Finite Group Databasebeta

← #20280084

Subgroups — #20280084

Order 6784.

484 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 161 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×#13133182C212yesmaximal
2#16810556C212yesmaximal
3#16810553C211yesmaximal, characteristic
4#16810527C212yesmaximal
5#16810527C212yesmaximal
6#16810520C212yesmaximal
7#16810518C212yesmaximal
8#16810492C211yesmaximal, characteristic
9C53×#398C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11#13133182—28no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C4×Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×Dic106—24no—
42C2×Dic106—24no—
43C2×Dic106Q818yescomplemented
44C4⋊C4×C53—24no—
45C4⋊C4×C53—41no—
46C4⋊C4×C53—41no—
47C4⋊Dic53—41no—
48C4⋊Dic53—41no—
49#11719919—24no—
50#11719919—24no—
51#11719919—24no—
52#11719919—24no—
53#11719919—42no—
54C4×Dic53—24no—
55C4×Dic53—24no—
56C4×Dic53D414yes—
57C23×C53C2411yescharacteristic
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×Q814yes—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53C2×D414yes—
80C2×Dic53C2×Q814yes—
81C2×Dic53C2×Q814yes—
82C2×Dic53#4114yes—
83Dic106—216no—
84C22×C53C22×D411yescharacteristic
85C22×C53C22×Q812yes—
86C22×C53C2×#4112yes—
87C22×C53H(5,𝔽2)11yescharacteristic
88C22×C532−1+411yescharacteristic
89C4×C53—24no—
90C4×C53—22no—
91C4×C53—42no—
92C4×C53—41no—
93C4×C53—41no—
94Dic53—24no—
95Dic53—42no—
96Dic53—42no—
97Dic53C22⋊Q818yes—
98#2582—531nomaximal, Sylow
99C2×C53C2×C22⋊Q812yes—
100C2×C53#53711yescharacteristic
101C2×C53#55712yes—
102C2×C53Q8211yescharacteristic
103C2×C53#55911yescharacteristic
104C2×C4×Q8—532no—
105#398—531no—
106#398—532no—
107#398D5311yescharacteristic
108#391—532no—
109#391—532no—
110#389—532no—
111#389—532no—
112#384—531no—
113C53#258211yescharacteristic, complemented, Sylow
114C22×Q8—532no—
115C4×Q8—1068no—
116C2×C4⋊C4—532no—
117C2×C4⋊C4—532no—
118C2×C4⋊C4—532no—
119C2×C4⋊C4—532no—
120C2×C4⋊C4—532no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—531no—
123C2×C4⋊C4C2×D5312yes—
124C2×C4⋊C4C2×D5311yescharacteristic
125C2×C4⋊C4C2×D5311yescharacteristic
126C2×C42—531no—
127C2×C42—532no—
128C2×C42—532no—
129#95—532no—
130#95—532no—
131#95—532no—
132#95—531no—
133#95—532no—
134#95C2×D5312yes—
135#95C2×D5311yescharacteristic
136C2×Q8—538no—
137C2×Q8—1064no—
138C2×Q8—1064no—
139C22×C4—532no—
140C22×C4—532no—
141C22×C4—532no—
142C22×C4—532no—
143C22×C4C22×D5311yescharacteristic
144C22×C4C22×D5311yescharacteristic
145C22×C4C22×D5312yes—
146C22×C4C22×D5311yescharacteristic
147C22×C4C22×D5312yes—
148C4⋊C4—24no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—1064no—
152C4⋊C4—1064no—
153C4⋊C4—1064no—
154C4⋊C4—1064no—
155C4⋊C4—2122no—
156C4⋊C4—2121no—
157C4⋊C4—2121no—
158C42—534no—
159C42—1064no—
160C42—1064no—
161C23C23×D5311yescharacteristic
162Q8—10616no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—534no—
175C2×C4—534no—
176C2×C4—534no—
177C2×C4—534no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4D4×D5312yes—
187C2×C4Q8×D5314yes—
188C22C2×D4×D5311yescharacteristic
189C22C2×Q8×D5312yes—
190C22#41×D5312yes—
191C22#1313326211yescharacteristic
192C22#1313326411yescharacteristic
193C4—24no—
194C4—22no—
195C4—41no—
196C4—41no—
197C4—42no—
198C4—538no—
199C4—1064no—
200C4—2122no—
201C4—2122no—
202C2C22⋊Q8×D5312yes—
203C2#1681120011yescharacteristic
204C2#1681135012yes—
205C2#1681135311yescharacteristic
206C2#1681135711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2582
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
43. C2×Dic106Q8semidirect
113. C53#2582semidirect