Finite Group Databasebeta

← #20273669

Subgroups — #20273669

Order 6784.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811185C211yesmaximal, characteristic
2#16810764C212yesmaximal, complemented
3#16810535C211yesmaximal, characteristic, complemented
4#16810358C212yesmaximal, complemented
5C53×#323C211yesmaximal, characteristic, complemented
6C22×Dic106C2211yescharacteristic, complemented
7C22×C22⋊D53—21no—
8C2×#11719913—22no—
9#13133201—22no—
10#13133191—21no—
11#13133191C2212yes—
12#13133187—22no—
13C2×C22⋊Dic53—21no—
14#13133142—22no—
15#13133124—21no—
16#13133124C2211yescharacteristic, complemented
17C2×C22⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yescomplemented
19C24×C53D411yescharacteristic
20C23×D53—21no—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53C2311yescharacteristic
23C2×C22⋊D53—22no—
24C2×C22⋊D53—22no—
25C2×C22⋊D53—42no—
26C2×C22⋊D53—42no—
27C22×Dic53—21no—
28C22×Dic53—21no—
29C22×Dic53—21no—
30C2×D212—21no—
31C2×D212D412yes—
32C2×Dic106—22no—
33C2×Dic106—42no—
34C2×C8×C53—22no—
35C22⋊C4×C53—22no—
36C22⋊Dic53—42no—
37#11719921—22no—
38#11719921—22no—
39C4⋊Dic53—21no—
40C4⋊Dic53D412yes—
41#11719913—44no—
42C23×C53—21no—
43C23×C53—21no—
44C23×C53—41no—
45C23×C53—41no—
46C23×C53SD1612yescomplemented
47C23×C53C2×D411yescharacteristic
48C22×D53—22no—
49C22×D53—42no—
50C22×D53—81no—
51C22×D53—81no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53C2×D412yes—
56C22⋊D53—44no—
57C22⋊D53—42no—
58C22⋊D53—42no—
59C2×Dic53—22no—
60C2×Dic53—22no—
61C2×Dic53—41no—
62C2×Dic53—42no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67D212—42no—
68D212—81no—
69Dic106—24no—
70Dic106—42no—
71C8×C53—42no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53—22no—
75C22×C53—42no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—81no—
79C22×C53#12011yescharacteristic
80C22×C53C2×SD1611yescharacteristic
81C22×C53AGL(1,ℤ/8)11yescharacteristic
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—84no—
86C4×C53—22no—
87C4×C53—41no—
88Dic53—42no—
89Dic53—42no—
90Dic53—81no—
91#1601—531nomaximal, Sylow
92C2×C53—21no—
93C2×C53—21no—
94C2×C53—42no—
95C2×C53#45011yescharacteristic
96C2×C53#45511yescharacteristic
97C2×C53B(4,𝔽2)11yescharacteristic
98D53—82no—
99#534—531no—
100#448—532no—
101#393—531no—
102#327—532no—
103#323D5311yescharacteristic, complemented
104C53#160111yescharacteristic, complemented, Sylow
105C22×Q8—531no—
106C22×D4—1061no—
107C2×SD16—1062no—
108#123—1062no—
109#121—532no—
110#121—1061no—
111#120—1062no—
112C2×C22⋊C4—1061no—
113C2×C22⋊C4C2×D5311yescharacteristic
114#103—1062no—
115C22⋊C8C2×D5312yescomplemented
116#95—531no—
117#95—1061no—
118C24—1061no—
119C24D21211yescharacteristic
120C2×Q8—1062no—
121C2×Q8—2122no—
122C2×D4—532no—
123C2×D4—1061no—
124C2×D4—1062no—
125C2×D4—1062no—
126C2×D4—2122no—
127C2×D4—2122no—
128C22×C4—1061no—
129C22×C4—1061no—
130C22×C4—1061no—
131C22×C4D21211yescharacteristic
132C22×C4C22×D5311yescharacteristic
133SD16—2124no—
134C2×C8—22no—
135C4⋊C4—532no—
136C4⋊C4—1061no—
137C22⋊C4—22no—
138C22⋊C4—1062no—
139C22⋊C4—1062no—
140C22⋊C4—2122no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—1062no—
146C23—4241no—
147C23—4241no—
148C23—2122no—
149C23#1171991312yescomplemented
150C23C2×D21211yescharacteristic
151Q8—1064no—
152Q8—2122no—
153D4—4241no—
154D4—2122no—
155D4—2124no—
156D4—2122no—
157D4—2122no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—2121no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2122no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4D4×D5312yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—4244no—
179C22—2122no—
180C22—2122no—
181C22—2122no—
182C22#1313318711yescharacteristic
183C22C2×#1171991311yescharacteristic
184C22#1313324511yescharacteristic
185C4—22no—
186C4—41no—
187C4—4241no—
188C4—2122no—
189C4—2122no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—4242no—
194C2#1681077611yescharacteristic
195C2#1681080311yescharacteristic
196C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53#120
hypercenter#323D53
hypocenterC53#1601
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810764C2semidirect
3. #16810535C2semidirect
4. #16810358C2semidirect
5. C53×#323C2semidirect
6. C22×Dic106C22semidirect
16. #13133124C22semidirect
18. C22⋊C8×C53C22semidirect
46. C23×C53SD16semidirect
103. #323D53semidirect
104. C53#1601semidirect
115. C22⋊C8C2×D53semidirect
149. C23#11719913semidirect