Finite Group Databasebeta

← #20273244

Subgroups — #20273244

Order 6784.

326 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 81 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
2#16810486C211yesmaximal, characteristic, complemented
3#16810357C214yesmaximal, complemented
4C2×#121×C53C211yesmaximal, characteristic
5C2×#11719917—21no—
6C2×C22⋊Dic53—21no—
7C22⋊#88322—21no—
8C22⋊#88322C2212yescomplemented
9#13133124—21no—
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—42no—
42#11719917—42no—
43C2×#88322—22no—
44C23×C53—22no—
45C23×C53—22no—
46C23×C53—22no—
47C23×C53—41no—
48C23×C53—42no—
49C23×C53—42no—
50C23×C53—42no—
51C23×C53C22⋊C411yescharacteristic
52C23×C53C22⋊C411yescharacteristic
53C23×C53C2×D411yescharacteristic
54D4×C53—42no—
55D4×C53—42no—
56D4×C53—44no—
57D4×C53—44no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—42no—
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×Dic53—42no—
71C2×Dic53—42no—
72C2×Dic53—42no—
73#88322—42no—
74C22×C53—21no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53—44no—
82C22×C53—42no—
83C22×C53—42no—
84C22×C53—42no—
85C22×C53—84no—
86C22×C53#9912yes—
87C22×C53C2×C22⋊C411yescharacteristic
88C22×C53AGL(1,ℤ/8)12yes—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—42no—
94Dic53—82no—
95#1510—531nomaximal, Sylow
96C2×C53—22no—
97C2×C53—22no—
98C2×C53—41no—
99C2×C53—82no—
100C2×C53C2×#9911yescharacteristic
101C2×C53#41811yescharacteristic
102C2×C53#42111yescharacteristic
103C2×#121D5311yescharacteristic
104#407—531no—
105#381—531no—
106#327—534no—
107C53#151011yescharacteristic, complemented, Sylow
108C22×D4—21no—
109C22×D4Dic5311yescharacteristic, complemented
110C23×C4C2×D5311yescharacteristic
111C2×M16—1061no—
112#121—21no—
113#121—21no—
114#121Dic5312yescomplemented
115#121C2×D5312yes—
116C2×C4⋊C4Dic5311yescharacteristic, complemented
117C2×C22⋊C4—21no—
118C2×C22⋊C4—1061no—
119C22⋊C8—532no—
120C22⋊C8—1061no—
121#95—532no—
122#95—1061no—
123C24—21no—
124C24C22⋊D5311yescharacteristic
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4C2×Dic5312yes—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—1062no—
134C22×C4C2×Dic5311yescharacteristic
135C22×C4C2×Dic5311yescharacteristic
136C22×C4C22⋊D5311yescharacteristic
137C22×C4C22⋊D5311yescharacteristic
138C22×C4C22⋊D5311yescharacteristic
139C22×C4C22×D5311yescharacteristic
140M16—2122no—
141C2×C8—1062no—
142C4⋊C4—21no—
143C4⋊C4C2×Dic5312yes—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—2122no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23C22⋊Dic5311yescharacteristic
155C23C22⋊Dic5311yescharacteristic
156C23C2×C22⋊D5311yescharacteristic
157D4—42no—
158D4—42no—
159D4—44no—
160D4—44no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4C22⋊Dic5311yescharacteristic
173C2×C4C22⋊Dic5311yescharacteristic
174C2×C4C22×Dic5311yescharacteristic
175C2×C4C2×C22⋊D5311yescharacteristic
176C8—2122no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—84no—
189C22#1313313212yes—
190C22C2×C22⋊Dic5311yescharacteristic
191C22#1313324312yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—4242no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—82no—
202C2C2×#1313313211yescharacteristic
203C2#1681062811yescharacteristic
204C2#1681064011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#121×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#121D53
hypocenterC53#1510
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. #16810486C2semidirect
3. #16810357C2semidirect
8. C22⋊#88322C22semidirect
10. #13133124C22semidirect
12. C22×D4×C53C4semidirect
16. #121×C53C4semidirect
18. C2×C4⋊C4×C53C4semidirect
107. C53#1510semidirect
109. C22×D4Dic53semidirect
114. #121Dic53semidirect
116. C2×C4⋊C4Dic53semidirect