Finite Group Databasebeta

← #20280097

Subgroups — #20280097

Order 6784.

468 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 161 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×#13133207C211yesmaximal, characteristic
2C2×Q8×Dic53C211yesmaximal, characteristic
3#16810555C211yesmaximal, characteristic
4#16810553C211yesmaximal, characteristic
5#16810542C211yesmaximal, characteristic
6#16810522C212yesmaximal
7#16810520C212yesmaximal
8#16810520C212yesmaximal
9#16810518C212yesmaximal
10#16810474C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12#13133207—24no—
13Q8×Dic53—24no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×C4×Dic53C2212yes—
26C2×C4×Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C2×Q8×C53—22no—
38C2×Q8×C53—24no—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C4⋊C4×C53—22no—
51C4⋊C4×C53—24no—
52C4⋊Dic53—22no—
53C4⋊Dic53—24no—
54C4⋊Dic53—22no—
55C4⋊Dic53—42no—
56#11719919—22no—
57#11719919—24no—
58#11719919—22no—
59#11719919—22no—
60#11719919—22no—
61#11719919—42no—
62#11719919Q818yescomplemented
63C4×Dic53—24no—
64C4×Dic53—42no—
65C4×Dic53D414yes—
66C23×C53C2411yescharacteristic
67Q8×C53—28no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×Q814yes—
80C2×C4×C53#4114yes—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53C2×D412yes—
94C2×Dic53C2×D412yes—
95C2×Dic53C2×Q814yes—
96C2×Dic53C2×Q814yes—
97C22×C53C22×D411yescharacteristic
98C22×C53C22×Q812yes—
99C22×C53C2×#4112yes—
100C22×C53H(5,𝔽2)11yescharacteristic
101C22×C532−1+411yescharacteristic
102C4×C53—22no—
103C4×C53—24no—
104C4×C53—24no—
105C4×C53—41no—
106C4×C53—41no—
107Dic53—22no—
108Dic53—24no—
109Dic53—41no—
110Dic53—42no—
111Dic53—41no—
112Dic53C4⋊Q814yes—
113#2584—531nomaximal, Sylow
114C2×C53#52511yescharacteristic
115C2×C53C2×C4⋊Q811yescharacteristic
116C2×C53#53711yescharacteristic
117C2×C53#55412yes—
118C2×C53#55712yes—
119C2×#125—531no—
120C2×C4×Q8—531no—
121#398—531no—
122#398—531no—
123#395—531no—
124#395D5311yescharacteristic
125#389—532no—
126#389—532no—
127#389—532no—
128#389—532no—
129C4×C4⋊C4—531no—
130C53#258411yescharacteristic, complemented, Sylow
131C22×Q8C2×D5311yescharacteristic
132#125—1064no—
133C4×Q8—1064no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C4⋊C4C2×D5312yes—
147C2×C42—531no—
148C2×C42—532no—
149C2×C42—532no—
150#95—531no—
151#95—532no—
152#95—532no—
153#95—532no—
154#95C2×D5311yescharacteristic
155#95C2×D5312yes—
156C2×Q8—22no—
157C2×Q8—24no—
158C22×C4—531no—
159C22×C4—532no—
160C22×C4—531no—
161C22×C4—532no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5311yescharacteristic
166C22×C4C22×D5311yescharacteristic
167C22×C4C22×D5312yes—
168C22×C4C22×D5312yes—
169C4⋊C4—22no—
170C4⋊C4—24no—
171C4⋊C4—538no—
172C4⋊C4—1064no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1064no—
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C42—534no—
183C42—1064no—
184C42—2122no—
185C23C23×D5311yescharacteristic
186Q8—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—532no—
198C2×C4—534no—
199C2×C4—532no—
200C2×C4—534no—
201C2×C4—1062no—
202C2×C4—1061no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—1062no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4D4×D5312yes—
214C2×C4#1171994714yes—
215C2×C4Q8×D5314yes—
216C22C2×D4×D5311yescharacteristic
217C22C2×Q8×D5312yes—
218C22C2×#1171994712yes—
219C22#1313326311yescharacteristic
220C22#1313326511yescharacteristic
221C4—24no—
222C4—22no—
223C4—24no—
224C4—41no—
225C4—41no—
226C4—534no—
227C4—1064no—
228C4—1062no—
229C4—2121no—
230C4—2122no—
231C4—2121no—
232C2#1681112911yescharacteristic
233C2C4⋊Q8×D5311yescharacteristic
234C2#1681120311yescharacteristic
235C2#1681133112yes—
236C2#1681134812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2584
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
62. #11719919Q8semidirect
130. C53#2584semidirect