Finite Group Databasebeta

← #20273667

Subgroups — #20273667

Order 6784.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810761C212yesmaximal, complemented
2#16810535C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic, complemented
5C53×#534C211yesmaximal, characteristic
6C22×Dic106C2211yescharacteristic, complemented
7C2×#11719923—22no—
8C2×C22⋊Dic53—21no—
9#13133140—22no—
10C22⋊#88322C2212yescomplemented
11#13133124—21no—
12#13133124C2211yescharacteristic, complemented
13C22×D4×C53—21no—
14#123×C53—22no—
15#121×C53—21no—
16#121×C53C2212yes—
17#120×C53—22no—
18C2×C22⋊C4×C53C2211yescharacteristic
19C24×C53—21no—
20C24×C53D411yescharacteristic
21C2×D4×C53—22no—
22C2×D4×C53—22no—
23C2×D4×C53—21no—
24C2×D4×C53—42no—
25C2×D4×C53—42no—
26C2×D4×C53D412yes—
27C22×C4×C53—21no—
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C2×Dic106—22no—
33C2×Dic106—42no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53D412yes—
36C22⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊Dic53—42no—
40#11719923—44no—
41C2×#88322—22no—
42C23×C53—21no—
43C23×C53—21no—
44C23×C53—22no—
45C23×C53—41no—
46C23×C53—41no—
47C23×C53—42no—
48C23×C53—81no—
49C23×C53—81no—
50C23×C53SD1612yescomplemented
51C23×C53C2×D411yescharacteristic
52D4×C53—42no—
53D4×C53—44no—
54D4×C53—42no—
55D4×C53—42no—
56D4×C53—81no—
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—41no—
60C2×C4×C53—42no—
61C2×C4×C53—41no—
62C2×C4×C53C2×D412yes—
63C2×Dic53—22no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69Dic106—24no—
70Dic106—42no—
71#88322—42no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53—22no—
75C22×C53—42no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53—81no—
82C22×C53—84no—
83C22×C53#12011yescharacteristic
84C22×C53C2×SD1611yescharacteristic
85C22×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53—22no—
87C4×C53—42no—
88C4×C53—41no—
89Dic53—42no—
90Dic53—81no—
91#1601—531nomaximal, Sylow
92C2×C53—21no—
93C2×C53—21no—
94C2×C53—42no—
95C2×C53—82no—
96C2×C53#45011yescharacteristic
97C2×C53#45511yescharacteristic
98C2×C53B(4,𝔽2)11yescharacteristic
99#534D5311yescharacteristic
100#448—532no—
101#393—531no—
102#327—532no—
103#323—531no—
104C53#160111yescharacteristic, complemented, Sylow
105C22×Q8—531no—
106C22×D4—21no—
107C2×SD16—1062no—
108#123—22no—
109#121—21no—
110#121C2×D5312yes—
111#120—22no—
112C2×C22⋊C4—1061no—
113C2×C22⋊C4C2×D5311yescharacteristic
114#103—1062no—
115C22⋊C8—532no—
116#95—531no—
117#95—1061no—
118C24—21no—
119C24C22⋊D5311yescharacteristic
120C2×Q8—1062no—
121C2×Q8—2122no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D5312yes—
128C22×C4—21no—
129C22×C4—1061no—
130C22×C4—1061no—
131C22×C4C22⋊D5311yescharacteristic
132C22×C4C22×D5311yescharacteristic
133SD16—2124no—
134C2×C8—1062no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D5312yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—2122no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1171992312yescomplemented
150C23C2×C22⋊D5311yescharacteristic
151Q8—1064no—
152Q8—2122no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—1062no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4C2×C22⋊D5312yes—
170C8—2122no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#1313318611yescharacteristic
183C22C2×#1171992311yescharacteristic
184C22#1313324311yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—4241no—
189C4—2122no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#1681077411yescharacteristic
195C2#1681080111yescharacteristic
196C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#534C2
socleC22×C53#120
hypercenter#534D53
hypocenterC53#1601
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #16810761C2semidirect
2. #16810535C2semidirect
3. #16810357C2semidirect
4. #16810347C2semidirect
6. C22×Dic106C22semidirect
10. C22⋊#88322C22semidirect
12. #13133124C22semidirect
50. C23×C53SD16semidirect
104. C53#1601semidirect
149. C23#11719923semidirect