Finite Group Databasebeta

← #20273182

Subgroups — #20273182

Order 6784.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811170C211yesmaximal, characteristic
2#16810679C211yesmaximal, characteristic, complemented
3#16810674C211yesmaximal, characteristic, complemented
4#16810352C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719945—21no—
9#13133216C411yescharacteristic
10#13133213C411yescharacteristic
11#13133203—21no—
12#13133192C411yescharacteristic
13#13133190—21no—
14#13133188C411yescharacteristic
15C4×D212C2211yescharacteristic
16C4×C22⋊D53—21no—
17C42×D53C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133146—21no—
20#13133129—21no—
21#13133129—21no—
22#13133127—21no—
23C4×#88322C2211yescharacteristic
24C4×D4×C53C2211yescharacteristic
25C4⋊C8×C53C2211yescharacteristic
26C4×C8×C53C2211yescharacteristic
27C2×D4×C53C2×C411yescharacteristic
28C22×C4×C53—21no—
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C2×C22⋊D53—21no—
32#11719945—42no—
33#11719945—42no—
34C2×D212—21no—
35C2×D212C2×C411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53C2×C411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×Dic106—21no—
41C2×C8×C53—21no—
42C2×C8×C53—21no—
43C4⋊C4×C53C2×C411yescharacteristic
44C22⋊C4×C53—21no—
45C42×C53C2311yescharacteristic
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53C2×C411yescharacteristic
50#11719919—21no—
51C4×Dic53C2×C411yescharacteristic
52C4×Dic53D411yescharacteristic
53C4×Dic53D411yescharacteristic
54C2×#88322—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C22×D53—21no—
58C22×D53C22⋊C411yescharacteristic
59D4×C53—21no—
60D4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53C22×C411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C22⋊D53—42no—
68C22⋊D53—42no—
69C22⋊D53—42no—
70C2×Dic53—21no—
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C411yescharacteristic
74D212—21no—
75D212—21no—
76D212—42no—
77C4×D53—21no—
78C4×D53—21no—
79C4×D53—21no—
80C4×D53—21no—
81C4×D53—41no—
82C4×D53—41no—
83C4×D53—41no—
84Dic106—42no—
85C8×C53—41no—
86C8×C53—41no—
87#88322—41no—
88#88322—41no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53C2×C22⋊C411yescharacteristic
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—41no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53#13111yescharacteristic
102C4×C53#13111yescharacteristic
103C4×C53AGL(1,ℤ/8)11yescharacteristic
104C4×C53AGL(1,ℤ/8)11yescharacteristic
105Dic53—21no—
106Dic53—21no—
107Dic53—21no—
108Dic53—41no—
109#1496—531nomaximal, Sylow
110C2×C53—81no—
111C2×C53#40811yescharacteristic
112C2×C53#41811yescharacteristic
113C2×C53#42111yescharacteristic
114D53—41no—
115D53—81no—
116#532—531no—
117#432—531no—
118#431—531no—
119D4⋊C8—531no—
120D4⋊C8—531no—
121D4⋊C8—531no—
122D4⋊C8D5311yescharacteristic, complemented
123C53#149611yescharacteristic, complemented, Sylow
124C2×#41—1061no—
125C4⋊Q8—531no—
126#127—531no—
127#124—1061no—
128#121—531no—
129#121—531no—
130#121—1061no—
131C4×D4—531no—
132C4×D4—1061no—
133C4×D4C2×D5311yescharacteristic
134C2×C42—531no—
135C4⋊C8—531no—
136C4⋊C8—1061no—
137C4⋊C8C2×D5311yescharacteristic
138C22⋊C8—1061no—
139C22⋊C8—1061no—
140#97—1061no—
141C4×C8—531no—
142C4×C8C2×D5311yescharacteristic
143#41—2122no—
144#41—2122no—
145C2×Q8—1061no—
146C2×D4—531no—
147C2×D4—1061no—
148C2×D4—1061no—
149C2×D4—1061no—
150C2×D4—1061no—
151C2×D4C4×D5311yescharacteristic
152C22×C4—21no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—1061no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—1061no—
160C2×C8—1061no—
161C4⋊C4—531no—
162C4⋊C4—1061no—
163C4⋊C4C4×D5311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C42—531no—
169C42—531no—
170C42—531no—
171C42C22×D5311yescharacteristic
172C23—21no—
173C23—531no—
174C23—1061no—
175Q8—2122no—
176D4—21no—
177D4—21no—
178D4—1061no—
179D4—1061no—
180D4—2122no—
181D4—2122no—
182D4—2122no—
183D4—2122no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4C2×C4×D5311yescharacteristic
199C2×C4D4×D5311yescharacteristic
200C2×C4D4×D5311yescharacteristic
201C8—41no—
202C8—41no—
203C8—2121no—
204C8—2121no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—1061no—
209C22—1061no—
210C22—1061no—
211C22—2121no—
212C22—2121no—
213C22—2121no—
214C22C22⋊C4×D5311yescharacteristic
215C4—21no—
216C4—41no—
217C4—1061no—
218C4—1061no—
219C4—1061no—
220C4—2121no—
221C4#1313322311yescharacteristic
222C4#1313322511yescharacteristic
223C4#1313324111yescharacteristic
224C4#1313324411yescharacteristic
225C2—81no—
226C2—4241no—
227C2—2121no—
228C2#1681058811yescharacteristic
229C2#1681062611yescharacteristic
230C2#1681063811yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53C2×C22⋊C4
hypercenterD4⋊C8D53
hypocenterC53#1496
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. #16810679C2semidirect
3. #16810674C2semidirect
4. #16810352C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×D4⋊C8C2semidirect
122. D4⋊C8D53semidirect
123. C53#1496semidirect