Finite Group Databasebeta

← #20280609

Subgroups — #20280609

Order 6784.

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

normal subgroups only

This group has 562 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
1C2×#13133210C211yesmaximal, characteristic, complemented
2C2×D4×Dic53C211yesmaximal, characteristic, complemented
3#16810563C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810546C212yesmaximal, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810537C212yesmaximal, complemented
8#16810503C212yesmaximal, complemented
9#16810495C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic
12C23×Dic53C2212yes—
13#13133210—24no—
14D4×Dic53—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C22⋊Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C22×D4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C24×C53C2312yes—
39C2×D4×C53—24no—
40C2×D4×C53—24no—
41C2×D4×C53—22no—
42C2×D4×C53—42no—
43C2×D4×C53—42no—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×Dic53—24no—
48C22×Dic53—42no—
49C22×Dic53—42no—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22⋊C4×C53—22no—
57C22⋊C4×C53—42no—
58C22⋊Dic53—24no—
59C22⋊Dic53—24no—
60C22⋊Dic53—22no—
61C22⋊Dic53—22no—
62C22⋊Dic53—42no—
63C22⋊Dic53—42no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66#11719919—22no—
67#11719919—22no—
68#11719919—42no—
69#11719919—42no—
70C4×Dic53D414yescomplemented
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53#4114yes—
83C23×C53C2411yescharacteristic
84D4×C53—28no—
85D4×C53—48no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D412yes—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—48no—
108C2×Dic53C2×D412yes—
109C2×Dic53C2×D412yes—
110C22×C53—24no—
111C22×C53—24no—
112C22×C53—24no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53C22×D411yescharacteristic
123C22×C53C2×#4112yes—
124C22×C53H(5,𝔽2)12yes—
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C532−1+411yescharacteristic
127C4×C53—22no—
128C4×C53—41no—
129C4×C53—42no—
130Dic53—22no—
131Dic53—22no—
132Dic53—41no—
133Dic53—42no—
134Dic53—41no—
135Dic53—42no—
136#2636—531nomaximal, Sylow
137C2×C53—44no—
138C2×C53—81no—
139C2×C53#52511yescharacteristic
140C2×C53#53511yescharacteristic
141C2×C53#53711yescharacteristic
142C2×C53#53812yes—
143C2×C53#53912yes—
144C2×#126—531no—
145C2×C4×D4—531no—
146#400—531no—
147#399—531no—
148#396—532no—
149#394—531no—
150#394—532no—
151#392D5311yescharacteristic
152#386—532no—
153#385—532no—
154#382—531no—
155C53#263611yescharacteristic, complemented, Sylow
156C22×D4C2×D5311yescharacteristic
157C22×D4C2×D5312yes—
158C23×C4—532no—
159#126—1064no—
160C4×D4—1064no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4—532no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4C2×D5311yescharacteristic
174C2×C22⋊C4C2×D5312yes—
175C2×C42—531no—
176#95—532no—
177#95—531no—
178#95—532no—
179#95—532no—
180#95C2×D5311yescharacteristic
181C24C22×D5311yescharacteristic
182C24C22×D5312yes—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—532no—
193C22×C4—532no—
194C22×C4—1064no—
195C22×C4—2122no—
196C22×C4—2122no—
197C22×C4C22×D5312yes—
198C22×C4C22×D5311yescharacteristic
199C22×C4C22×D5311yescharacteristic
200C4⋊C4—1062no—
201C4⋊C4—1062no—
202C4⋊C4—1062no—
203C4⋊C4—1062no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—1064no—
209C22⋊C4—1062no—
210C22⋊C4—1062no—
211C22⋊C4—1064no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C42—534no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23#1171994714yes—
227C23C23×D5311yescharacteristic
228D4—28no—
229D4—48no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—532no—
237C2×C4—532no—
238C2×C4—1061no—
239C2×C4—1062no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1061no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—2128no—
253C2×C4D4×D5312yes—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22C2×D4×D5311yescharacteristic
267C22C2×#1171994712yes—
268C22#1313326212yes—
269C22#1313326211yescharacteristic
270C22#1313326411yescharacteristic
271C4—22no—
272C4—41no—
273C4—42no—
274C4—1062no—
275C4—1062no—
276C4—2121no—
277C4—2122no—
278C4—2121no—
279C4—2122no—
280C2—44no—
281C2—81no—
282C2#1681112911yescharacteristic
283C2#1681118611yescharacteristic
284C2#1681120211yescharacteristic
285C2#1681120812yes—
286C2#1681121812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2636
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133210C2semidirect
2. C2×D4×Dic53C2semidirect
3. #16810563C2semidirect
4. #16810559C2semidirect
5. #16810546C2semidirect
6. #16810537C2semidirect
7. #16810537C2semidirect
8. #16810503C2semidirect
9. #16810495C2semidirect
10. #16810487C2semidirect
70. C4×Dic53D4semidirect
155. C53#2636semidirect