Finite Group Databasebeta

← #20280644

Subgroups — #20280644

Order 6784.

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

normal subgroups only

This group has 576 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×#13133193C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic
3#16810564C212yesmaximal, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810538C212yesmaximal, complemented
7#16810534C211yesmaximal, characteristic, complemented
8#16810501C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C4×D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133193—24no—
18#13133190—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C2×Q8×C53—22no—
41C2×Q8×C53—42no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—22no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—42no—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C2×D212—22no—
55C2×D212—24no—
56C2×C4×D53—24no—
57C2×C4×D53—42no—
58C2×C4×D53—42no—
59C2×C4×D53—42no—
60C2×C4×D53D414yes—
61C22⋊C4×C53—22no—
62C22⋊C4×C53—42no—
63C22⋊Dic53—42no—
64#11719921—24no—
65#11719921—22no—
66#11719921—42no—
67#11719921—42no—
68C4⋊Dic53—22no—
69#11719919—22no—
70#11719919—24no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—21no—
84C22×D53—21no—
85C22×D53C2×D412yes—
86C22×D53#4114yes—
87Q8×C53—44no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53C2×D412yes—
98C22⋊D53—44no—
99C22⋊D53—44no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53C2×D412yes—
110D212—44no—
111C4×D53—44no—
112C4×D53—48no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53C22×D411yescharacteristic
119C22×C53C2×#4112yes—
120C22×C53H(5,𝔽2)12yes—
121C22×C53H(5,𝔽2)11yescharacteristic
122C22×C532−1+411yescharacteristic
123C2×D53—22no—
124C2×D53—22no—
125C2×D53—24no—
126C2×D53—24no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53#12414yes—
133C4×C53—22no—
134C4×C53—41no—
135C4×C53—42no—
136C4×C53—42no—
137Dic53—22no—
138Dic53—41no—
139Dic53—42no—
140Dic53—42no—
141#2639—531nomaximal, Sylow
142C2×C53—81no—
143C2×C53C2×#12411yescharacteristic
144C2×C53#53411yescharacteristic
145C2×C53#53711yescharacteristic
146C2×C53#53812yes—
147C2×C53#53912yes—
148D53—24no—
149D53—81no—
150C2×C22⋊Q8—531no—
151C2×#121—531no—
152#400—532no—
153#397—531no—
154#397—531no—
155#394—532no—
156#393—531no—
157#393D5311yescharacteristic, complemented
158#386—532no—
159#386—532no—
160C2×#95—531no—
161C53#263911yescharacteristic, complemented, Sylow
162C22×Q8C2×D5311yescharacteristic
163C22×D4—531no—
164C22×D4—531no—
165C22×D4—531no—
166C23×C4—531no—
167C23×C4—532no—
168C22⋊Q8—1064no—
169#121—1064no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—531no—
173C2×C22⋊C4—531no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4C2×D5311yescharacteristic
179C2×C22⋊C4C2×D5312yes—
180#95—531no—
181#95—532no—
182#95—532no—
183#95—532no—
184#95—532no—
185#95C2×D5311yescharacteristic
186#95C2×D5312yes—
187C24—531no—
188C24—531no—
189C24C22×D5311yescharacteristic
190C2×Q8—22no—
191C2×Q8—42no—
192C2×D4—1062no—
193C2×D4—1064no—
194C2×D4—1062no—
195C2×D4—1062no—
196C2×D4—1064no—
197C2×D4—2122no—
198C22×C4—531no—
199C22×C4—534no—
200C22×C4—532no—
201C22×C4—531no—
202C22×C4—532no—
203C22×C4—1064no—
204C22×C4—2122no—
205C22×C4—2122no—
206C22×C4—2122no—
207C22×C4C22×D5312yes—
208C22×C4C22×D5311yescharacteristic
209C22×C4C22×D5312yes—
210C22×C4C22×D5311yescharacteristic
211C4⋊C4—1062no—
212C4⋊C4—1064no—
213C4⋊C4—1062no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—1064no—
217C22⋊C4—1062no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—532no—
227C23—534no—
228C23—1062no—
229C23—1061no—
230C23—1061no—
231C23—1061no—
232C23—1062no—
233C23—1062no—
234C23—1061no—
235C23—1061no—
236C23C23×D5311yescharacteristic
237Q8—44no—
238D4—2124no—
239D4—2124no—
240D4—2124no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—532no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2124no—
261C2×C4—2128no—
262C2×C4D4×D5312yes—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—534no—
269C22—1062no—
270C22—1062no—
271C22—1064no—
272C22—1064no—
273C22—2121no—
274C22—2121no—
275C22—2121no—
276C22—2122no—
277C22—2122no—
278C22C2×D4×D5311yescharacteristic
279C22#41×D5312yes—
280C22#1313326212yes—
281C22#1313326311yescharacteristic
282C22#1313326511yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—42no—
287C4—1062no—
288C4—2121no—
289C4—2122no—
290C4—2122no—
291C2—81no—
292C2—1064no—
293C2—4241no—
294C2#124×D5311yescharacteristic
295C2#1681118311yescharacteristic
296C2#1681120311yescharacteristic
297C2#1681120712yes—
298C2#1681121912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2639
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
3. #16810564C2semidirect
4. #16810550C2semidirect
5. #16810548C2semidirect
6. #16810538C2semidirect
7. #16810534C2semidirect
8. #16810501C2semidirect
9. #16810499C2semidirect
10. #95×D53C2semidirect
11. C53×#393C2semidirect
157. #393D53semidirect
161. C53#2639semidirect