Finite Group Databasebeta

← #20276136

Subgroups — #20276136

Order 6784.

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

normal subgroups only

This group has 222 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
1C2×#13133141C211yesmaximal, characteristic
2#16810530C211yesmaximal, characteristic
3#16810488C211yesmaximal, characteristic
4#16810386C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810377C211yesmaximal, characteristic
7C2×#103×C53C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C2×C4⋊Dic53C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4×Dic53C2211yescharacteristic
12#13133141—22no—
13#13133124—21no—
14#13133124—21no—
15#13133124—21no—
16#13133124—21no—
17C22×Q8×C53C2211yescharacteristic
18C22×C8×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20#103×C53—22no—
21C2×Q8×C53—21no—
22C2×Q8×C53—21no—
23C2×Q8×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53D412yes—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33C22×Dic53D411yescharacteristic
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4×Dic53—21no—
44C4×Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53C2×D411yescharacteristic
49Q8×C53—22no—
50Q8×C53—22no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C8×C53—41no—
78#88322—41no—
79C22×C53#12111yescharacteristic
80C22×C53#12311yescharacteristic
81C22×C53#12411yescharacteristic
82C22×C53#13511yescharacteristic
83C22×C53#13511yescharacteristic
84C22×C53#13711yescharacteristic
85C22×C53#13711yescharacteristic
86C4×C53—42no—
87C4×C53—81no—
88C4×C53#12311yescharacteristic
89C4×C53#12311yescharacteristic
90C4×C53#12611yescharacteristic
91C4×C53#12611yescharacteristic
92Dic53—41no—
93Dic53—41no—
94Dic53—81no—
95Dic53—81no—
96#2074—531nomaximal, Sylow
97C2×C53#39711yescharacteristic
98C2×C53#46411yescharacteristic
99C2×C53#46411yescharacteristic
100C2×C53#48511yescharacteristic
101C2×C53#48511yescharacteristic
102C2×C53#48811yescharacteristic
103C2×C53#49111yescharacteristic
104C2×#103—531no—
105C2×#103D5311yescharacteristic
106#391—531no—
107#382—531no—
108#340—531no—
109#340—531no—
110#336—531no—
111C53#207411yescharacteristic, complemented, Sylow
112C22×Q8C2×D5311yescharacteristic
113C22×C8—531no—
114C22×C8C2×D5311yescharacteristic
115C2×C4⋊C4—531no—
116C2×C4⋊C4—531no—
117C2×C4⋊C4C2×D5311yescharacteristic
118C2×C42—531no—
119#103—22no—
120#103—1062no—
121#95—1061no—
122#95—1061no—
123#95—1061no—
124#95—1061no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C22⋊D5312yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—1061no—
136C22×C4—1061no—
137C22×C4C22×D5311yescharacteristic
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—1061no—
142C2×C8—1061no—
143C2×C8—1061no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C42—1061no—
151C42—1061no—
152C23D4×D5311yescharacteristic
153Q8—22no—
154Q8—22no—
155Q8—41no—
156Q8—41no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—1061no—
164C2×C4—1061no—
165C2×C4—1061no—
166C2×C4—1061no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4—2121no—
173C2×C4—2121no—
174C2×C4—2121no—
175C2×C4#1171994511yescharacteristic
176C2×C4#1171994511yescharacteristic
177C2×C4#1171994711yescharacteristic
178C2×C4#1171994711yescharacteristic
179C2×C4#1171994711yescharacteristic
180C2×C4C2×C22⋊D5311yescharacteristic
181C8—41no—
182C8—2121no—
183C22#1313318911yescharacteristic
184C22#1313319911yescharacteristic
185C22#1313320311yescharacteristic
186C22#1313323811yescharacteristic
187C22#1313324011yescharacteristic
188C22#1313324611yescharacteristic
189C22#1313324911yescharacteristic
190C4—42no—
191C4—81no—
192C4—4241no—
193C4—4241no—
194C4—2121no—
195C4—2121no—
196C4#1313319811yescharacteristic
197C4#1313320011yescharacteristic
198C4#1313321011yescharacteristic
199C4#1313321011yescharacteristic
200C2#1681055111yescharacteristic
201C2#1681084311yescharacteristic
202C2#1681084711yescharacteristic
203C2#1681096211yescharacteristic
204C2#1681096311yescharacteristic
205C2#1681097911yescharacteristic
206C2#1681099011yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC2×#103×C53C2
socleC23×C53C2×D4
hypercenterC2×#103D53
hypocenterC53#2074
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
111. C53#2074semidirect