Finite Group Databasebeta

← #20285871

Subgroups — #20285871

Order 6784.

326 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 117 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
2#16811178C211yesmaximal, characteristic
3#16811028C211yesmaximal, characteristic, complemented
4#16811019C211yesmaximal, characteristic, complemented
5#16810973C211yesmaximal, characteristic, complemented
6#16810963C214yesmaximal
7#16810932C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic
9C53×#487C211yesmaximal, characteristic, complemented
10#13133217C2211yescharacteristic
11#13133217C2211yescharacteristic
12#13133208—21no—
13#13133208C2212yes—
14#13133197—21no—
15#13133197—21no—
16#13133193—21no—
17#13133193C2212yes—
18Q8×Dic53—21no—
19Q8×Dic53C2212yes—
20#13133182—21no—
21#13133182—21no—
22#13133175—21no—
23#13133175C2212yes—
24C42×D53C2211yescharacteristic
25#13133151C2214yes—
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—
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53C2311yescharacteristic
40C2×Dic106—21no—
41C2×Dic106—21no—
42C2×C8×C53C2312yes—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53C2312yes—
47C42×C53C2311yescharacteristic
48#11719921—22no—
49#11719921—22no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4⋊Dic53—22no—
53C4⋊Dic53—22no—
54C4⋊Dic53C2312yes—
55C4⋊Dic53C2312yes—
56#11719919—22no—
57#11719919—22no—
58C4×Dic53—22no—
59C4×Dic53—22no—
60C4×Dic53D411yescharacteristic
61C4×Dic53D411yescharacteristic
62C4×Dic53C2311yescharacteristic
63C2×#88322C2312yes—
64C22×D53C2×D411yescharacteristic
65Q8×C53—24no—
66Q8×C53—42no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2411yescharacteristic
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53C2×D411yescharacteristic
75C2×Dic53C2×D411yescharacteristic
76C2×Dic53C2×D411yescharacteristic
77C4×D53—22no—
78C4×D53—22no—
79C4×D53—22no—
80Dic106—42no—
81Dic106—42no—
82C8×C53—22no—
83#88322—22no—
84C22×C53C22×D411yescharacteristic
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—21no—
88C4×C53—42no—
89C4×C53—42no—
90C4×C53#13512yes—
91C4×C53#13712yes—
92C4×C532−1+412yes—
93Dic53—21no—
94Dic53—21no—
95Dic53—21no—
96Dic53—42no—
97Dic53—42no—
98#3231—531nomaximal, Sylow
99C2×C53#53611yescharacteristic
100C2×C53C2×#13511yescharacteristic
101C2×C53C2×#13711yescharacteristic
102D53—41no—
103#533—531no—
104#533—531no—
105#500—531no—
106#498—531no—
107#487—531no—
108#487D5311yescharacteristic, complemented
109#485—534no—
110#479—534no—
111#431—531no—
112C53#323111yescharacteristic, complemented, Sylow
113C4⋊Q8—531no—
114C4⋊Q8—531no—
115C4⋊Q8C2×D5311yescharacteristic
116C4⋊Q8C2×D5311yescharacteristic
117#125—532no—
118#125—1061no—
119C22⋊Q8—532no—
120C22⋊Q8—1061no—
121C22⋊Q8—1061no—
122C22⋊Q8—1061no—
123C4×Q8—532no—
124C4×Q8—1061no—
125C4×Q8—1061no—
126C4×Q8—1061no—
127#117—532no—
128#117—1061no—
129C2×C42—531no—
130#107—534no—
131#106—534no—
132C4⋊C8—532no—
133#103—534no—
134#103C2×D5314yes—
135C22⋊C8—532no—
136C4×C8—531no—
137C4×C8C2×D5311yescharacteristic
138C2×Q8—21no—
139C2×Q8—1061no—
140C2×Q8—1061no—
141C2×Q8C22×D5312yes—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—531no—
145C2×C8—532no—
146C2×C8C22×D5312yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—532no—
151C4⋊C4—532no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4—1062no—
155C4⋊C4—1062no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4C22×D5312yes—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C42—531no—
162C42—531no—
163C42—531no—
164C42—1062no—
165C42—1062no—
166C42C22×D5311yescharacteristic
167C23—531no—
168Q8—24no—
169Q8—42no—
170Q8—2122no—
171Q8—2122no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—531no—
175C2×C4—531no—
176C2×C4—531no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4D4×D5311yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C2×C4C23×D5311yescharacteristic
185C8—22no—
186C8—1062no—
187C22—1061no—
188C22—1061no—
189C22—1061no—
190C22C2×D4×D5311yescharacteristic
191C4—42no—
192C4—42no—
193C4—1061no—
194C4—1061no—
195C4—1061no—
196C4—2122no—
197C4—2122no—
198C4#1313324012yes—
199C4#1313324612yes—
200C4#1313326412yes—
201C2—2121no—
202C2#1681119311yescharacteristic
203C2C2×#1313324011yescharacteristic
204C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3231
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. #16811028C2semidirect
4. #16811019C2semidirect
5. #16810973C2semidirect
7. #16810932C2semidirect
9. C53×#487C2semidirect
108. #487D53semidirect
112. C53#3231semidirect