Finite Group Databasebeta

← #20289993

Subgroups — #20289993

Order 6784.

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

normal subgroups only

This group has 270 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
2D53⋊(C22⋊C8)C211yesmaximal, characteristic
3#16811468C214yesmaximal, complemented
4C2×#13133197C211yesmaximal, characteristic
5C2×#11719936—21no—
6C2×D53⋊C8—21no—
7#13133274—21no—
8#13133274—21no—
9C22⋊#88324C2212yes—
10C22⋊#88324C2212yes—
11C22×Dic106C411yescharacteristic
12C22×C4×D53C2211yescharacteristic
13#13133197—21no—
14#13133197—21no—
15#13133197—21no—
16#13133197—21no—
17C2×C4⋊Dic53C2212yes—
18C2×#11719921C412yes—
19C2×C4⋊C4×C53C411yescharacteristic
20C23×D53C2×C411yescharacteristic
21C22×C4×C53C2×C412yes—
22C22×C4×C53C2×C411yescharacteristic
23C22×Dic53C2×C412yes—
24C22×Dic53C2311yescharacteristic
25C2×C4×D53—42no—
26C2×C4×D53D411yescharacteristic
27C2×C4×D53D411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—42no—
33C2×#88324—22no—
34C2×#88324—22no—
35#11719936—42no—
36D53⋊C8—42no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
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×D53C22⋊C411yescharacteristic
49C2×C4×C53—22no—
50C2×C4×C53—22no—
51C2×C4×C53—24no—
52C2×C4×C53C22⋊C411yescharacteristic
53C2×C4×C53C22⋊C411yescharacteristic
54C2×Dic53—22no—
55C2×Dic53—41no—
56C2×Dic53—42no—
57C2×Dic53C2×D411yescharacteristic
58C2×Dic53C2×D411yescharacteristic
59C4×D53—22no—
60C4×D53—22no—
61C4×D53—41no—
62C4×D53—41no—
63Dic106—42no—
64Dic106—42no—
65#88324—42no—
66#88324—42no—
67C22×C53—21no—
68C22×C53—21no—
69C22×C53C2×C22⋊C411yescharacteristic
70C22×C53#13112yes—
71C2×D53—21no—
72C2×D53—21no—
73C2×D53—21no—
74C2×D53—21no—
75C2×D53—21no—
76C2×D53—44no—
77C2×D53#10112yes—
78C4×C53—21no—
79C4×C53—42no—
80C4×C53#9912yes—
81Dic53—21no—
82Dic53—21no—
83Dic53—42no—
84#1472—531nomaximal, Sylow
85C2×C53—22no—
86C2×C53#40811yescharacteristic
87C2×C53C2×#9911yescharacteristic
88C2×C53C2×#10111yescharacteristic
89D53—22no—
90D53—41no—
91C2×C22⋊Q8—531no—
92#407—531no—
93C2×C22⋊C8—531no—
94#324—534no—
95C53#147211yescharacteristic, complemented, Sylow
96C22×Q8—531no—
97C23×C4—531no—
98C2×M16—1061no—
99C22×C8—1061no—
100C22⋊Q8—1061no—
101C22⋊Q8—1061no—
102C22⋊Q8—1061no—
103C22⋊Q8—1061no—
104C2×C4⋊C4—532no—
105C2×C4⋊C4#615711yescharacteristic
106C2×C22⋊C4—532no—
107C22⋊C8—532no—
108C22⋊C8—532no—
109C22⋊C8—1061no—
110C22⋊C8—1061no—
111C24—531no—
112C2×Q8—1061no—
113C2×Q8—1061no—
114C2×Q8—2122no—
115C22×C4—531no—
116C22×C4—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—531no—
120C22×C4—532no—
121C22×C4—2122no—
122C22×C4C2×#615712yes—
123C22×C4C2×#615711yescharacteristic
124M16—2122no—
125C2×C8—1062no—
126C2×C8—1062no—
127C2×C8—2122no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1062no—
131C4⋊C4—1062no—
132C22⋊C4—1062no—
133C22⋊C4—1062no—
134C23—531no—
135C23—531no—
136C23—1062no—
137C23—1062no—
138C23—2121no—
139C23C22×#615711yescharacteristic
140Q8—2122no—
141Q8—2122no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—531no—
146C2×C4—531no—
147C2×C4—1062no—
148C2×C4—1062no—
149C2×C4—1062no—
150C2×C4—2121no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4—2122no—
154C2×C4C22⋊#615711yescharacteristic
155C2×C4C22⋊#615711yescharacteristic
156C8—2122no—
157C8—2122no—
158C22—21no—
159C22—21no—
160C22—532no—
161C22—1061no—
162C22—1061no—
163C22—1061no—
164C22—1061no—
165C22—1061no—
166C22—2124no—
167C22C2×C22⋊#615711yescharacteristic
168C22#1313330412yes—
169C4—21no—
170C4—42no—
171C4—1061no—
172C4—1061no—
173C4—2122no—
174C4#1313327612yes—
175C2—22no—
176C2—1062no—
177C2—2121no—
178C2#1681159711yescharacteristic
179C2C2×#1313327611yescharacteristic
180C2D53⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1472
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16811468C2semidirect
95. C53#1472semidirect