Finite Group Databasebeta

← #20289972

Subgroups — #20289972

Order 6784.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#16811594C212yesmaximal, complemented
2#16811468C212yesmaximal, complemented
3#16811467C212yesmaximal, complemented
4C2×C4×D212C211yesmaximal, characteristic
5C2×#11719936—22no—
6#13133274—22no—
7C22⋊#88324C2214yes—
8C22×D212C411yescharacteristic
9C22×C4×D53C411yescharacteristic
10C22×C4×D53C2211yescharacteristic
11C4×D212—22no—
12C4×D212—22no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719921C411yescharacteristic
15C2×#11719921C2211yescharacteristic
16C2×C42×C53C411yescharacteristic
17C23×D53C2×C411yescharacteristic
18C23×D53C2×C411yescharacteristic
19C22×C4×C53C2×C411yescharacteristic
20C22×C4×C53C2×C411yescharacteristic
21C22×C4×C53C2×C411yescharacteristic
22C22×Dic53C2×C411yescharacteristic
23C22×Dic53C2311yescharacteristic
24C2×D212—22no—
25C2×D212—22no—
26C2×D212—22no—
27C2×C4×D53—21no—
28C2×C4×D53—21no—
29C2×C4×D53—42no—
30C2×C4×D53—42no—
31C2×C4×D53D412yes—
32C2×C4×D53D412yes—
33C2×#88324—24no—
34#11719936—44no—
35C42×C53—21no—
36C42×C53—21no—
37#11719921—22no—
38#11719921—22no—
39C4⋊Dic53—22no—
40C23×C53C22×C411yescharacteristic
41C22×D53—21no—
42C22×D53—21no—
43C22×D53—21no—
44C22×D53—21no—
45C22×D53—41no—
46C22×D53—41no—
47C22×D53—41no—
48C22×D53—41no—
49C22×D53C22⋊C412yes—
50C22×D53M1612yescomplemented
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—22no—
54C2×C4×C53—21no—
55C2×C4×C53—41no—
56C2×C4×C53C22⋊C411yescharacteristic
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53M1612yescomplemented
59C2×Dic53—21no—
60C2×Dic53—41no—
61C2×Dic53—41no—
62C2×Dic53C2×D412yes—
63D212—28no—
64C4×D53—24no—
65C4×D53—42no—
66C4×D53—42no—
67C4×D53—42no—
68#88324—44no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53C2×C22⋊C411yescharacteristic
72C22×C53C2×M1611yescharacteristic
73C22×C53C2×M1611yescharacteristic
74C2×D53—22no—
75C2×D53—22no—
76C2×D53—22no—
77C2×D53—22no—
78C2×D53—42no—
79C2×D53—42no—
80C2×D53—42no—
81C2×D53—41no—
82C2×D53—41no—
83C2×D53—42no—
84C4×C53—22no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53#9912yes—
88Dic53—22no—
89Dic53—41no—
90#1465—531nomaximal, Sylow
91C2×C53—21no—
92C2×C53—21no—
93C2×C53#40711yescharacteristic
94C2×C53C2×#9911yescharacteristic
95C2×C53#41311yescharacteristic
96D53—42no—
97D53—42no—
98C2×C4×D4—531no—
99#407—532no—
100#324—532no—
101#323—532no—
102C53#146511yescharacteristic, complemented, Sylow
103C22×D4—531no—
104C23×C4—531no—
105C23×C4—531no—
106C2×M16—1062no—
107C4×D4—1062no—
108C4×D4—1062no—
109C2×C4⋊C4—531no—
110C2×C22⋊C4—531no—
111C2×C22⋊C4—531no—
112C2×C42#615711yescharacteristic
113C22⋊C8—534no—
114C22⋊C8—1062no—
115C24—531no—
116C24—531no—
117C2×D4—1062no—
118C2×D4—1062no—
119C2×D4—1062no—
120C22×C4—531no—
121C22×C4—532no—
122C22×C4—532no—
123C22×C4—531no—
124C22×C4—1061no—
125C22×C4—1061no—
126C22×C4—2122no—
127C22×C4—2122no—
128C22×C4C2×#615711yescharacteristic
129C22×C4C2×#615711yescharacteristic
130C22×C4C2×#615711yescharacteristic
131M16—2124no—
132C2×C8—1064no—
133C4⋊C4—1062no—
134C22⋊C4—1062no—
135C22⋊C4—1062no—
136C42—21no—
137C42—21no—
138C23—532no—
139C23—532no—
140C23—1061no—
141C23—1061no—
142C23—1061no—
143C23—1061no—
144C23—2121no—
145C23—2121no—
146C23—2121no—
147C23—2121no—
148C23C22×#615711yescharacteristic
149D4—1068no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—532no—
156C2×C4—1064no—
157C2×C4—1061no—
158C2×C4—2122no—
159C2×C4—2122no—
160C2×C4—2121no—
161C2×C4—2122no—
162C2×C4—2121no—
163C2×C4#1171993612yescomplemented
164C2×C4C22⋊#615711yescharacteristic
165C2×C4C22⋊#615711yescharacteristic
166C8—2124no—
167C22—21no—
168C22—21no—
169C22—1062no—
170C22—1062no—
171C22—1062no—
172C22—1062no—
173C22—2122no—
174C22—2122no—
175C22—2122no—
176C22—2122no—
177C22—2121no—
178C22—2121no—
179C22C2×C22⋊#615711yescharacteristic
180C22D53⋊M1611yescharacteristic
181C22C2×#1171993611yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—1062no—
186C4—2121no—
187C4#1313327612yes—
188C2—21no—
189C2—21no—
190C2—2122no—
191C2—2122no—
192C2#1681159411yescharacteristic
193C2C2×#1313327611yescharacteristic
194C2#1681160911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C42×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C42#6157
hypocenterC53#1465
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #16811594C2semidirect
2. #16811468C2semidirect
3. #16811467C2semidirect
50. C22×D53M16semidirect
58. C2×C4×C53M16semidirect
102. C53#1465semidirect
163. C2×C4#11719936semidirect