Finite Group Databasebeta

← #20280683

Subgroups — #20280683

Order 6784.

352 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C53⋊#401C212yesmaximal
2#16810565C212yesmaximal
3#16810562C212yesmaximal
4#16810561C212yesmaximal
5#16810529C211yesmaximal, characteristic
6#16810528C211yesmaximal, characteristic
7#16810528C211yesmaximal, characteristic
8#16810522C211yesmaximal, characteristic
9#16810519C211yesmaximal, characteristic
10#16810469C211yesmaximal, characteristic
11C53×#389C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2212yes—
16C2×C4×Dic53C2211yescharacteristic
17C2×C4×Dic53C2211yescharacteristic
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29#13133124C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C4⋊C4×C53C2212yes—
33C2×C42×C53C2211yescharacteristic
34#95×C53C2212yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2311yescharacteristic
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—42no—
52C42×C53—22no—
53C4⋊Dic53—22no—
54C4⋊Dic53—42no—
55#11719919—42no—
56C4×Dic53—22no—
57C4×Dic53—22no—
58C23×C53C2411yescharacteristic
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53#4112yes—
72C2×C4×C53#4112yes—
73C2×C4×C53#4112yes—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92Dic106—44no—
93C22×C53C2×#4111yescharacteristic
94C22×C53C2×#4111yescharacteristic
95C22×C53C2×#4111yescharacteristic
96C22×C53H(5,𝔽2)11yescharacteristic
97C22×C532−1+411yescharacteristic
98C22×C532−1+411yescharacteristic
99C22×C532−1+411yescharacteristic
100C4×C53—41no—
101C4×C53—42no—
102C4×C53—41no—
103C4×C53—42no—
104C4×C53#12614yes—
105Dic53—42no—
106Dic53—41no—
107Dic53—41no—
108Dic53—41no—
109Dic53—41no—
110Dic53—42no—
111#2645—531nomaximal, Sylow
112C2×C53C2×#12611yescharacteristic
113C2×C53#54111yescharacteristic
114C2×C53#54111yescharacteristic
115C2×C53#54111yescharacteristic
116C2×C53#54211yescharacteristic
117C2×C53#54211yescharacteristic
118C2×C53#54211yescharacteristic
119#401—532no—
120#400—532no—
121#400—532no—
122#400—532no—
123#391—531no—
124#391—531no—
125#391—531no—
126#389—531no—
127#389—531no—
128#389D5311yescharacteristic
129#376—531no—
130C53#264511yescharacteristic, complemented, Sylow
131C22×Q8—531no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4C2×D5311yescharacteristic
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C4⋊C4C2×D5312yes—
138C2×C42—531no—
139C2×C42—531no—
140C2×C42C2×D5311yescharacteristic
141#95—531no—
142#95—531no—
143#95—531no—
144#95—531no—
145#95—532no—
146#95—532no—
147#95—532no—
148#95—532no—
149#95—532no—
150#95—532no—
151#95—532no—
152#95—532no—
153#95C2×D5312yes—
154C2×Q8—1062no—
155C2×Q8—1062no—
156C2×Q8—1062no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—532no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—532no—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5312yes—
166C22×C4C22×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—1062no—
172C4⋊C4—2122no—
173C4⋊C4—2122no—
174C42—22no—
175C42—1062no—
176C42—1062no—
177C23C23×D5311yescharacteristic
178Q8—2124no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—1062no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1062no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1062no—
209C2×C4#1171994512yes—
210C2×C4#1171994712yes—
211C2×C4#1171994912yes—
212C22C2×#1171994711yescharacteristic
213C22C2×#1171994511yescharacteristic
214C22C2×#1171994911yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326411yescharacteristic
217C22#1313326411yescharacteristic
218C22#1313326511yescharacteristic
219C4—41no—
220C4—42no—
221C4—41no—
222C4—42no—
223C4—2122no—
224C4—2121no—
225C4—2121no—
226C4—2121no—
227C4—2121no—
228C4—2122no—
229C4#1313321214yes—
230C2C2×#1313321211yescharacteristic
231C2#1681123111yescharacteristic
232C2#1681123311yescharacteristic
233C2#1681123611yescharacteristic
234C2#1681123811yescharacteristic
235C2#1681124011yescharacteristic
236C2#1681124711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2645
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
130. C53#2645semidirect