Finite Group Databasebeta

← #20273061

Subgroups — #20273061

Order 6784.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#13133200C211yesmaximal, characteristic
2#16810587C211yesmaximal, characteristic, complemented
3#16810348C212yesmaximal, complemented
4#16810346C212yesmaximal, complemented
5C2×C22⋊C8×C53C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C411yescharacteristic
7C2×#11719917—21no—
8#13133200—21no—
9#13133200—21no—
10#13133200—21no—
11#13133200—21no—
12C2×#11719919C411yescharacteristic
13C2×#11719919C2211yescharacteristic
14C2×C22⋊Dic53C411yescharacteristic
15C2×#11719921C411yescharacteristic
16C2×#11719921C2211yescharacteristic
17C22⋊#88322—21no—
18C22⋊#88322C2212yes—
19C23×C4×C53C2211yescharacteristic
20C22×C8×C53—21no—
21C22⋊C8×C53—21no—
22C22⋊C8×C53C2212yes—
23C24×C53C2×C411yescharacteristic
24C23×D53C2×C411yescharacteristic
25C22×C4×C53—42no—
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C22×C4×C53D411yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—42no—
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C411yescharacteristic
37C22×Dic53C2×C411yescharacteristic
38C2×C8×C53—22no—
39C2×C8×C53—42no—
40C22⋊Dic53—21no—
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46#11719919—21no—
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50#11719917—42no—
51C2×#88322—22no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—41no—
55C23×C53C22⋊C411yescharacteristic
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C22×D53—21no—
59C22×D53—22no—
60C22×D53—41no—
61C22×D53—41no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C22⋊D53—42no—
73C22⋊D53—42no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—22no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C8×C53—42no—
81#88322—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—44no—
90C22×C53#9912yes—
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53#13112yes—
93C2×D53—41no—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—82no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101Dic53—41no—
102Dic53—41no—
103Dic53—41no—
104#1468—531nomaximal, Sylow
105C2×C53—22no—
106C2×C53—22no—
107C2×C53—41no—
108C2×C53#40811yescharacteristic
109C2×C53C2×#9911yescharacteristic
110C2×C53#41311yescharacteristic
111D53—81no—
112C2×#123—531no—
113#407—531no—
114C2×C22⋊C8D5311yescharacteristic, complemented
115#324—532no—
116#323—532no—
117C53#146811yescharacteristic, complemented, Sylow
118C22×D4—531no—
119C23×C4C2×D5311yescharacteristic
120C2×M16—1061no—
121C22×C8—21no—
122#123—1061no—
123#123—1061no—
124#123—1061no—
125#123—1061no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—531no—
128C2×C22⋊C4—531no—
129C2×C22⋊C4—531no—
130C2×C22⋊C4—531no—
131C22⋊C8—21no—
132C22⋊C8—532no—
133C22⋊C8—1061no—
134C22⋊C8C2×D5312yes—
135C24—531no—
136C24C4×D5311yescharacteristic
137C2×D4—1061no—
138C2×D4—1061no—
139C2×D4—2122no—
140C22×C4—42no—
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4C4×D5311yescharacteristic
145C22×C4D21211yescharacteristic
146C22×C4D21211yescharacteristic
147C22×C4C22⋊D5311yescharacteristic
148C22×C4C22⋊D5311yescharacteristic
149C22×C4C22×D5311yescharacteristic
150M16—2122no—
151C2×C8—22no—
152C2×C8—42no—
153C2×C8—1062no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—1062no—
168C23—1061no—
169C23—2121no—
170C23—2121no—
171C23#1171992111yescharacteristic
172C23#1171992111yescharacteristic
173C23C2×C4×D5311yescharacteristic
174D4—2122no—
175D4—2122no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4—1062no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4#1171992111yescharacteristic
189C2×C4#1171992111yescharacteristic
190C2×C4C2×D21211yescharacteristic
191C2×C4C2×C22⋊D5311yescharacteristic
192C8—42no—
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—4242no—
203C22—2121no—
204C22—2121no—
205C22—2121no—
206C22#1313313112yes—
207C22C2×#1171992111yescharacteristic
208C22#1313322312yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2121no—
214C4—2121no—
215C4—2121no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—4241no—
220C2#1681058911yescharacteristic
221C2C2×#1313313111yescharacteristic
222C2#1681060511yescharacteristic
subgroupG/H
centerC22C2×#11719921
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×C22⋊C8×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C8D53
hypocenterC53#1468
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #16810587C2semidirect
3. #16810348C2semidirect
4. #16810346C2semidirect
5. C2×C22⋊C8×C53C2semidirect
114. C2×C22⋊C8D53semidirect
117. C53#1468semidirect