Finite Group Databasebeta

← #20279706

Subgroups — #20279706

Order 6784.

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

normal subgroups only

This group has 604 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133203C212yesmaximal
3C2×#13133177C211yesmaximal, characteristic
4#16810550C211yesmaximal, characteristic, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810527C212yesmaximal, complemented
8#16810482C212yesmaximal, complemented
9#16810481C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C22×Dic106C2212yes—
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133216—24no—
17#13133203—28no—
18#13133177—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53D414yes—
47C22×Dic53—22no—
48C22×Dic53—24no—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—24no—
56C2×Dic106—24no—
57C2×Dic106—24no—
58C2×Dic106—24no—
59C4⋊C4×C53—22no—
60C22⋊C4×C53—24no—
61C22⋊C4×C53—22no—
62C22⋊Dic53—24no—
63#11719921—24no—
64#11719921—24no—
65#11719921—22no—
66C4⋊Dic53—22no—
67#11719919—22no—
68#11719919—24no—
69#11719919—24no—
70C4×Dic53—24no—
71C4×Dic53D414yes—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53C2×D412yes—
77C23×C53C2411yescharacteristic
78C22×D53—21no—
79C22×D53—22no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53C2×D412yes—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D412yes—
93C22⋊D53—28no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—44no—
103C2×Dic53C2×D412yes—
104C2×Dic53C2×D412yes—
105C2×Dic53C2×D412yes—
106C2×Dic53#4112yes—
107C2×Dic53#4114yes—
108C4×D53—44no—
109Dic106—48no—
110C22×C53—22no—
111C22×C53—24no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53C22×D411yescharacteristic
116C22×C53C22×D411yescharacteristic
117C22×C53C2×#4111yescharacteristic
118C22×C53C2×#4112yes—
119C22×C532−1+412yes—
120C2×D53—22no—
121C2×D53—24no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—42no—
125C4×C53—22no—
126C4×C53—42no—
127C4×C53—41no—
128C4×C53—42no—
129Dic53—42no—
130Dic53—41no—
131Dic53—42no—
132Dic53#12114yes—
133Dic53#12414yescomplemented
134#2547—531nomaximal, Sylow
135C2×C53—42no—
136C2×C53C2×#12111yescharacteristic
137C2×C53C2×#12411yescharacteristic
138C2×C53#53611yescharacteristic
139C2×C53#54712yes—
140C2×C53#55612yes—
141D53—42no—
142C2×C4⋊Q8—531no—
143C2×#124—532no—
144C2×C4×D4—531no—
145#397—531no—
146#397—531no—
147#397—531no—
148#397D5311yescharacteristic, complemented
149#391—532no—
150#380—532no—
151#380—532no—
152C4×C4⋊C4—531no—
153C53#254711yescharacteristic, complemented, Sylow
154C22×Q8—532no—
155C22×D4—531no—
156C23×C4—531no—
157C23×C4—531no—
158C4⋊Q8—1064no—
159#124—1068no—
160C4×D4—1064no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4C2×D5311yescharacteristic
166C2×C22⋊C4—532no—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—532no—
170C2×C22⋊C4C2×D5311yescharacteristic
171C2×C22⋊C4C2×D5312yes—
172C2×C42—531no—
173C2×C42—532no—
174#95—531no—
175#95—532no—
176#95—532no—
177#95—532no—
178#95C2×D5311yescharacteristic
179#95C2×D5312yes—
180C24—531no—
181C24C22×D5311yescharacteristic
182C2×Q8—1064no—
183C2×Q8—1064no—
184C2×Q8—1064no—
185C2×D4—534no—
186C2×D4—1062no—
187C2×D4—1062no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—532no—
192C22×C4—532no—
193C22×C4—1062no—
194C22×C4—1064no—
195C22×C4—1062no—
196C22×C4—1064no—
197C22×C4C22×D5311yescharacteristic
198C22×C4C22×D5312yes—
199C22×C4C22×D5312yes—
200C22×C4C22×D5311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—1062no—
203C4⋊C4—1064no—
204C4⋊C4—1064no—
205C4⋊C4—1062no—
206C22⋊C4—24no—
207C22⋊C4—22no—
208C22⋊C4—1064no—
209C22⋊C4—1064no—
210C22⋊C4—1064no—
211C22⋊C4—1062no—
212C42—534no—
213C42—1064no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—532no—
219C23—1061no—
220C23—1062no—
221C23—1061no—
222C23—1062no—
223C23D4×D5312yes—
224C23C23×D5311yescharacteristic
225Q8—2128no—
226D4—1068no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—532no—
237C2×C4—532no—
238C2×C4—532no—
239C2×C4—534no—
240C2×C4—532no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—2124no—
250C2×C4—2124no—
251C2×C4D4×D5312yes—
252C22—24no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—1064no—
258C22—1062no—
259C22—2122no—
260C22—2121no—
261C22—2121no—
262C22C2×D4×D5311yescharacteristic
263C22C2×D4×D5311yescharacteristic
264C22#41×D5312yes—
265C22#41×D5311yescharacteristic
266C22#1313326412yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—534no—
272C4—534no—
273C4—2122no—
274C4—2122no—
275C4—2121no—
276C2—42no—
277C2—2122no—
278C2#121×D5311yescharacteristic
279C2#124×D5311yescharacteristic
280C2#1681119511yescharacteristic
281C2#1681127112yes—
282C2#1681134012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2547
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
4. #16810550C2semidirect
5. #16810549C2semidirect
6. #16810548C2semidirect
7. #16810527C2semidirect
8. #16810482C2semidirect
9. #16810481C2semidirect
10. #16810474C2semidirect
11. C53×#397C2semidirect
133. Dic53#124semidirect
148. #397D53semidirect
153. C53#2547semidirect