Finite Group Databasebeta

← #20279195

Subgroups — #20279195

Order 6784.

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

normal subgroups only

This group has 512 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
1C2×#13133195C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810563C211yesmaximal, characteristic, complemented
4#16810562C211yesmaximal, characteristic, complemented
5#16810561C211yesmaximal, characteristic, complemented
6#16810527C212yesmaximal, complemented
7#16810487C212yesmaximal, complemented
8#16810485C212yesmaximal
9#16810485C212yesmaximal
10#16810470C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2212yes—
14C22×C4×D53C2211yescharacteristic
15#13133195—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C4⋊C4×D53—24no—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C2×C4×D53—24no—
49C2×C4×D53—24no—
50C2×C4×D53—28no—
51C2×C4×D53D414yes—
52C2×Dic106—22no—
53C2×Dic106—24no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56#11719921—22no—
57#11719921—22no—
58#11719921—24no—
59C4⋊Dic53—22no—
60#11719919—22no—
61#11719919—24no—
62#11719919—22no—
63C4×Dic53—24no—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53C2×D412yes—
69C22×D53#4114yes—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—24no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53C2×D412yes—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—24no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53C2×D412yes—
93C2×Dic53#4114yes—
94C4×D53—28no—
95C4×D53—48no—
96Dic106—44no—
97C22×C53C22×D411yescharacteristic
98C22×C53C2×#4112yes—
99C22×C53C2×#4112yes—
100C22×C532−1+411yescharacteristic
101C22×C532−1+411yescharacteristic
102C2×D53—22no—
103C2×D53—22no—
104C2×D53—24no—
105C2×D53—24no—
106C2×D53#12414yes—
107C4×C53—22no—
108C4×C53—41no—
109C4×C53—42no—
110C4×C53—41no—
111C4×C53—42no—
112Dic53—41no—
113Dic53—42no—
114Dic53—41no—
115Dic53—42no—
116Dic53#12314yes—
117#2503—531nomaximal, Sylow
118C2×C53C2×#12311yescharacteristic
119C2×C53C2×#12411yescharacteristic
120C2×C53#53611yescharacteristic
121C2×C53#55212yes—
122C2×C53#55212yes—
123D53—24no—
124C2×C22⋊Q8—531no—
125C22×C4⋊C4—531no—
126#400—531no—
127#400—531no—
128#400—531no—
129#400D5311yescharacteristic, complemented
130#391—532no—
131#382—532no—
132#381—532no—
133#381—532no—
134C4×C22⋊C4—531no—
135C53#250311yescharacteristic, complemented, Sylow
136C22×Q8—531no—
137C23×C4—531no—
138C23×C4—532no—
139C22⋊Q8—1064no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—1064no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C22⋊C4—531no—
148C2×C22⋊C4—531no—
149C2×C22⋊C4—532no—
150C2×C42—532no—
151#95—531no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156#95—532no—
157#95—532no—
158#95C2×D5311yescharacteristic
159#95C2×D5312yes—
160#95C2×D5312yes—
161C24—531no—
162C2×Q8—1062no—
163C2×Q8—1064no—
164C22×C4—531no—
165C22×C4—534no—
166C22×C4—532no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—532no—
170C22×C4—1064no—
171C22×C4—1064no—
172C22×C4—1068no—
173C22×C4C22×D5312yes—
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—1064no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—1064no—
187C42—1064no—
188C23—532no—
189C23—534no—
190C23—1062no—
191C23—1061no—
192C23—1061no—
193C23C23×D5311yescharacteristic
194Q8—2124no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—532no—
207C2×C4—534no—
208C2×C4—1068no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1064no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—2128no—
220C2×C4D4×D5312yes—
221C22—534no—
222C22—1064no—
223C22—1064no—
224C22—1062no—
225C22—1062no—
226C22C2×D4×D5311yescharacteristic
227C22#41×D5312yes—
228C22#41×D5312yes—
229C22#1313326411yescharacteristic
230C22#1313326511yescharacteristic
231C4—22no—
232C4—42no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—534no—
237C4—2122no—
238C4—2121no—
239C4—2122no—
240C4—2121no—
241C2—1064no—
242C2#123×D5311yescharacteristic
243C2#124×D5311yescharacteristic
244C2#1681119611yescharacteristic
245C2#1681130512yes—
246C2#1681130912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2503
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #16810563C2semidirect
4. #16810562C2semidirect
5. #16810561C2semidirect
6. #16810527C2semidirect
7. #16810487C2semidirect
11. C53×#400C2semidirect
129. #400D53semidirect
135. C53#2503semidirect