Finite Group Databasebeta

← #20290695

Subgroups — #20290695

Order 6784.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#11719936C211yesmaximal, characteristic, complemented
2C2×#13133274C212yesmaximal, complemented
3C2×#13133274C211yesmaximal, characteristic, complemented
4#16811489C212yesmaximal, complemented
5C2×#13133179C211yesmaximal, characteristic
6C2×#11719936—22no—
7C2×#11719936—24no—
8C22×#88324C2212yes—
9C22×#88324C2212yes—
10#13133274—22no—
11#13133274—24no—
12C22×D212C411yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133179—22no—
16#13133179—22no—
17C2×#11719921C411yescharacteristic
18C2×#11719921C411yescharacteristic
19C2×C4×Dic53C2211yescharacteristic
20C2×C4⋊C4×C53C411yescharacteristic
21C23×D53C2×C411yescharacteristic
22C23×D53C2×C411yescharacteristic
23C22×C4×C53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C22×Dic53C2×C411yescharacteristic
27C22×Dic53C2311yescharacteristic
28C2×D212—22no—
29C2×D212—22no—
30C2×D212—22no—
31C2×C4×D53—22no—
32C2×C4×D53—22no—
33C2×C4×D53—21no—
34C2×C4×D53—21no—
35C2×C4×D53—22no—
36C2×C4×D53—22no—
37C2×C4×D53D412yes—
38C2×C4×D53D412yes—
39C2×#88324—22no—
40C2×#88324—22no—
41C2×#88324—22no—
42C2×#88324—22no—
43C2×#88324—22no—
44C2×#88324D414yescomplemented
45#11719936—28no—
46C4⋊C4×C53—22no—
47#11719921—22no—
48#11719921—22no—
49C4×Dic53—21no—
50C4×Dic53—21no—
51C23×C53C22×C411yescharacteristic
52C22×D53—21no—
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—21no—
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—22no—
59C22×D53—22no—
60C22×D53C22⋊C412yes—
61C22×D53M1612yescomplemented
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53C22⋊C412yes—
68C2×C4×C53M1612yescomplemented
69C2×Dic53—21no—
70C2×Dic53—22no—
71C2×Dic53C2×D412yes—
72C2×Dic53C2×D411yescharacteristic
73C2×Dic53C2×D411yescharacteristic
74C2×Dic53#4112yes—
75D212—28no—
76C4×D53—22no—
77C4×D53—24no—
78C4×D53—22no—
79C4×D53—41no—
80C4×D53—42no—
81C4×D53—41no—
82#88324—24no—
83#88324—42no—
84C22×C53C2×C22⋊C411yescharacteristic
85C22×C53C4×D412yes—
86C22×C53C2×M1611yescharacteristic
87C22×C53C2×M1611yescharacteristic
88C22×C53#13112yes—
89C2×D53—22no—
90C2×D53—22no—
91C2×D53—22no—
92C2×D53—22no—
93C2×D53—24no—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—41no—
102Dic53—41no—
103Dic53#12011yescharacteristic
104Dic53#12112yes—
105Dic53#12311yescharacteristic
106#1870—531nomaximal, Sylow
107C2×C53#38611yescharacteristic
108C2×C53#40711yescharacteristic
109C2×C53#40811yescharacteristic
110C2×C53#43512yes—
111C2×C53#43612yes—
112D53—42no—
113D53—42no—
114C22×M16—531no—
115C2×C4×D4—531no—
116C2×C22⋊C8—532no—
117C2×C22⋊C8—531no—
118#336—532no—
119C53#187011yescharacteristic, complemented, Sylow
120C22×D4—531no—
121C23×C4—531no—
122C23×C4—531no—
123C2×M16—1064no—
124C2×M16—1062no—
125C22×C8—532no—
126C22×C8—532no—
127C4×D4—1062no—
128C4×D4—1062no—
129C2×C4⋊C4#615711yescharacteristic
130C2×C22⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C42—531no—
133C22⋊C8—1064no—
134C22⋊C8—1062no—
135C24—531no—
136C24—531no—
137C2×D4—1062no—
138C2×D4—1062no—
139C2×D4—1062no—
140C22×C4—531no—
141C22×C4—532no—
142C22×C4—532no—
143C22×C4—531no—
144C22×C4—1062no—
145C22×C4—1062no—
146C22×C4—1062no—
147C22×C4—1061no—
148C22×C4—1061no—
149C22×C4—1062no—
150C22×C4C2×#615711yescharacteristic
151C22×C4C2×#615711yescharacteristic
152C22×C4C2×#615711yescharacteristic
153M16—1068no—
154C2×C8—534no—
155C2×C8—1062no—
156C2×C8—1062no—
157C2×C8—1062no—
158C2×C8—1062no—
159C2×C8—1062no—
160C4⋊C4—22no—
161C22⋊C4—1062no—
162C22⋊C4—1062no—
163C42—1061no—
164C42—1061no—
165C23—532no—
166C23—532no—
167C23—1061no—
168C23—1062no—
169C23—1062no—
170C23—1061no—
171C23—1061no—
172C23—1061no—
173C23—1062no—
174C23—1062no—
175C23C22×#615711yescharacteristic
176D4—1068no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—532no—
185C2×C4—532no—
186C2×C4—1062no—
187C2×C4—1064no—
188C2×C4—1062no—
189C2×C4—1061no—
190C2×C4—1062no—
191C2×C4—2121no—
192C2×C4—2122no—
193C2×C4—2121no—
194C2×C4#1171993612yescomplemented
195C2×C4C22⋊#615712yes—
196C8—1064no—
197C8—2122no—
198C22—1062no—
199C22—1062no—
200C22—1062no—
201C22—1062no—
202C22—1064no—
203C22—2121no—
204C22—2121no—
205C22—2122no—
206C22—2122no—
207C22—2122no—
208C22C2×C22⋊#615711yescharacteristic
209C22D4×#615712yes—
210C22D53⋊M1611yescharacteristic
211C22C2×#1171993611yescharacteristic
212C22#1313330512yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—531no—
217C4—532no—
218C4—531no—
219C4—2121no—
220C2—2122no—
221C2—2122no—
222C2#1681157111yescharacteristic
223C2#1681159411yescharacteristic
224C2#1681159811yescharacteristic
225C2#1681165312yes—
226C2#1681165712yes—
subgroupG/H
center, FrattiniC23C22×#6157
commutator, socleC23×C53C22×C4
FittingC2×C4⋊C4×C53C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1870
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#11719936C2semidirect
2. C2×#13133274C2semidirect
3. C2×#13133274C2semidirect
4. #16811489C2semidirect
44. C2×#88324D4semidirect
61. C22×D53M16semidirect
68. C2×C4×C53M16semidirect
119. C53#1870semidirect
194. C2×C4#11719936semidirect