Finite Group Databasebeta

← #20289999

Subgroups — #20289999

Order 6784.

214 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811643C211yesmaximal, characteristic, complemented
2#16811639C211yesmaximal, characteristic, complemented
3#16811468C212yesmaximal, complemented
4#16811467C212yesmaximal
5C2×#13133212C211yesmaximal, characteristic
6#13133284—21no—
7C4⋊#88324—21no—
8C22⋊#88324C2212yes—
9C22⋊#88324C2212yes—
10#13133272—21no—
11C4×#88324—21no—
12#13133212—21no—
13#13133212—21no—
14#13133212—21no—
15#13133212—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C411yescharacteristic
18C2×#11719921C411yescharacteristic
19C2×#11719921C411yescharacteristic
20C2×#11719921C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22C2×C4⋊C4×C53C411yescharacteristic
23C23×D53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×Dic53C2×C411yescharacteristic
28C22×Dic53C2×C411yescharacteristic
29C22×Dic53C2311yescharacteristic
30C2×#88324—22no—
31C2×#88324—22no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34#11719921—21no—
35#11719921—21no—
36#11719921—21no—
37#11719921—21no—
38#11719921—21no—
39#11719921—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42#11719919—21no—
43#11719919—21no—
44C4×Dic53D411yescharacteristic
45C4×Dic53D411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C23×C53C22×C411yescharacteristic
49C22×D53—21no—
50C22×D53—22no—
51C22×D53—41no—
52C22×D53—41no—
53C2×C4×C53—22no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—22no—
57C2×C4×C53—41no—
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22⋊C411yescharacteristic
60C2×Dic53—21no—
61C2×Dic53—41no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53C22⋊C411yescharacteristic
65C2×Dic53C22⋊C411yescharacteristic
66C2×Dic53C2×D411yescharacteristic
67C2×Dic53C2×D411yescharacteristic
68#88324—42no—
69#88324—42no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53C2×C22⋊C411yescharacteristic
73C22×C53#13112yes—
74C2×D53—41no—
75C2×D53—41no—
76C2×D53—41no—
77C2×D53—82no—
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—41no—
81C4×C53—41no—
82Dic53—21no—
83Dic53—21no—
84Dic53—21no—
85Dic53—21no—
86Dic53—41no—
87#1473—531nomaximal, Sylow
88C2×C53—22no—
89C2×C53#40811yescharacteristic
90C2×C53#41011yescharacteristic
91C2×C53#41311yescharacteristic
92D53—81no—
93C2×#126—531no—
94#432—531no—
95#431—531no—
96#324—532no—
97#323—532no—
98C53#147311yescharacteristic, complemented, Sylow
99#126—1061no—
100#126—1061no—
101#126—1061no—
102#126—1061no—
103C2×C4⋊C4—531no—
104C2×C4⋊C4—531no—
105C2×C4⋊C4#615711yescharacteristic
106C2×C22⋊C4—531no—
107C2×C22⋊C4—531no—
108C2×C22⋊C4—531no—
109C2×C42—531no—
110C4⋊C8—1061no—
111C4⋊C8—1061no—
112C22⋊C8—532no—
113C22⋊C8—532no—
114#97—1061no—
115C4×C8—1061no—
116C24—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—531no—
120C22×C4C2×#615711yescharacteristic
121C22×C4C2×#615711yescharacteristic
122C22×C4C2×#615711yescharacteristic
123C2×C8—1062no—
124C2×C8—1062no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C22⋊C4—1061no—
132C22⋊C4—1061no—
133C22⋊C4—1061no—
134C22⋊C4—1061no—
135C22⋊C4—1061no—
136C22⋊C4—1061no—
137C42—531no—
138C42—531no—
139C42—531no—
140C42—531no—
141C23—1062no—
142C23—1061no—
143C23—2121no—
144C23—2121no—
145C23C22×#615711yescharacteristic
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—531no—
152C2×C4—531no—
153C2×C4—531no—
154C2×C4—531no—
155C2×C4—1061no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4C22⋊#615711yescharacteristic
160C2×C4C22⋊#615711yescharacteristic
161C8—2122no—
162C8—2122no—
163C22—21no—
164C22—21no—
165C22—4242no—
166C22—2121no—
167C22—2121no—
168C22—2121no—
169C22C2×C22⋊#615711yescharacteristic
170C22#1313330412yes—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—1061no—
176C4—1061no—
177C4—1061no—
178C4—1061no—
179C4—2121no—
180C2—22no—
181C2—4241no—
182C2#1681159611yescharacteristic
183C2#1681160411yescharacteristic
184C2#1681160911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1473
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #16811643C2semidirect
2. #16811639C2semidirect
3. #16811468C2semidirect
98. C53#1473semidirect