Finite Group Databasebeta

← #20281746

Subgroups — #20281746

Order 6784.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133216C212yesmaximal
2#16810556C212yesmaximal
3#16810544C212yesmaximal
4#16810543C212yesmaximal
5#16810529C211yesmaximal, characteristic
6#16810521C211yesmaximal, characteristic
7#16810509C212yesmaximal
8#16810506C212yesmaximal
9C53×#389C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C22×Dic106C2211yescharacteristic
12#13133216—28no—
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×C4×Dic53C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C2×Dic106—24no—
42C2×Dic106—41no—
43C2×Dic106—41no—
44C2×Dic106—42no—
45C2×Dic106—42no—
46C2×Dic106D414yes—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—41no—
49C4⋊C4×C53Q814yescomplemented
50C42×C53—41no—
51C4⋊Dic53—24no—
52C4⋊Dic53—41no—
53C4⋊Dic53—41no—
54C4⋊Dic53Q814yescomplemented
55#11719919—24no—
56#11719919—24no—
57#11719919—42no—
58C4×Dic53—24no—
59C23×C53C2411yescharacteristic
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×Q812yes—
73C2×C4×C53C2×Q814yes—
74C2×C4×C53C2×Q812yes—
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—
84C2×Dic53—22no—
85C2×Dic53C2×D414yes—
86C2×Dic53C2×Q814yes—
87Dic106—44no—
88Dic106—48no—
89C22×C53C22×D411yescharacteristic
90C22×C53C22×Q811yescharacteristic
91C22×C53C22×Q811yescharacteristic
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C53H(5,𝔽2)11yescharacteristic
94C22×C532−1+411yescharacteristic
95C22×C532−1+411yescharacteristic
96C4×C53—22no—
97C4×C53—22no—
98C4×C53—22no—
99C4×C53—24no—
100C4×C53—42no—
101Dic53—24no—
102Dic53—24no—
103Dic53—42no—
104Dic53—42no—
105#2734—531nomaximal, Sylow
106C2×C53#53411yescharacteristic
107C2×C53#54411yescharacteristic
108C2×C53#54411yescharacteristic
109C2×C53D4×Q811yescharacteristic
110C2×C53D4×Q811yescharacteristic
111C2×C53Q8211yescharacteristic
112C2×C53#56311yescharacteristic
113C2×C4⋊Q8—532no—
114#398—532no—
115#395—532no—
116#395—532no—
117#391—531no—
118#389—531no—
119#389D5311yescharacteristic
120#387—532no—
121#387—532no—
122C53#273411yescharacteristic, complemented, Sylow
123C22×Q8—531no—
124C22×Q8—532no—
125C4⋊Q8—1068no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4C2×D5312yes—
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C4⋊C4C2×D5311yescharacteristic
136C2×C42—532no—
137C2×C42C2×D5311yescharacteristic
138#95—532no—
139#95—532no—
140#95—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95C2×D5312yes—
145C2×Q8—534no—
146C2×Q8—1064no—
147C2×Q8—2121no—
148C2×Q8—2121no—
149C2×Q8—2122no—
150C2×Q8—2122no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4C22×D5312yes—
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C22×C4C22×D5311yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—534no—
163C4⋊C4—1064no—
164C4⋊C4—1064no—
165C4⋊C4—1064no—
166C4⋊C4—2122no—
167C4⋊C4—2121no—
168C4⋊C4—2121no—
169C4⋊C4Dic10614yescomplemented
170C42—41no—
171C42—1064no—
172C23C23×D5311yescharacteristic
173Q8—2124no—
174Q8—2128no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—534no—
187C2×C4—534no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4C2×Dic10614yes—
199C2×C4C2×Dic10612yes—
200C2×C4D4×D5312yes—
201C2×C4Q8×D5312yes—
202C22C2×D4×D5311yescharacteristic
203C22C2×Q8×D5311yescharacteristic
204C22C22×Dic10611yescharacteristic
205C22#1313326211yescharacteristic
206C22#1313326311yescharacteristic
207C22#1313326411yescharacteristic
208C22#1313326511yescharacteristic
209C4—24no—
210C4—22no—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—1064no—
215C4—1064no—
216C4—2122no—
217C4—2122no—
218C2#1681118111yescharacteristic
219C2#1681125211yescharacteristic
220C2#1681125411yescharacteristic
221C2#1681128811yescharacteristic
222C2D4×Dic10611yescharacteristic
223C2Q8×Dic10611yescharacteristic
224C2#1681137711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2734
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
49. C4⋊C4×C53Q8semidirect
54. C4⋊Dic53Q8semidirect
122. C53#2734semidirect
169. C4⋊C4Dic106semidirect