Finite Group Databasebeta

← #20289992

Subgroups — #20289992

Order 6784.

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

normal subgroups only

This group has 322 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
2D53⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
3#16811467C214yesmaximal, complemented
4C2×#13133192C211yesmaximal, characteristic
5C2×#11719936—21no—
6C2×D53⋊C8—21no—
7#13133274—21no—
8#13133274—21no—
9C22⋊#88324C2212yes—
10C22⋊#88324C2212yes—
11C22×D212C412yes—
12C22×D212C411yescharacteristic
13C22×C4×D53C2211yescharacteristic
14#13133192—21no—
15#13133192—21no—
16#13133192—21no—
17#13133192—21no—
18C2×#11719921C2212yes—
19C2×C4⋊C4×C53C411yescharacteristic
20C23×D53C2×C411yescharacteristic
21C23×D53C2×C412yes—
22C22×C4×C53C2×C411yescharacteristic
23C22×C4×C53C2×C412yes—
24C22×Dic53C2311yescharacteristic
25C2×D212—22no—
26C2×D212—22no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212—42no—
30C2×D212—44no—
31C2×C4×D53—42no—
32C2×C4×D53D411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×#88324—22no—
37C2×#88324—22no—
38#11719936—42no—
39D53⋊C8—42no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42#11719921—22no—
43#11719921—22no—
44C23×C53C22×C411yescharacteristic
45C22×D53—22no—
46C22×D53—22no—
47C22×D53—22no—
48C22×D53—24no—
49C22×D53—41no—
50C22×D53—42no—
51C22×D53—42no—
52C22×D53C22⋊C411yescharacteristic
53C22×D53C22⋊C411yescharacteristic
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—42no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22⋊C411yescharacteristic
59C2×Dic53—41no—
60C2×Dic53C2×D411yescharacteristic
61C2×Dic53C2×D411yescharacteristic
62D212—42no—
63D212—44no—
64D212—44no—
65D212—42no—
66C4×D53—22no—
67C4×D53—22no—
68C4×D53—41no—
69C4×D53—41no—
70#88324—42no—
71#88324—42no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53C2×C22⋊C411yescharacteristic
75C22×C53#13112yes—
76C2×D53—21no—
77C2×D53—21no—
78C2×D53—21no—
79C2×D53—21no—
80C2×D53—21no—
81C2×D53—44no—
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—84no—
86C2×D53#10012yes—
87C4×C53—21no—
88C4×C53—42no—
89C4×C53#9912yes—
90Dic53—21no—
91Dic53—21no—
92#1471—531nomaximal, Sylow
93C2×C53—22no—
94C2×C53#40811yescharacteristic
95C2×C53C2×#9911yescharacteristic
96C2×C53C2×#10011yescharacteristic
97D53—22no—
98D53—41no—
99D53—82no—
100C2×#121—531no—
101#407—531no—
102C2×C22⋊C8—531no—
103#323—534no—
104C53#147111yescharacteristic, complemented, Sylow
105C22×D4—532no—
106C22×D4—531no—
107C23×C4—531no—
108C2×M16—1061no—
109C22×C8—1061no—
110#121—1061no—
111#121—1061no—
112#121—1061no—
113#121—1061no—
114C2×C4⋊C4#615711yescharacteristic
115C2×C22⋊C4—532no—
116C22⋊C8—532no—
117C22⋊C8—532no—
118C22⋊C8—1061no—
119C22⋊C8—1061no—
120C24—531no—
121C24—532no—
122C2×D4—1062no—
123C2×D4—1062no—
124C2×D4—1061no—
125C2×D4—1061no—
126C2×D4—2122no—
127C2×D4—2124no—
128C22×C4—531no—
129C22×C4—531no—
130C22×C4—531no—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—2122no—
134C22×C4C2×#615712yes—
135C22×C4C2×#615711yescharacteristic
136M16—2122no—
137C2×C8—1062no—
138C2×C8—1062no—
139C2×C8—2122no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C23—531no—
145C23—531no—
146C23—1062no—
147C23—1062no—
148C23—1064no—
149C23—1062no—
150C23—2121no—
151C23—2122no—
152C23—2122no—
153C23C22×#615711yescharacteristic
154D4—2122no—
155D4—2124no—
156D4—2124no—
157D4—2122no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—531no—
162C2×C4—531no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4C22⋊#615711yescharacteristic
169C2×C4C22⋊#615711yescharacteristic
170C8—2122no—
171C8—2122no—
172C22—21no—
173C22—21no—
174C22—532no—
175C22—1061no—
176C22—1061no—
177C22—1061no—
178C22—1061no—
179C22—1061no—
180C22—4244no—
181C22—2124no—
182C22—2122no—
183C22—2122no—
184C22—2122no—
185C22C2×C22⋊#615711yescharacteristic
186C22#1313330512yes—
187C4—21no—
188C4—42no—
189C4—1061no—
190C4—1061no—
191C4#1313327612yes—
192C2—22no—
193C2—1062no—
194C2—4242no—
195C2—2121no—
196C2#1681159811yescharacteristic
197C2C2×#1313327611yescharacteristic
198C2D53⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1471
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
1. #16811594C2semidirect
2. D53⋊(C22⋊C8)C2semidirect
3. #16811467C2semidirect
104. C53#1471semidirect