Finite Group Databasebeta

← #20282762

Subgroups — #20282762

Order 6784.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133204C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic, complemented
3#16810538C212yesmaximal, complemented
4#16810537C212yesmaximal, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810535C212yesmaximal, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810531C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16#13133204—24no—
17#13133203—24no—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C22×D4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C24×C53C2312yes—
39C23×D53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—42no—
42C2×D4×C53—42no—
43C2×D4×C53—42no—
44C2×D4×C53—42no—
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—22no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2312yes—
56C22×Dic53C2311yescharacteristic
57C22×Dic53C2311yescharacteristic
58C2×Dic106—22no—
59C2×Dic106—22no—
60C2×Dic106—42no—
61C2×Dic106—42no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—42no—
64C22⋊Dic53—22no—
65C22⋊Dic53—24no—
66C22⋊Dic53—24no—
67C22⋊Dic53—42no—
68C22⋊Dic53—42no—
69#11719921—22no—
70#11719921—22no—
71#11719921—42no—
72#11719919—42no—
73C4×Dic53D414yescomplemented
74C23×C53—21no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—22no—
86C23×C53C2411yescharacteristic
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53—21no—
92D4×C53—44no—
93D4×C53—48no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53C2×D412yes—
101C22⋊D53—44no—
102C22⋊D53—48no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53C2×D412yes—
115C2×Dic53C2×D412yes—
116Dic106—44no—
117Dic106—44no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53C22×D411yescharacteristic
131C22×C53H(5,𝔽2)12yes—
132C22×C53H(5,𝔽2)12yes—
133C22×C53H(5,𝔽2)11yescharacteristic
134C22×C532−1+411yescharacteristic
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—42no—
139C2×D53—42no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—42no—
143Dic53—22no—
144Dic53—22no—
145Dic53—41no—
146Dic53—42no—
147Dic53—42no—
148#2819—531nomaximal, Sylow
149C2×C53—81no—
150C2×C53—82no—
151C2×C53#53511yescharacteristic
152C2×C53#53511yescharacteristic
153C2×C53#53711yescharacteristic
154C2×C53H(3,𝔽4)12yes—
155C2×C53#56212yes—
156D53—81no—
157C2×#124—531no—
158C2×#124—531no—
159#394—532no—
160#394—532no—
161#393—532no—
162#393—531no—
163#393—531no—
164#392—532no—
165#392—531no—
166#392D5311yescharacteristic, complemented
167#382—531no—
168C53#281911yescharacteristic, complemented, Sylow
169C22×Q8—531no—
170C22×Q8—531no—
171C22×D4—531no—
172C22×D4—532no—
173C22×D4C2×D5311yescharacteristic
174C22×D4C2×D5312yes—
175#124—1064no—
176#124—1064no—
177C2×C4⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—531no—
182C2×C22⋊C4—531no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4C2×D5312yes—
188C2×C42—531no—
189#95—532no—
190#95—531no—
191#95—532no—
192#95—532no—
193#95C2×D5311yescharacteristic
194C24—531no—
195C24C22×D5311yescharacteristic
196C24C22×D5312yes—
197C2×Q8—1062no—
198C2×Q8—1062no—
199C2×Q8—2122no—
200C2×Q8—2122no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—1062no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2122no—
210C2×D4—2122no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4—531no—
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—2122no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—1062no—
223C22⋊C4—1062no—
224C22⋊C4—1064no—
225C22⋊C4—1064no—
226C22⋊C4—1062no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C42—534no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—1061no—
244C23—1062no—
245C23—1061no—
246C23—1062no—
247C23—1061no—
248C23C23×D5311yescharacteristic
249Q8—2124no—
250Q8—2124no—
251D4—44no—
252D4—48no—
253D4—2124no—
254D4—2128no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—532no—
262C2×C4—532no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4D4×D5312yes—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—2122no—
288C22—2122no—
289C22—2121no—
290C22—2121no—
291C22—2121no—
292C22C2×D4×D5311yescharacteristic
293C22#1313326211yescharacteristic
294C22#1313326212yes—
295C22#1313326212yes—
296C22#1313326411yescharacteristic
297C4—22no—
298C4—42no—
299C4—41no—
300C4—1062no—
301C4—1062no—
302C4—2122no—
303C4—2121no—
304C4—2122no—
305C2—81no—
306C2—82no—
307C2—4241no—
308C2#1681118611yescharacteristic
309C2#1681118811yescharacteristic
310C2#1681120211yescharacteristic
311C2#1681136612yes—
312C2#1681136812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2819
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133204C2semidirect
2. C2×#13133203C2semidirect
3. #16810538C2semidirect
4. #16810537C2semidirect
5. #16810536C2semidirect
6. #16810535C2semidirect
7. #16810535C2semidirect
8. #16810531C2semidirect
9. #16810531C2semidirect
10. #16810487C2semidirect
11. C53×#392C2semidirect
73. C4×Dic53D4semidirect
166. #392D53semidirect
168. C53#2819semidirect