Finite Group Databasebeta

← #20273138

Subgroups — #20273138

Order 6784.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×D4×D53C211yesmaximal, characteristic
2#16810674C211yesmaximal, characteristic, complemented
3#16810645C211yesmaximal, characteristic, complemented
4#16810352C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×D4×D53C411yescharacteristic
9C22×C4×D53—21no—
10C2×#11719912—21no—
11C4×D212C2211yescharacteristic
12D4×Dic53C411yescharacteristic
13#13133179C411yescharacteristic
14C4×C22⋊D53—21no—
15#13133177—21no—
16C4⋊C4×D53C411yescharacteristic
17C22⋊C4×D53—21no—
18C42×D53C2211yescharacteristic
19C4⋊#88322C2211yescharacteristic, complemented
20#13133146—21no—
21#13133146—21no—
22#13133129—21no—
23C4×#88322C2211yescharacteristic, complemented
24C4×D4×C53C2211yescharacteristic
25C4⋊C8×C53C2211yescharacteristic, complemented
26C4×C8×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32D4×D53—22no—
33D4×D53—22no—
34C2×D212C2×C411yescharacteristic
35C2×C4×D53—21no—
36C2×C4×D53—21no—
37C2×C4×D53—21no—
38C2×C4×D53—42no—
39C2×C4×D53—42no—
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C8×C53—21no—
44C2×C8×C53—21no—
45C4⋊C4×C53C2×C411yescharacteristic
46C22⋊C4×C53—21no—
47C42×C53C2311yescharacteristic
48C22⋊Dic53—21no—
49#11719921—21no—
50#11719921—21no—
51C4⋊Dic53C2×C411yescharacteristic
52#11719919—21no—
53C4×Dic53C2×C411yescharacteristic
54C4×Dic53D411yescharacteristic
55C4×Dic53D411yescharacteristic
56C2×#88322—21no—
57C2×#88322—21no—
58#11719912—42no—
59C23×C53—21no—
60C22×D53—24no—
61C22×D53—21no—
62C22×D53—42no—
63C22×D53—42no—
64C22×D53C22⋊C411yescharacteristic
65D4×C53—21no—
66D4×C53M1612yescomplemented
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53C22×C411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C22⋊D53—24no—
74C2×Dic53—21no—
75C2×Dic53—42no—
76C2×Dic53C22⋊C411yescharacteristic
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C22⋊C411yescharacteristic
79D212—21no—
80D212M1612yescomplemented
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—41no—
84C4×D53—42no—
85C4×D53—41no—
86C4×D53—41no—
87C4×D53—42no—
88C4×D53D812yes—
89C4×D53SD1612yes—
90C8×C53—41no—
91C8×C53—41no—
92#88322—41no—
93#88322—41no—
94C22×C53—22no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53C2×C22⋊C411yescharacteristic
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—22no—
101C2×D53—48no—
102C2×D53—41no—
103C2×D53—41no—
104C2×D53#10212yes—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53C2×M1611yescharacteristic
108C4×C53C2×M1611yescharacteristic
109C4×C53C2×D811yescharacteristic
110C4×C53C2×SD1611yescharacteristic
111Dic53—21no—
112Dic53—21no—
113Dic53—41no—
114Dic53#10212yes—
115#1486—531nomaximal, Sylow
116C2×C53—42no—
117C2×C53#40711yescharacteristic
118C2×C53C2×#10211yescharacteristic
119C2×C53#42111yescharacteristic
120D53—22no—
121D53—42no—
122C2×C4×D4—531no—
123#431—531no—
124#423—531no—
125D4⋊C8—531no—
126D4⋊C8—531no—
127D4⋊C8—531no—
128D4⋊C8D5311yescharacteristic, complemented
129C53#148611yescharacteristic, complemented, Sylow
130C22×D4—531no—
131C23×C4—1061no—
132C2×M16—1061no—
133C4×D4—531no—
134C4×D4—531no—
135C4×D4—531no—
136C4×D4—1061no—
137C4×D4—1061no—
138C4×D4C2×D5311yescharacteristic
139C2×C4⋊C4—531no—
140C2×C22⋊C4—1061no—
141C2×C42—531no—
142C4⋊C8—531no—
143C4⋊C8—1061no—
144C4⋊C8—1061no—
145C4⋊C8C2×D5311yescharacteristic, complemented
146C22⋊C8—1061no—
147C4×C8—531no—
148C4×C8C2×D5311yescharacteristic, complemented
149C24—1061no—
150C2×D4—531no—
151C2×D4—1061no—
152C2×D4—1062no—
153C2×D4—1062no—
154C2×D4C4×D5311yescharacteristic
155C22×C4—21no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—1061no—
160C22×C4—1061no—
161C22×C4—1061no—
162C22×C4—1061no—
163C22×C4—2122no—
164C22×C4—2122no—
165M16—2122no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—1061no—
169C2×C8—1061no—
170C4⋊C4—531no—
171C4⋊C4—1061no—
172C4⋊C4C4×D5311yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—1061no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C42—531no—
178C42—531no—
179C42—531no—
180C42C22×D5311yescharacteristic
181C23—21no—
182C23—531no—
183C23—1064no—
184C23—1061no—
185C23—2122no—
186C23—2122no—
187D4—21no—
188D4—532no—
189D4—1064no—
190D4—1061no—
191D4#1171991212yescomplemented
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—531no—
196C2×C4—532no—
197C2×C4—532no—
198C2×C4—531no—
199C2×C4—531no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—2121no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2121no—
208C2×C4—2121no—
209C2×C4C2×C4×D5311yescharacteristic
210C2×C4D4×D5311yescharacteristic
211C2×C4D4×D5311yescharacteristic
212C8—41no—
213C8—41no—
214C8—2121no—
215C8—2121no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—532no—
220C22—1061no—
221C22—1061no—
222C22—1062no—
223C22—2128no—
224C22—2121no—
225C22—2121no—
226C22C22⋊C4×D5311yescharacteristic
227C4—21no—
228C4—41no—
229C4—532no—
230C4—1061no—
231C4—1061no—
232C4—2121no—
233C4M16×D5311yescharacteristic
234C4C2×#1171991211yescharacteristic
235C4D8×D5311yescharacteristic
236C4SD16×D5311yescharacteristic
237C2—42no—
238C2—1062no—
239C2—2122no—
240C2#1681058511yescharacteristic
241C2#102×D5311yescharacteristic
242C2#1681063811yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53C2×C22⋊C4
hypercenterD4⋊C8D53
hypocenterC53#1486
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16810674C2semidirect
3. #16810645C2semidirect
4. #16810352C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×D4⋊C8C2semidirect
19. C4⋊#88322C22semidirect
23. C4×#88322C22semidirect
25. C4⋊C8×C53C22semidirect
26. C4×C8×C53C22semidirect
66. D4×C53M16semidirect
80. D212M16semidirect
128. D4⋊C8D53semidirect
129. C53#1486semidirect
145. C4⋊C8C2×D53semidirect
148. C4×C8C2×D53semidirect
191. D4#11719912semidirect