Finite Group Databasebeta

← #20290111

Subgroups — #20290111

Order 6784.

346 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 64 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#16811594C211yesmaximal, characteristic, complemented
2C4×C22⋊#6157C211yesmaximal, characteristic, complemented
3#16811474C214yesmaximal, complemented
4C2×#13133188C211yesmaximal, characteristic
5C2×#11719936—21no—
6C2×C22⋊#6157—21no—
7C2×C4×#6157—21no—
8#13133274C2212yescomplemented
9C22⋊#88324—21no—
10(C2×#6157)⋊C4C2212yescomplemented
11C22×C22⋊D53—21no—
12C22×C4×D53C2211yescharacteristic
13#13133188—21no—
14#13133188—21no—
15#13133188C412yescomplemented
16#13133188C2212yes—
17C2×C4⋊Dic53C411yescharacteristic, complemented
18C2×C22⋊Dic53—21no—
19C22×D4×C53C411yescharacteristic, complemented
20C24×C53—21no—
21C23×D53D411yescharacteristic
22C22×#6157—22no—
23C2×D4×C53—21no—
24C2×D4×C53—42no—
25C2×D4×C53C2×C412yes—
26C22×C4×C53C2×C411yescharacteristic
27C2×C22⋊D53—22no—
28C2×C22⋊D53—22no—
29C2×C22⋊D53—44no—
30C22×Dic53—21no—
31C22×Dic53D411yescharacteristic
32C2×C4×D53—42no—
33C2×C4×D53C2×C411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53C2311yescharacteristic
37C22⋊#6157—24no—
38C2×#88324—22no—
39C4×#6157—42no—
40#11719936—42no—
41C22⋊Dic53—22no—
42C22⋊Dic53—22no—
43C4⋊Dic53—21no—
44C4⋊Dic53C2×C412yes—
45C23×C53—22no—
46C23×C53—42no—
47C23×C53—42no—
48C23×C53—42no—
49C23×C53C22⋊C411yescharacteristic
50C22×D53—22no—
51C22×D53—22no—
52C22×D53—41no—
53C22×D53C22⋊C411yescharacteristic
54C22×D53C2×D411yescharacteristic
55C2×#6157—24no—
56C2×#6157—42no—
57C2×#6157—42no—
58D4×C53—42no—
59D4×C53—42no—
60C2×C4×C53—22no—
61C2×C4×C53C22⋊C411yescharacteristic
62C2×C4×C53C22×C411yescharacteristic
63C22⋊D53—44no—
64C22⋊D53—44no—
65C2×Dic53—22no—
66C2×Dic53—41no—
67C2×Dic53—42no—
68C2×Dic53C22⋊C411yescharacteristic
69C2×Dic53C2×D411yescharacteristic
70C4×D53—22no—
71C4×D53—22no—
72C4×D53—41no—
73C4×D53—41no—
74#88324—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—42no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—84no—
81C22×C53#10412yes—
82C22×C53C2×C22⋊C411yescharacteristic
83C2×D53—21no—
84C2×D53—21no—
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—21no—
88C2×D53—44no—
89C2×D53AGL(1,ℤ/8)12yes—
90#6157—44no—
91C4×C53—21no—
92C4×C53#9912yes—
93Dic53—21no—
94Dic53—21no—
95Dic53—42no—
96#1507—531nomaximal, Sylow
97C2×C53—22no—
98C2×C53—82no—
99C2×C53C2×#9911yescharacteristic
100C2×C53#41811yescharacteristic
101C2×C53C2×#10411yescharacteristic
102D53—22no—
103D53—41no—
104C2×#121—531no—
105#407—531no—
106C4×C22⋊C4—531no—
107#327—534no—
108C53#150711yescharacteristic, complemented, Sylow
109C22×D4—1061no—
110C22×D4#615711yescharacteristic, complemented
111C23×C4—531no—
112C2×M16—1061no—
113#121—532no—
114#121—532no—
115#121—1061no—
116#121—1061no—
117C2×C4⋊C4—531no—
118C2×C22⋊C4—1061no—
119C2×C22⋊C4—1061no—
120C2×C42—1061no—
121C22⋊C8—532no—
122C22⋊C8—1061no—
123#95—532no—
124C24—21no—
125C24—531no—
126C2×D4—21no—
127C2×D4—42no—
128C2×D4—1062no—
129C2×D4—1062no—
130C2×D4—2124no—
131C2×D4C2×#615712yes—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—1062no—
138C22×C4—1061no—
139C22×C4—2122no—
140C22×C4C2×#615711yescharacteristic
141M16—2122no—
142C2×C8—1062no—
143C4⋊C4—532no—
144C4⋊C4—1061no—
145C22⋊C4—1062no—
146C22⋊C4—1062no—
147C22⋊C4—1064no—
148C42—2122no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—531no—
154C23—531no—
155C23—1062no—
156C23—1062no—
157C23—2121no—
158C23C22⋊#615711yescharacteristic
159D4—42no—
160D4—42no—
161D4—2124no—
162D4—2124no—
163C2×C4—22no—
164C2×C4—531no—
165C2×C4—531no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1064no—
169C2×C4—1062no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4—2121no—
173C2×C4—2122no—
174C2×C4—2122no—
175C2×C4—2122no—
176C2×C4C22⋊#615711yescharacteristic
177C2×C4C22×#615711yescharacteristic
178C8—2122no—
179C22—21no—
180C22—21no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22—532no—
186C22—1061no—
187C22—1061no—
188C22—1061no—
189C22—1061no—
190C22—1061no—
191C22—2124no—
192C22#1313328112yes—
193C22C2×C22⋊#615711yescharacteristic
194C4—21no—
195C4—1061no—
196C4—1061no—
197C4—2124no—
198C4—2122no—
199C4#1313327512yes—
200C2—22no—
201C2—82no—
202C2—1062no—
203C2—2121no—
204C2C2×#1313327511yescharacteristic
205C2#1681161511yescharacteristic
206C2C2×#1313328111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×D4#6157
hypocenterC53#1507
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #16811594C2semidirect
2. C4×C22⋊#6157C2semidirect
3. #16811474C2semidirect
8. #13133274C22semidirect
10. (C2×#6157)⋊C4C22semidirect
15. #13133188C4semidirect
17. C2×C4⋊Dic53C4semidirect
19. C22×D4×C53C4semidirect
108. C53#1507semidirect
110. C22×D4#6157semidirect