Finite Group Databasebeta

← #20279018

Subgroups — #20279018

Order 6784.

624 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133202C212yesmaximal
2C2×#13133179C211yesmaximal, characteristic
3C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810553C211yesmaximal, characteristic, complemented
5#16810550C211yesmaximal, characteristic, complemented
6#16810549C211yesmaximal, characteristic, complemented
7#16810518C212yesmaximal, complemented
8#16810485C212yesmaximal, complemented
9#16810482C211yesmaximal, characteristic, complemented
10#16810482C212yesmaximal, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53C2212yes—
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133202—28no—
17#13133179—24no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C4⋊C4×D53—24no—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36#95×C53C2211yescharacteristic
37#95×C53C2212yes—
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C2×D212—22no—
50C2×D212—22no—
51C2×D212D414yes—
52C2×C4×D53—24no—
53C2×C4×D53—24no—
54C2×C4×D53—24no—
55C2×C4×D53—24no—
56C2×C4×D53—22no—
57C2×C4×D53—24no—
58C2×C4×D53D414yes—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—24no—
62#11719921—22no—
63#11719921—24no—
64#11719921—24no—
65#11719921—24no—
66#11719921—22no—
67C4⋊Dic53—22no—
68#11719919—24no—
69#11719919—24no—
70#11719919—24no—
71C4×Dic53—22no—
72C23×C53C2411yescharacteristic
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—21no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—22no—
79C22×D53C2×D412yes—
80C22×D53C2×D412yes—
81C22×D53C2×D412yes—
82C22×D53#4112yes—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53C2×D412yes—
95C2×C4×C53C2×D412yes—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53C2×D412yes—
105C2×Dic53#4114yes—
106D212—28no—
107C4×D53—28no—
108C4×D53—48no—
109C4×D53—44no—
110C22×C53C22×D411yescharacteristic
111C22×C53C22×D411yescharacteristic
112C22×C53C2×#4111yescharacteristic
113C22×C53C2×#4112yes—
114C22×C532−1+412yes—
115C2×D53—22no—
116C2×D53—22no—
117C2×D53—24no—
118C2×D53—24no—
119C2×D53—22no—
120C2×D53—24no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—42no—
124C2×D53#12114yes—
125C4×C53—22no—
126C4×C53—22no—
127C4×C53—42no—
128C4×C53—41no—
129C4×C53—42no—
130Dic53—42no—
131Dic53—41no—
132Dic53—42no—
133Dic53#12314yescomplemented
134#2486—531nomaximal, Sylow
135C2×C53C2×#12111yescharacteristic
136C2×C53C2×#12311yescharacteristic
137C2×C53#53611yescharacteristic
138C2×C53#54712yes—
139C2×C53#55212yes—
140D53—24no—
141D53—42no—
142C2×#123—532no—
143C2×C4×D4—531no—
144C22×C4⋊C4—531no—
145#398—531no—
146#398D5311yescharacteristic, complemented
147#397—531no—
148#397—531no—
149#389—532no—
150#381—532no—
151#380—532no—
152#380—531no—
153C53#248611yescharacteristic, complemented, Sylow
154C22×D4—531no—
155C23×C4—531no—
156C23×C4—532no—
157C23×C4—531no—
158#123—1068no—
159C4×D4—1064no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—532no—
164C2×C4⋊C4—1064no—
165C2×C4⋊C4C2×D5311yescharacteristic
166C2×C4⋊C4C2×D5311yescharacteristic
167C2×C4⋊C4C2×D5312yes—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C42—531no—
174#95—532no—
175#95—531no—
176#95—532no—
177#95—532no—
178#95C2×D5312yes—
179#95C2×D5311yescharacteristic
180C24—531no—
181C24—531no—
182C2×D4—534no—
183C2×D4—1062no—
184C2×D4—1062no—
185C22×C4—531no—
186C22×C4—534no—
187C22×C4—531no—
188C22×C4—532no—
189C22×C4—532no—
190C22×C4—1064no—
191C22×C4—1062no—
192C22×C4—1064no—
193C22×C4—1064no—
194C22×C4—1064no—
195C22×C4—1064no—
196C22×C4C22×D5311yescharacteristic
197C22×C4C22×D5312yes—
198C22×C4C22×D5312yes—
199C22×C4C22×D5311yescharacteristic
200C22×C4C22×D5311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—1064no—
205C4⋊C4—1064no—
206C4⋊C4—1062no—
207C4⋊C4—1064no—
208C22⋊C4—1062no—
209C22⋊C4—1064no—
210C22⋊C4—1064no—
211C22⋊C4—1064no—
212C22⋊C4—1062no—
213C42—1062no—
214C23—532no—
215C23—532no—
216C23—532no—
217C23—532no—
218C23—1062no—
219C23—1062no—
220C23—1061no—
221C23—1062no—
222C23—1061no—
223C23—1062no—
224C23C23×D5311yescharacteristic
225D4—1068no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—532no—
238C2×C4—534no—
239C2×C4—1068no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—2124no—
249C2×C4—2128no—
250C2×C4D4×D5312yes—
251C2×C4D4×D5312yes—
252C22—534no—
253C22—1064no—
254C22—1064no—
255C22—1062no—
256C22—1062no—
257C22—1064no—
258C22—1062no—
259C22—2122no—
260C22—2121no—
261C22—2121no—
262C22C2×D4×D5311yescharacteristic
263C22C2×D4×D5311yescharacteristic
264C22#41×D5312yes—
265C22#41×D5311yescharacteristic
266C22#1313326512yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—534no—
273C4—2122no—
274C4—2122no—
275C4—2121no—
276C2—1064no—
277C2—2122no—
278C2#121×D5311yescharacteristic
279C2#123×D5311yescharacteristic
280C2#1681119811yescharacteristic
281C2#1681127512yes—
282C2#1681130512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2486
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D53C2semidirect
4. #16810553C2semidirect
5. #16810550C2semidirect
6. #16810549C2semidirect
7. #16810518C2semidirect
8. #16810485C2semidirect
9. #16810482C2semidirect
10. #16810482C2semidirect
11. C53×#398C2semidirect
133. Dic53#123semidirect
146. #398D53semidirect
153. C53#2486semidirect