Finite Group Databasebeta

← #20281844

Subgroups — #20281844

Order 6784.

416 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 141 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#16810562C212yesmaximal, complemented
2#16810559C211yesmaximal, characteristic
3#16810544C212yesmaximal, complemented
4#16810535C211yesmaximal, characteristic
5#16810522C212yesmaximal, complemented
6#16810516C212yesmaximal
7#16810506C212yesmaximal, complemented
8#16810483C212yesmaximal
9C53×#385C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C4×Dic53C2212yes—
19#13133124C2211yescharacteristic
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C22×C4×C53—24no—
33C22×C4×C53—41no—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C2×Dic106—41no—
43C2×Dic106—42no—
44C4⋊C4×C53—24no—
45C22⋊C4×C53—41no—
46C22⋊C4×C53Q814yescomplemented
47C22⋊Dic53—42no—
48C22⋊Dic53—42no—
49C4⋊Dic53—24no—
50C4⋊Dic53—42no—
51C4⋊Dic53—42no—
52C4⋊Dic53—41no—
53#11719919—42no—
54C4×Dic53—42no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53—22no—
58C23×C53—21no—
59C23×C53C2×Q812yes—
60C23×C53C2411yescharacteristic
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—21no—
68C2×C4×C53—21no—
69C2×C4×C53—44no—
70C2×C4×C53C2×Q814yes—
71C2×C4×C53#4114yes—
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—22no—
83C2×Dic53—22no—
84Dic106—44no—
85C22×C53—22no—
86C22×C53—24no—
87C22×C53—42no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53C22×Q811yescharacteristic
91C22×C53C2×#4112yes—
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C53H(5,𝔽2)12yes—
94C22×C532−1+411yescharacteristic
95C4×C53—24no—
96C4×C53—24no—
97C4×C53—41no—
98C4×C53—41no—
99Dic53—42no—
100Dic53—42no—
101Dic53—42no—
102Dic53—42no—
103#2742—531nomaximal, Sylow
104C2×C53—42no—
105C2×C53#54212yes—
106C2×C53#54411yescharacteristic
107C2×C53#55111yescharacteristic
108C2×C53#55412yes—
109C2×C53#56011yescharacteristic
110#400—532no—
111#399—531no—
112#395—532no—
113#393—531no—
114#389—532no—
115#388—532no—
116#387—532no—
117#385D5311yescharacteristic
118#380—532no—
119C53#274211yescharacteristic, complemented, Sylow
120C22×Q8—531no—
121C23×C4C2×D5311yescharacteristic
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4C2×D5312yes—
128C2×C22⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4C2×D5311yescharacteristic
131C2×C22⋊C4C2×D5311yescharacteristic
132C2×C42—532no—
133#95—532no—
134#95—532no—
135#95—531no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95—531no—
140#95C2×D5312yes—
141C24C22×D5311yescharacteristic
142C2×Q8—2121no—
143C2×Q8—2122no—
144C22×C4—24no—
145C22×C4—41no—
146C22×C4—532no—
147C22×C4—532no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4C22×D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C22×C4C22×D5312yes—
153C22×C4C22×D5312yes—
154C4⋊C4—24no—
155C4⋊C4—1064no—
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C4⋊C4—2122no—
159C4⋊C4—2121no—
160C22⋊C4—41no—
161C22⋊C4—2122no—
162C22⋊C4—2122no—
163C22⋊C4Dic10614yescomplemented
164C42—2122no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—21no—
169C23C2×Dic10612yes—
170C23C23×D5311yescharacteristic
171Q8—2124no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—44no—
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×C4—1062no—
192C2×C4—1062no—
193C2×C4C2×Dic10614yes—
194C2×C4#1171994714yes—
195C22—24no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22C22×Dic10611yescharacteristic
201C22C2×#1171994712yes—
202C22#1313326211yescharacteristic
203C22#1313326312yes—
204C22#1313326511yescharacteristic
205C4—24no—
206C4—24no—
207C4—41no—
208C4—41no—
209C4—2122no—
210C4—2122no—
211C4—2122no—
212C4—2122no—
213C2—42no—
214C2#1681124612yes—
215C2#1681125411yescharacteristic
216C2#1681130411yescharacteristic
217C2#1681133012yes—
218C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2742
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #16810562C2semidirect
3. #16810544C2semidirect
5. #16810522C2semidirect
7. #16810506C2semidirect
46. C22⋊C4×C53Q8semidirect
119. C53#2742semidirect
163. C22⋊C4Dic106semidirect