Finite Group Databasebeta

← #20273231

Subgroups — #20273231

Order 6784.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 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#16810587C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
3#16810357C214yesmaximal, complemented
4C2×#121×C53C211yesmaximal, characteristic
5C2×#11719917—21no—
6C2×C22⋊Dic53—21no—
7C2×C4×Dic53—21no—
8C22⋊#88322—21no—
9C22⋊#88322C2212yescomplemented
10#13133124C2212yescomplemented
11C22×D4×C53—21no—
12C22×D4×C53C411yescharacteristic, complemented
13C23×C4×C53C2211yescharacteristic
14#121×C53—21no—
15#121×C53—21no—
16#121×C53C412yescomplemented
17#121×C53C2212yes—
18C2×C4⋊C4×C53C411yescharacteristic, complemented
19C2×C22⋊C4×C53—21no—
20C24×C53—21no—
21C24×C53D411yescharacteristic
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53—21no—
25C2×D4×C53—42no—
26C2×D4×C53—44no—
27C2×D4×C53C2×C412yes—
28C22×C4×C53—21no—
29C22×C4×C53—42no—
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53C2×C411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53—22no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53C2×C412yes—
39C22⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊Dic53—24no—
42C4×Dic53—42no—
43#11719917—42no—
44C2×#88322—22no—
45C23×C53—22no—
46C23×C53—22no—
47C23×C53—22no—
48C23×C53—41no—
49C23×C53—42no—
50C23×C53—42no—
51C23×C53—42no—
52C23×C53C22⋊C411yescharacteristic
53C23×C53C22⋊C411yescharacteristic
54C23×C53C2×D411yescharacteristic
55D4×C53—42no—
56D4×C53—44no—
57D4×C53—44no—
58D4×C53—42no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—42no—
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22×C411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×Dic53—24no—
72C2×Dic53—42no—
73C2×Dic53—42no—
74#88322—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53—44no—
83C22×C53—42no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53—84no—
87C22×C53#10412yes—
88C22×C53C2×C22⋊C411yescharacteristic
89C22×C53AGL(1,ℤ/8)12yes—
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—42no—
94C4×C53#9912yes—
95Dic53—44no—
96#1507—531nomaximal, Sylow
97C2×C53—22no—
98C2×C53—22no—
99C2×C53—41no—
100C2×C53—82no—
101C2×C53C2×#9911yescharacteristic
102C2×C53#41811yescharacteristic
103C2×C53C2×#10411yescharacteristic
104C2×#121D5311yescharacteristic
105#407—531no—
106C4×C22⋊C4—531no—
107#327—534no—
108C53#150711yescharacteristic, complemented, Sylow
109C22×D4—21no—
110C22×D4Dic5311yescharacteristic, complemented
111C23×C4C2×D5311yescharacteristic
112C2×M16—1061no—
113#121—21no—
114#121—21no—
115#121Dic5312yescomplemented
116#121C2×D5312yes—
117C2×C4⋊C4Dic5311yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—1061no—
120C2×C42—1061no—
121C22⋊C8—532no—
122C22⋊C8—1061no—
123#95—532no—
124C24—21no—
125C24C22⋊D5311yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×Dic5312yes—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—1062no—
135C22×C4C2×Dic5311yescharacteristic
136C22×C4C2×Dic5311yescharacteristic
137C22×C4C22⋊D5311yescharacteristic
138C22×C4C22⋊D5311yescharacteristic
139C22×C4C22⋊D5311yescharacteristic
140C22×C4C22×D5311yescharacteristic
141M16—2122no—
142C2×C8—1062no—
143C4⋊C4—21no—
144C4⋊C4C2×Dic5312yes—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—1064no—
148C42—2122no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊Dic5311yescharacteristic
157C23C22⋊Dic5311yescharacteristic
158C23C2×C22⋊D5311yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—1064no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4C22⋊Dic5311yescharacteristic
175C2×C4C22⋊Dic5311yescharacteristic
176C2×C4C22×Dic5311yescharacteristic
177C2×C4C2×C22⋊D5311yescharacteristic
178C8—2122no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22#1313314412yes—
192C22C2×C22⋊Dic5311yescharacteristic
193C22#1313324312yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—2124no—
199C4#1313313212yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×#1313313211yescharacteristic
205C2#1681062811yescharacteristic
206C2C2×#1313314411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#121×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#121D53
hypocenterC53#1507
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #16810587C2semidirect
2. C4×C22⋊Dic53C2semidirect
3. #16810357C2semidirect
9. C22⋊#88322C22semidirect
10. #13133124C22semidirect
12. C22×D4×C53C4semidirect
16. #121×C53C4semidirect
18. C2×C4⋊C4×C53C4semidirect
108. C53#1507semidirect
110. C22×D4Dic53semidirect
115. #121Dic53semidirect
117. C2×C4⋊C4Dic53semidirect