no—
123#327—531no—
124#327—531no—
125#327—531no—
126#327D5311yescharacteristic, complemented
127C53#151711yescharacteristic, complemented, Sylow
128C2×#41—1061no—
129C4⋊Q8—531no—
130#127—531no—
131#124—1061no—
132#121—531no—
133#121—1061no—
134#121C2×D5311yescharacteristic
135C4×D4—531no—
136C4×D4—531no—
137C4×D4—1061no—
138C2×C4⋊C4—1061no—
139C2×C42—531no—
140C4⋊C8—1061no—
141C22⋊C8—531no—
142C22⋊C8C2×D5311yescharacteristic
143C4×C8—1061no—
144#95—531no—
145#95—1061no—
146#95C2×D5311yescharacteristic
147#41—2122no—
148#41—2122no—
149C2×Q8—1061no—
150C2×D4—21no—
151C2×D4—531no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4—1061no—
155C2×D4C4×D5311yescharacteristic
156C22×C4—21no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—1061no—
160C22×C4—1061no—
161C22×C4—1061no—
162C22×C4C22×D5311yescharacteristic
163C2×C8—21no—
164C2×C8—1061no—
165C4⋊C4—531no—
166C4⋊C4—1061no—
167C4⋊C4—2122no—
168C4⋊C4C4×D5311yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C42—531no—
174C42—531no—
175C42—531no—
176C42—531no—
177C23—21no—
178C23—1061no—
179C23D4×D5311yescharacteristic
180Q8—2122no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—2121no—
185D4—2122no—
186D4—2122no—
187D4—2121no—
188D4—2121no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—532no—
197C2×C4—532no—
198C2×C4—531no—
199C2×C4—531no—
200C2×C4—1061no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4—2121no—
207C2×C4C2×C4×D5311yescharacteristic
208C2×C4D4×D5311yescharacteristic
209C8—41no—
210C8—2121no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—2121no—
217C22—2121no—
218C22—2121no—
219C22C22⋊C4×D5311yescharacteristic
220C22D8×D5311yescharacteristic
221C22SD16×D5311yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—532no—
227C4—532no—
228C4—1061no—
229C4—1061no—
230C4—4241no—
231C4—2121no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—4241no—
236C2#1681059511yescharacteristic
237C2#102×D5311yescharacteristic
238C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1517
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810673C2semidirect
3. #16810487C2semidirect
4. #16810358C2semidirect
5. #16810357C2semidirect
6. #16810356C2semidirect
7. C53×#327C2semidirect
9. #13133216C4semidirect
10. #13133213C4semidirect
14. D4×Dic53C4semidirect
15. #13133179C4semidirect
126. #327D53semidirect
127. C53#1517semidirect