Finite Group Databasebeta

← #20290544

Subgroups — #20290544

Order 6784.

612 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 86 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×#13133276C212yesmaximal, complemented
2C2×#13133275C212yesmaximal, complemented
3#16811566C212yesmaximal, complemented
4C22×#11719945C211yesmaximal, characteristic
5C2×C22⋊#6157C2214yes—
6#13133276—24no—
7#13133275—24no—
8(C2×#6157)⋊C4—24no—
9C2×#11719945—24no—
10C2×#11719945—24no—
11C2×#11719945—24no—
12C22×Dic106C411yescharacteristic, complemented
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C411yescharacteristic, complemented
15C22×C22⋊D53C2211yescharacteristic
16C22×C4×D53C411yescharacteristic, complemented
17C22×C4×D53C2211yescharacteristic
18C23×C4×C53C411yescharacteristic, complemented
19C24×C53C2×C411yescharacteristic
20C23×D53C2×C411yescharacteristic
21C23×D53C2311yescharacteristic
22C22×#6157—24no—
23C22×C4×C53—22no—
24C22×C4×C53—22no—
25C22×C4×C53—22no—
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C2×C22⋊D53—22no—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32C2×C22⋊D53—22no—
33C2×C22⋊D53D414yes—
34C22×Dic53C2×C411yescharacteristic
35C22×Dic53C2×C411yescharacteristic
36#11719945—232no—
37C2×D212—22no—
38C2×D212—22no—
39C2×D212D414yes—
40C2×C4×D53—22no—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53—22no—
44C2×C4×D53—22no—
45C2×C4×D53D414yes—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106—22no—
49C22⋊#6157—24no—
50C22⋊#6157—24no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—42no—
54C23×C53C22⋊C412yes—
55C23×C53C22×C411yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—42no—
59C22×D53—42no—
60C22×D53C22⋊C412yes—
61C22×D53C2×D412yes—
62C22×D53C2×D412yes—
63C22×D53C2×D412yes—
64C2×#6157—48no—
65C2×#6157—44no—
66C2×C4×C53—28no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53C22⋊C412yes—
72C2×C4×C53C22⋊C412yes—
73C22⋊D53—28no—
74C22⋊D53—28no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53C22⋊C412yes—
80C2×Dic53C22⋊C412yes—
81D212—28no—
82C4×D53—28no—
83C4×D53—28no—
84Dic106—28no—
85C22×C53—22no—
86C22×C53—22no—
87C22×C53—42no—
88C22×C53—42no—
89C22×C53—42no—
90C22×C53C2×C22⋊C411yescharacteristic
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53C2×C22⋊C411yescharacteristic
93C2×D53—24no—
94C2×D53—22no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53#12014yes—
101#6157—84no—
102C4×C53—22no—
103C4×C53—22no—
104Dic53—22no—
105Dic53—22no—
106#1782—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—42no—
110C2×C53#37911yescharacteristic
111C2×C53#41012yes—
112D53—24no—
113D53—42no—
114C22×#41—531no—
115C2×#99—532no—
116C2×#99—532no—
117#381—532no—
118C53#178211yescharacteristic, complemented, Sylow
119C2×#41—1064no—
120C2×#41—1064no—
121C2×#41—1064no—
122C22×Q8—531no—
123C22×D4—531no—
124C22×D4—531no—
125C22×D4—531no—
126C23×C4—531no—
127C23×C4—531no—
128C23×C4#615711yescharacteristic, complemented
129C2×C22⋊C4—534no—
130#99—1064no—
131#99—1064no—
132#95—1064no—
133C24—531no—
134C24—531no—
135C24C2×#615711yescharacteristic
136#41—10632no—
137C2×Q8—1062no—
138C2×Q8—1062no—
139C2×Q8—1062no—
140C2×D4—534no—
141C2×D4—534no—
142C2×D4—1062no—
143C2×D4—1062no—
144C2×D4—1062no—
145C2×D4—1062no—
146C2×D4—1062no—
147C2×D4—1062no—
148C2×D4—1062no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—534no—
155C22×C4—1062no—
156C22×C4—1062no—
157C22×C4—1062no—
158C22×C4—1062no—
159C22×C4—1062no—
160C22×C4—1064no—
161C22×C4C2×#615711yescharacteristic
162C22×C4C2×#615711yescharacteristic
163C22⋊C4—1064no—
164C22⋊C4—1064no—
165C23—21no—
166C23—21no—
167C23—42no—
168C23—532no—
169C23—532no—
170C23—532no—
171C23—532no—
172C23—1061no—
173C23—1061no—
174C23—2122no—
175C23—2122no—
176C23C22⋊#615712yes—
177C23C22×#615711yescharacteristic
178Q8—1068no—
179D4—1068no—
180D4—1068no—
181D4—1068no—
182C2×C4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—532no—
188C2×C4—532no—
189C2×C4—1068no—
190C2×C4—1068no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—2124no—
196C2×C4—2128no—
197C2×C4C22⋊#615712yes—
198C2×C4C22⋊#615712yes—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—534no—
205C22—1064no—
206C22—1062no—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22C2×C22⋊#615711yescharacteristic
213C22C2×C22⋊#615711yescharacteristic
214C22C2×C22⋊#615711yescharacteristic
215C4—22no—
216C4—22no—
217C4—1062no—
218C4—1062no—
219C4—4244no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—1064no—
224C2—2122no—
225C2#1681156411yescharacteristic
226C2#1681160212yes—
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC23×C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC23×C4#6157
hypocenterC53#1782
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133276C2semidirect
2. C2×#13133275C2semidirect
3. #16811566C2semidirect
12. C22×Dic106C4semidirect
14. C22×C22⋊D53C4semidirect
16. C22×C4×D53C4semidirect
18. C23×C4×C53C4semidirect
118. C53#1782semidirect
128. C23×C4#6157semidirect