Finite Group Databasebeta

← #20278050

Subgroups — #20278050

Order 6784.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#11719945C211yesmaximal, characteristic
2C2×#13133200C212yesmaximal, complemented
3#16810543C212yesmaximal, complemented
4#16810540C212yesmaximal, complemented
5#16810502C212yesmaximal, complemented
6#16810494C214yesmaximal, complemented
7#16810486C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C53C211yesmaximal, characteristic, complemented
9C2×#11719945—24no—
10C2×#11719945—28no—
11C22×Dic106C2211yescharacteristic
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C4×D53C2212yes—
15#13133200—28no—
16C2×#11719919C2214yes—
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2214yes—
19C2×#11719921C2214yes—
20#13133124C2214yes—
21#13133124C2212yes—
22#13133124C2212yes—
23C23×C4×C53C2211yescharacteristic
24C23×C4×C53C2211yescharacteristic
25C23×C4×C53C2211yescharacteristic
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2212yes—
29C24×C53C2311yescharacteristic
30C23×D53C2312yes—
31C22×C4×C53—22no—
32C22×C4×C53—24no—
33C22×C4×C53—24no—
34C22×C4×C53—22no—
35C22×C4×C53—24no—
36C22×C4×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C2×C22⋊D53—24no—
42C2×C22⋊D53—24no—
43C2×C22⋊D53D418yes—
44C22×Dic53C2314yes—
45C22×Dic53C2312yes—
46#11719945—232no—
47C2×D212—24no—
48C2×D212—22no—
49C2×C4×D53—24no—
50C2×C4×D53—24no—
51C2×C4×D53—24no—
52C2×Dic106—22no—
53C2×Dic106—24no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53—24no—
56C22⋊Dic53—24no—
57#11719921—28no—
58#11719921—28no—
59#11719919—28no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53C2×D414yes—
63C23×C53C2×D412yes—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—24no—
68C22×D53—24no—
69C22×D53C2×D414yes—
70C2×C4×C53—28no—
71C2×C4×C53—28no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—24no—
77C2×C4×C53—22no—
78C2×C4×C53—44no—
79C2×C4×C53C2×D412yes—
80C2×C4×C53C2×D412yes—
81C2×C4×C53#4114yescomplemented
82C22⋊D53—216no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—24no—
86C2×Dic53—24no—
87C2×Dic53—24no—
88C2×Dic53C2×D414yes—
89D212—28no—
90C4×D53—216no—
91Dic106—28no—
92C22×C53—22no—
93C22×C53—22no—
94C22×C53—28no—
95C22×C53#12014yes—
96C22×C53C22×D412yes—
97C22×C53C22×D411yescharacteristic
98C22×C53C2×#4112yes—
99C22×C532−1+412yes—
100C2×D53—24no—
101C2×D53—28no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—44no—
105C4×C53—22no—
106C4×C53—22no—
107C4×C53—24no—
108C4×C53—42no—
109Dic53—24no—
110Dic53—44no—
111#2406—531nomaximal, Sylow
112C2×C53—24no—
113C2×C53C2×#12011yescharacteristic
114C2×C53#52511yescharacteristic
115C2×C53#53611yescharacteristic
116C2×C53#54714yes—
117D53—44no—
118C22×#41—531no—
119C2×#123—532no—
120C22×C4⋊C4D5311yescharacteristic, complemented
121#395—532no—
122#394—532no—
123#386—532no—
124#385—534no—
125#381—531no—
126C53#240611yescharacteristic, complemented, Sylow
127C2×#41—1068no—
128C2×#41—1064no—
129C22×Q8—531no—
130C22×D4—532no—
131C22×D4—531no—
132C23×C4—532no—
133C23×C4C2×D5311yescharacteristic
134C23×C4C2×D5311yescharacteristic
135C23×C4C2×D5311yescharacteristic
136#123—1068no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—534no—
139C2×C4⋊C4C2×D5312yes—
140C2×C4⋊C4C2×D5312yes—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—534no—
143C2×C22⋊C4—534no—
144#95—532no—
145#95—532no—
146#95—534no—
147C24—532no—
148C24C22×D5311yescharacteristic
149#41—10632no—
150C2×Q8—1064no—
151C2×Q8—1062no—
152C2×D4—538no—
153C2×D4—1064no—
154C2×D4—1064no—
155C2×D4—1064no—
156C2×D4—1062no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—532no—
163C22×C4—534no—
164C22×C4—1064no—
165C22×C4—1064no—
166C22×C4—1064no—
167C22×C4C22⋊D5314yes—
168C22×C4C22×D5312yes—
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—1068no—
175C22⋊C4—1068no—
176C22⋊C4—1064no—
177C22⋊C4—1068no—
178C23—22no—
179C23—22no—
180C23—534no—
181C23—1062no—
182C23—1062no—
183C23—1064no—
184C23—1064no—
185C23D4×D5314yes—
186C23C2×C22⋊D5312yes—
187C23C23×D5311yescharacteristic
188Q8—1068no—
189D4—1068no—
190D4—10616no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—534no—
201C2×C4—10616no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1064no—
205C2×C4—1064no—
206C2×C4—1064no—
207C2×C4#1171994514yescomplemented
208C2×C4C2×C22⋊D5312yes—
209C2×C4C2×C22⋊D5312yes—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—1068no—
214C22—1064no—
215C22—2124no—
216C22—2122no—
217C22—2122no—
218C22#1313318514yes—
219C22C2×D4×D5312yes—
220C22C22×C22⋊D5311yescharacteristic
221C22C2×#1171994512yes—
222C22#1313326512yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—1064no—
228C4—2124no—
229C2—24no—
230C2—2124no—
231C2C2×#1313318511yescharacteristic
232C2#1681113411yescharacteristic
233C2#1681119711yescharacteristic
234C2#1681127414yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2406
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#13133200C2semidirect
3. #16810543C2semidirect
4. #16810540C2semidirect
5. #16810502C2semidirect
6. #16810494C2semidirect
7. #16810486C2semidirect
8. C22×C4⋊C4×C53C2semidirect
81. C2×C4×C53#41semidirect
120. C22×C4⋊C4D53semidirect
126. C53#2406semidirect
207. C2×C4#11719945semidirect