Finite Group Databasebeta

← #20280887

Subgroups — #20280887

Order 6784.

424 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 137 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#16810565C212yesmaximal, complemented
2#16810546C212yesmaximal
3#16810544C212yesmaximal, complemented
4#16810535C212yesmaximal
5#16810527C212yesmaximal, complemented
6#16810506C212yesmaximal, complemented
7#16810483C211yesmaximal, characteristic
8C4×C22⋊Dic53C211yesmaximal, characteristic
9C53×#381C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C4×Dic53C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yes—
28#95×C53C2211yescharacteristic
29#95×C53C2211yescharacteristic
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C22×C4×C53—42no—
33C22×C4×C53Q814yescomplemented
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C2×Dic106—24no—
43C2×Dic106—42no—
44C2×Dic106—42no—
45C22⋊C4×C53—42no—
46C22⋊Dic53—24no—
47C22⋊Dic53—41no—
48C22⋊Dic53—41no—
49C4⋊Dic53—24no—
50C4⋊Dic53—42no—
51#11719919—42no—
52#11719919—42no—
53C4×Dic53—42no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53C2×Q812yes—
59C23×C53C2411yescharacteristic
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—44no—
69C2×C4×C53C2×Q812yes—
70C2×C4×C53C2×Q812yes—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53#4114yes—
83Dic106—48no—
84C22×C53—22no—
85C22×C53—22no—
86C22×C53—22no—
87C22×C53—42no—
88C22×C53—42no—
89C22×C53C22×Q811yescharacteristic
90C22×C53C2×#4112yes—
91C22×C53H(5,𝔽2)12yes—
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C532−1+411yescharacteristic
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—42no—
97C4×C53—42no—
98Dic53—24no—
99Dic53—42no—
100Dic53—42no—
101Dic53—42no—
102#2665—531nomaximal, Sylow
103C2×C53—42no—
104C2×C53#53311yescharacteristic
105C2×C53#53812yes—
106C2×C53#54212yes—
107C2×C53#54311yescharacteristic
108C2×C53#54411yescharacteristic
109#400—532no—
110#396—532no—
111#395—532no—
112#393—532no—
113#391—532no—
114#387—532no—
115#381D5311yescharacteristic
116#380—531no—
117C4×C22⋊C4—531no—
118C53#266511yescharacteristic, complemented, Sylow
119C22×Q8—532no—
120C23×C4C2×D5311yescharacteristic
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C22⋊C4—532no—
126C2×C22⋊C4—531no—
127C2×C22⋊C4—531no—
128C2×C22⋊C4C2×D5312yes—
129C2×C42—532no—
130#95—531no—
131#95—531no—
132#95—532no—
133#95—532no—
134#95—532no—
135#95—532no—
136#95—532no—
137#95C2×D5311yescharacteristic
138#95C2×D5311yescharacteristic
139#95C2×D5312yes—
140C24C22×D5311yescharacteristic
141C2×Q8—1064no—
142C2×Q8—2122no—
143C2×Q8—2122no—
144C22×C4—42no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—532no—
148C22×C4—532no—
149C22×C4Dic10614yescomplemented
150C22×C4C22×D5312yes—
151C22×C4C22×D5312yes—
152C22×C4C22×D5311yescharacteristic
153C22×C4C22×D5311yescharacteristic
154C4⋊C4—1064no—
155C4⋊C4—2122no—
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C22⋊C4—42no—
159C22⋊C4—1064no—
160C22⋊C4—2121no—
161C22⋊C4—2121no—
162C42—2122no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23C2×Dic10612yes—
168C23C23×D5311yescharacteristic
169Q8—2128no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—534no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4C2×Dic10612yes—
192C2×C4C2×Dic10612yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C22×Dic10611yescharacteristic
199C22#41×D5312yes—
200C22#1313326212yes—
201C22#1313326311yescharacteristic
202C22#1313326411yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—1064no—
208C4—2122no—
209C4—2122no—
210C4—2122no—
211C2—42no—
212C2#1681117611yescharacteristic
213C2#1681120912yes—
214C2#1681124112yes—
215C2#1681124911yescharacteristic
216C2#1681125111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2665
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
3. #16810544C2semidirect
5. #16810527C2semidirect
6. #16810506C2semidirect
33. C22×C4×C53Q8semidirect
118. C53#2665semidirect
149. C22×C4Dic106semidirect