Finite Group Databasebeta

← #20275966

Subgroups — #20275966

Order 6784.

382 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 69 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#16811185C211yesmaximal, characteristic
2C2×#13133131C212yesmaximal, complemented
3#16810502C212yesmaximal, complemented
4#16810390C211yesmaximal, characteristic, complemented
5C53×#385C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C2212yes—
7#13133201—22no—
8#13133191—21no—
9#13133191—21no—
10#13133187—22no—
11C2×C22⋊Dic53C2212yes—
12C2×#11719921—22no—
13#13133131—24no—
14#13133124—22no—
15C23×C4×C53C2211yescharacteristic
16C2×C4⋊C4×C53—21no—
17C2×C22⋊C4×C53C2211yescharacteristic
18C2×C22⋊C4×C53C2211yescharacteristic
19#95×C53—21no—
20C24×C53C2311yescharacteristic
21C23×D53D412yes—
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53—22no—
25C22×C4×C53—42no—
26C22×C4×C53D411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C2×C22⋊D53—22no—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32C2×C22⋊D53—22no—
33C2×C22⋊D53—22no—
34C22×Dic53—22no—
35C22×Dic53D412yes—
36C2×D212—21no—
37C2×D212—21no—
38C4⋊C4×C53—42no—
39C22⋊C4×C53—22no—
40C22⋊C4×C53—21no—
41C22⋊C4×C53D412yescomplemented
42C22⋊Dic53—22no—
43C22⋊Dic53—22no—
44#11719921—22no—
45#11719921—22no—
46#11719921—44no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C23×C53—41no—
50C23×C53—41no—
51C23×C53C2×D411yescharacteristic
52C23×C53C2×D412yes—
53C23×C53C2×D411yescharacteristic
54C23×C53#4111yescharacteristic
55C23×C53#4112yes—
56C22×D53—22no—
57C22×D53—22no—
58C22×D53—22no—
59C22×D53—42no—
60C22×D53—42no—
61C2×C4×C53—24no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—41no—
67C2×C4×C53—42no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C22⋊D53—42no—
74C22⋊D53—42no—
75C22⋊D53—42no—
76C22⋊D53—42no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83D212—81no—
84D212—81no—
85C22×C53—22no—
86C22×C53—21no—
87C22×C53—22no—
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—42no—
92C22×C53#12011yescharacteristic
93C22×C53#12111yescharacteristic
94C22×C53#12112yes—
95C22×C53#12312yes—
96C22×C53#12411yescharacteristic
97C2×D53—44no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—82no—
102C4×C53—42no—
103C4×C53—41no—
104C4×C53—42no—
105C4×C53—41no—
106Dic53—42no—
107Dic53—82no—
108#2044—531nomaximal, Sylow
109C2×C53—22no—
110C2×C53—21no—
111C2×C53—22no—
112C2×C53—21no—
113C2×C53#39411yescharacteristic
114C2×C53B(4,𝔽2)11yescharacteristic
115C2×C53#45811yescharacteristic
116D53—82no—
117#534—531no—
118C2×#99—532no—
119#386—532no—
120#385D5311yescharacteristic, complemented
121#342—531no—
122C53#204411yescharacteristic, complemented, Sylow
123C22×D4—532no—
124C23×C4C2×D5311yescharacteristic
125#123—1062no—
126#121—1061no—
127#121—1061no—
128#120—1062no—
129C2×C4⋊C4—21no—
130C2×C22⋊C4—532no—
131C2×C22⋊C4—1062no—
132C2×C22⋊C4C2×D5311yescharacteristic
133C2×C22⋊C4C2×D5311yescharacteristic
134#99—1064no—
135#95—21no—
136#95—1062no—
137C24—532no—
138C24C22×D5311yescharacteristic
139C2×D4—1061no—
140C2×D4—1061no—
141C2×D4—1062no—
142C2×D4—1062no—
143C2×D4—1062no—
144C2×D4—1062no—
145C2×D4—1062no—
146C2×D4—1062no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—532no—
152C22×C4—1062no—
153C22×C4C22⋊D5311yescharacteristic
154C22×C4C22⋊D5311yescharacteristic
155C4⋊C4—42no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C22⋊C4—21no—
159C22⋊C4—22no—
160C22⋊C4—1062no—
161C22⋊C4—1062no—
162C22⋊C4—1062no—
163C22⋊C4—1062no—
164C22⋊C4—2124no—
165C22⋊C4D21212yescomplemented
166C23—41no—
167C23—41no—
168C23—1062no—
169C23—1062no—
170C23—1062no—
171C23—2122no—
172C23—2122no—
173C23C2×D21211yescharacteristic
174C23#1171994511yescharacteristic
175C23D4×D5312yes—
176C23#1171994912yes—
177C23C2×C22⋊D5311yescharacteristic
178D4—4241no—
179D4—4241no—
180D4—2122no—
181D4—2122no—
182D4—2122no—
183D4—2122no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—4242no—
210C22—2124no—
211C22—2122no—
212C22—2122no—
213C22—2122no—
214C22#1313318511yescharacteristic
215C22#1313319111yescharacteristic
216C22#1313319212yes—
217C22#1313320212yes—
218C22#1313320611yescharacteristic
219C4—42no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—4242no—
224C4—2122no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—4242no—
230C2#1681054011yescharacteristic
231C2#1681080911yescharacteristic
232C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC53×#385C2
socleC22×C53#120
hypercenter#385D53
hypocenterC53#2044
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#13133131C2semidirect
3. #16810502C2semidirect
4. #16810390C2semidirect
5. C53×#385C2semidirect
41. C22⋊C4×C53D4semidirect
120. #385D53semidirect
122. C53#2044semidirect
165. C22⋊C4D212semidirect