Finite Group Databasebeta

← #20290123

Subgroups — #20290123

Order 6784.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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#16811595C211yesmaximal, characteristic
2#16811566C211yesmaximal, characteristic
3#16811476C214yesmaximal, complemented
4C2×#13133197C211yesmaximal, characteristic
5C2×#11719940—21no—
6C2×C22⋊#6157—21no—
7#13133274—21no—
8#13133274C2212yes—
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4C2212yes—
11C22×Dic106C411yescharacteristic, complemented
12C22×C4×D53C2211yescharacteristic
13#13133197—21no—
14#13133197—21no—
15#13133197C412yescomplemented
16#13133197C2212yes—
17C2×C4⋊Dic53—21no—
18C2×#11719921—21no—
19C2×C4⋊C4×C53C411yescharacteristic, complemented
20C23×D53D411yescharacteristic
21C22×#6157—22no—
22C22×C4×C53—21no—
23C22×C4×C53C2×C411yescharacteristic
24C22×Dic53—21no—
25C22×Dic53D411yescharacteristic
26C2×C4×D53—42no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53C2311yescharacteristic
31C2×Dic106—21no—
32C2×Dic106—42no—
33C2×Dic106C2×C412yes—
34C22⋊#6157—42no—
35#11719940—42no—
36C2×#88324—22no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53C2×C412yes—
39#11719921—22no—
40#11719921—22no—
41C4⋊Dic53—22no—
42C4⋊Dic53—22no—
43C23×C53C22⋊C411yescharacteristic
44C22×D53—22no—
45C22×D53—22no—
46C22×D53—41no—
47C22×D53C22⋊C411yescharacteristic
48C22×D53C2×D411yescharacteristic
49C2×#6157—42no—
50C2×#6157—42no—
51C2×#6157—42no—
52C2×C4×C53—22no—
53C2×C4×C53—22no—
54C2×C4×C53—42no—
55C2×C4×C53C22⋊C411yescharacteristic
56C2×C4×C53C22×C411yescharacteristic
57C2×Dic53—22no—
58C2×Dic53—22no—
59C2×Dic53—42no—
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C2×D411yescharacteristic
62C4×D53—41no—
63C4×D53—41no—
64C4×D53—41no—
65C4×D53—41no—
66Dic106—42no—
67Dic106—42no—
68#88324—42no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53#9912yes—
72C22×C53C2×C22⋊C411yescharacteristic
73C2×D53—21no—
74C2×D53—21no—
75C2×D53—21no—
76C2×D53—21no—
77C2×D53—21no—
78C2×D53—44no—
79C2×D53#13712yes—
80#6157—82no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—42no—
84Dic53—21no—
85Dic53—21no—
86Dic53—42no—
87#1511—531nomaximal, Sylow
88C2×C53—22no—
89C2×C53C2×#9911yescharacteristic
90C2×C53#41911yescharacteristic
91C2×C53#42111yescharacteristic
92D53—22no—
93D53—41no—
94C2×C22⋊Q8—531no—
95#407—531no—
96#381—531no—
97#328—534no—
98C53#151111yescharacteristic, complemented, Sylow
99C22×Q8—531no—
100C23×C4—531no—
101C2×M16—1061no—
102C22⋊Q8—532no—
103C22⋊Q8—532no—
104C22⋊Q8—1061no—
105C22⋊Q8—1061no—
106C2×C4⋊C4—1061no—
107C2×C4⋊C4#615711yescharacteristic, complemented
108C2×C22⋊C4—1061no—
109C2×C22⋊C4—1061no—
110C22⋊C8—532no—
111C22⋊C8—1061no—
112#95—532no—
113#95—1061no—
114C24—531no—
115C2×Q8—532no—
116C2×Q8—1061no—
117C2×Q8—2122no—
118C22×C4—21no—
119C22×C4—531no—
120C22×C4—531no—
121C22×C4—531no—
122C22×C4—531no—
123C22×C4—531no—
124C22×C4—1061no—
125C22×C4—1062no—
126C22×C4—2122no—
127C22×C4C2×#615711yescharacteristic
128M16—2122no—
129C2×C8—1062no—
130C4⋊C4—21no—
131C4⋊C4—1062no—
132C4⋊C4—1062no—
133C4⋊C4C2×#615712yes—
134C22⋊C4—1062no—
135C22⋊C4—1062no—
136C22⋊C4—2122no—
137C23—531no—
138C23—531no—
139C23—1062no—
140C23—1062no—
141C23—2121no—
142C23C22⋊#615711yescharacteristic
143Q8—2122no—
144Q8—2122no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—531no—
149C2×C4—531no—
150C2×C4—1062no—
151C2×C4—1062no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4—2121no—
155C2×C4—2121no—
156C2×C4—2122no—
157C2×C4—2122no—
158C2×C4—2122no—
159C2×C4—2122no—
160C2×C4C22⋊#615711yescharacteristic
161C2×C4C22×#615711yescharacteristic
162C8—2122no—
163C22—21no—
164C22—21no—
165C22—532no—
166C22—1061no—
167C22—1061no—
168C22—1061no—
169C22—1061no—
170C22—1061no—
171C22—2124no—
172C22#1313327612yes—
173C22C2×C22⋊#615711yescharacteristic
174C4—21no—
175C4—21no—
176C4—42no—
177C4—1061no—
178C4—1061no—
179C4—4242no—
180C4—2122no—
181C2—22no—
182C2—1062no—
183C2—2121no—
184C2C2×#1313327611yescharacteristic
185C2#1681161611yescharacteristic
186C2#1681162111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1511
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16811476C2semidirect
11. C22×Dic106C4semidirect
15. #13133197C4semidirect
19. C2×C4⋊C4×C53C4semidirect
98. C53#1511semidirect
107. C2×C4⋊C4#6157semidirect