Finite Group Databasebeta

← #20285855

Subgroups — #20285855

Order 6784.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811178C211yesmaximal, characteristic
2C4⋊Q8×D53C211yesmaximal, characteristic
3#16811021C211yesmaximal, characteristic, complemented
4#16811019C211yesmaximal, characteristic, complemented
5#16810973C211yesmaximal, characteristic, complemented
6#16810952C214yesmaximal
7#16810906C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic
9C53×#487C211yesmaximal, characteristic, complemented
10C2×Q8×D53—21no—
11#13133217C2211yescharacteristic
12#13133217C2211yescharacteristic
13#13133216—21no—
14#13133216C2212yes—
15#13133215—21no—
16#13133208—21no—
17#13133197—21no—
18#13133193C2212yes—
19Q8×Dic53C2212yes—
20#13133182—21no—
21#13133175—21no—
22C4⋊C4×D53—21no—
23C4⋊C4×D53C2212yes—
24C42×D53C2211yescharacteristic
25#13133149C2214yes—
26#13133148C2214yes—
27#13133146C2212yes—
28#13133141C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32C4⋊Q8×C53C2211yescharacteristic
33#103×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×Q8×C53C2312yes—
37Q8×D53—44no—
38C2×C4×D53—22no—
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×Dic106—21no—
43C2×Dic106—21no—
44C2×Dic106—22no—
45C2×C8×C53C2312yes—
46C4⋊C4×C53—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51#11719921—22no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C4⋊Dic53—22no—
55C4⋊Dic53C2312yes—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65C22×D53C2×D411yescharacteristic
66Q8×C53—24no—
67Q8×C53—42no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2411yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53C2×D411yescharacteristic
76C2×Dic53C2×D411yescharacteristic
77C2×Dic53C2×D411yescharacteristic
78C4×D53—22no—
79C4×D53—22no—
80C4×D53—44no—
81C4×D53SD1614yes—
82Dic106—42no—
83Dic106—42no—
84Dic106—44no—
85C8×C53—22no—
86#88322—22no—
87C22×C53C22×D411yescharacteristic
88C2×D53—21no—
89C2×D53—21no—
90C2×D53C2×SD1612yes—
91C4×C53—42no—
92C4×C53—42no—
93C4×C53C2×SD1612yes—
94C4×C53#13712yes—
95C4×C532−1+412yes—
96Dic53—21no—
97Dic53—21no—
98Dic53—42no—
99Dic53—42no—
100Dic53C2×SD1612yes—
101#3229—531nomaximal, Sylow
102C2×C53#53611yescharacteristic
103C2×C53C22×SD1611yescharacteristic
104C2×C53C2×#13711yescharacteristic
105D53—22no—
106#533—531no—
107C2×C4⋊Q8—531no—
108#498—531no—
109#498—531no—
110#487—531no—
111#487D5311yescharacteristic, complemented
112#483—534no—
113#475—534no—
114#431—531no—
115C53#322911yescharacteristic, complemented, Sylow
116C22×Q8—1061no—
117C4⋊Q8—532no—
118C4⋊Q8—531no—
119C4⋊Q8—531no—
120C4⋊Q8—1061no—
121C4⋊Q8—1061no—
122C4⋊Q8C2×D5311yescharacteristic
123C4⋊Q8C2×D5311yescharacteristic
124#125—1061no—
125C22⋊Q8—532no—
126C22⋊Q8—1061no—
127C4×Q8—532no—
128C4×Q8—1061no—
129#117—1061no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—1061no—
132C2×C42—531no—
133#106—534no—
134#106—534no—
135C4⋊C8—532no—
136#103—534no—
137#103C2×D5314yes—
138C22⋊C8—532no—
139C4×C8—531no—
140C4×C8C2×D5311yescharacteristic
141C2×Q8—21no—
142C2×Q8—1062no—
143C2×Q8—1061no—
144C2×Q8—1061no—
145C2×Q8—2124no—
146C2×Q8C22×D5312yes—
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—1062no—
151C2×C8—532no—
152C2×C8C22×D5312yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—532no—
157C4⋊C4—532no—
158C4⋊C4—1062no—
159C4⋊C4—1062no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—1061no—
163C4⋊C4—1061no—
164C4⋊C4C22×D5312yes—
165C22⋊C4—1062no—
166C42—531no—
167C42—531no—
168C42—531no—
169C42—1062no—
170C42C22×D5311yescharacteristic
171C23—531no—
172Q8—24no—
173Q8—42no—
174Q8—2124no—
175Q8—2122no—
176Q8—2122no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—531no—
180C2×C4—534no—
181C2×C4—531no—
182C2×C4—531no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—2124no—
188C2×C4D4×D5311yescharacteristic
189C2×C4D4×D5311yescharacteristic
190C2×C4C23×D5311yescharacteristic
191C8—22no—
192C8—1062no—
193C22—532no—
194C22—1061no—
195C22—1061no—
196C22C2×D4×D5311yescharacteristic
197C4—42no—
198C4—42no—
199C4—532no—
200C4—1061no—
201C4—1061no—
202C4—2122no—
203C4—2122no—
204C4SD16×D5312yes—
205C4#1313324912yes—
206C4#1313326412yes—
207C2—1062no—
208C2#1681119311yescharacteristic
209C2C2×SD16×D5311yescharacteristic
210C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3229
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #16811021C2semidirect
4. #16811019C2semidirect
5. #16810973C2semidirect
7. #16810906C2semidirect
9. C53×#487C2semidirect
111. #487D53semidirect
115. C53#3229semidirect