Finite Group Databasebeta

← #20279881

Subgroups — #20279881

Order 6784.

532 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133209C211yesmaximal, characteristic, complemented
2#16810561C212yesmaximal, complemented
3#16810537C212yesmaximal
4#16810517C214yesmaximal, complemented
5#16810483C212yesmaximal, complemented
6#16810483C212yesmaximal, complemented
7C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
8C2×C4×D4×C53C211yesmaximal, characteristic
9#13133209—24no—
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2214yes—
16C2×C22⋊Dic53C2214yes—
17C2×C4×Dic53C2212yes—
18#13133124C2214yes—
19#13133124C2214yes—
20#13133124C2212yes—
21C22×D4×C53C2211yescharacteristic
22C23×C4×C53C2212yes—
23C4×D4×C53—24no—
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C22⋊C4×C53C2212yes—
26C2×C42×C53C2211yescharacteristic
27C24×C53C2312yes—
28C2×D4×C53—24no—
29C2×D4×C53—22no—
30C22×C4×C53—24no—
31C22×C4×C53—24no—
32C22×C4×C53—24no—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2314yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C4⋊C4×C53—22no—
41C22⋊C4×C53—24no—
42C42×C53D414yes—
43C22⋊Dic53—28no—
44C22⋊Dic53—44no—
45C4⋊Dic53—22no—
46C4⋊Dic53—22no—
47C4⋊Dic53—24no—
48#11719919—24no—
49#11719919—24no—
50C4×Dic53—42no—
51C23×C53—22no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—22no—
57C23×C53#4114yes—
58C23×C53C2411yescharacteristic
59D4×C53—28no—
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—48no—
67C2×C4×C53C2×D412yes—
68C2×C4×C53C2×D412yes—
69C2×C4×C53C2×D412yes—
70C2×C4×C53#4114yes—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—24no—
74C2×Dic53—24no—
75C2×Dic53—24no—
76C2×Dic53—24no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C22×C53—24no—
80C22×C53—24no—
81C22×C53—24no—
82C22×C53—42no—
83C22×C53—42no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53C22×D411yescharacteristic
87C22×C53C2×#4112yes—
88C22×C53C2×#4112yes—
89C22×C53H(5,𝔽2)11yescharacteristic
90C22×C532−1+411yescharacteristic
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—42no—
94C4×C53#12414yes—
95Dic53—42no—
96Dic53—42no—
97Dic53—44no—
98#2562—531nomaximal, Sylow
99C2×C53—44no—
100C2×C53#52511yescharacteristic
101C2×C53C2×#12411yescharacteristic
102C2×C53#53711yescharacteristic
103C2×C53#55212yes—
104C2×C53#55312yes—
105C2×#125—531no—
106C2×C4×D4D5311yescharacteristic
107#400—532no—
108#394—532no—
109#388—534no—
110#380—532no—
111#380—532no—
112C4×C4⋊C4—531no—
113C53#256211yescharacteristic, complemented, Sylow
114C22×D4C2×D5311yescharacteristic
115C23×C4C2×D5312yes—
116#125—1064no—
117C4×D4—24no—
118C2×C4⋊C4—532no—
119C2×C4⋊C4—532no—
120C2×C4⋊C4—531no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4C2×D5311yescharacteristic
124C2×C22⋊C4—534no—
125C2×C22⋊C4—534no—
126C2×C22⋊C4C2×D5312yes—
127C2×C42—532no—
128C2×C42C2×D5311yescharacteristic
129#95—532no—
130#95—534no—
131#95—534no—
132C24C22×D5312yes—
133C2×D4—24no—
134C2×D4—22no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—534no—
139C22×C4—532no—
140C22×C4—532no—
141C22×C4C22×D5312yes—
142C22×C4C22×D5311yescharacteristic
143C22×C4C22×D5311yescharacteristic
144C22×C4C22×D5311yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—1064no—
147C4⋊C4—1064no—
148C4⋊C4—1064no—
149C4⋊C4—1062no—
150C4⋊C4—1062no—
151C22⋊C4—24no—
152C22⋊C4—1068no—
153C22⋊C4—2124no—
154C42—2122no—
155C42C22⋊D5314yes—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23#1171994514yes—
163C23C23×D5311yescharacteristic
164D4—28no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—48no—
172C2×C4—1062no—
173C2×C4—1062no—
174C2×C4—1064no—
175C2×C4—1064no—
176C2×C4—1064no—
177C2×C4—1064no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4#1171994714yes—
181C2×C4C2×C22⋊D5312yes—
182C2×C4C2×C22⋊D5312yes—
183C2×C4C2×C22⋊D5312yes—
184C22—24no—
185C22—24no—
186C22—24no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D5311yescharacteristic
192C22C2×#1171994712yes—
193C22C2×#1171994512yes—
194C22#1313326311yescharacteristic
195C22#1313326411yescharacteristic
196C4—22no—
197C4—22no—
198C4—42no—
199C4—2124no—
200C4—2122no—
201C4—2122no—
202C4#1313320414yes—
203C2—44no—
204C2#1681113411yescharacteristic
205C2C2×#1313320411yescharacteristic
206C2#1681120111yescharacteristic
207C2#1681131312yes—
208C2#1681131912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×D4×C53C2
hypercenterC2×C4×D4D53
hypocenterC53#2562
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133209C2semidirect
2. #16810561C2semidirect
4. #16810517C2semidirect
5. #16810483C2semidirect
6. #16810483C2semidirect
7. C4×C4⋊Dic53C2semidirect
113. C53#2562semidirect