Finite Group Databasebeta

← #20280492

Subgroups — #20280492

Order 6784.

702 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133185C211yesmaximal, characteristic
2C22×#11719921C211yesmaximal, characteristic, complemented
3#16810552C212yesmaximal, complemented
4#16810550C212yesmaximal, complemented
5#16810539C212yesmaximal, complemented
6#16810538C212yesmaximal, complemented
7#16810503C211yesmaximal, characteristic, complemented
8#16810498C211yesmaximal, characteristic, complemented
9#16810496C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic, complemented
11C53×#386C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133185—24no—
17C2×C4⋊Dic53C2212yescomplemented
18C2×#11719919C2212yescomplemented
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yescomplemented
22C2×#11719921—24no—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32C22×D4×C53C2211yescharacteristic
33C23×C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yescomplemented
36#95×C53C2212yescomplemented
37C24×C53C2311yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53—22no—
40C23×D53—22no—
41C23×D53—42no—
42C23×D53—42no—
43C23×D53D414yes—
44C23×D53C2311yescharacteristic
45C23×D53C2311yescharacteristic
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—42no—
51C22×C4×C53C2311yescharacteristic
52C22×C4×C53C2312yes—
53C22×C4×C53C2312yes—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—42no—
57C2×C22⋊D53—42no—
58C22×Dic53—22no—
59C22×Dic53—42no—
60C22×Dic53C2312yes—
61C22×Dic53C2312yes—
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22⋊C4×C53—22no—
65C22⋊C4×C53—42no—
66C22⋊Dic53—22no—
67C22⋊Dic53—22no—
68C22⋊Dic53—42no—
69#11719921—24no—
70#11719921—24no—
71#11719921—22no—
72#11719921—22no—
73#11719921—42no—
74#11719921—42no—
75C4⋊Dic53—42no—
76#11719919—42no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53C2×D412yes—
85C23×C53#4112yes—
86C23×C53#4112yes—
87C23×C53C2411yescharacteristic
88C22×D53—28no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—44no—
98C22×D53—44no—
99C22×D53—48no—
100C22×D53—48no—
101C22×D53C2×D412yes—
102C22×D53C2×D412yes—
103D4×C53—44no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—44no—
114C22⋊D53—44no—
115C22⋊D53—44no—
116C2×Dic53—22no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—44no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53#12314yes—
137C22×C53C22×D411yescharacteristic
138C22×C53C2×#4111yescharacteristic
139C22×C53C2×#4111yescharacteristic
140C22×C53H(5,𝔽2)12yes—
141C22×C53H(5,𝔽2)12yes—
142C2×D53—22no—
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—22no—
148C2×D53—416no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—42no—
152C2×D53—42no—
153C4×C53—41no—
154C4×C53—42no—
155C4×C53—42no—
156Dic53—42no—
157Dic53—41no—
158Dic53—41no—
159Dic53—42no—
160#2622—531nomaximal, Sylow
161C2×C53—24no—
162C2×C53—81no—
163C2×C53C2×#12311yescharacteristic
164C2×C53#53411yescharacteristic
165C2×C53#53411yescharacteristic
166C2×C53#53812yes—
167C2×C53#53812yes—
168D53—42no—
169D53—42no—
170C2×#120—531no—
171C22×C22⋊C4—531no—
172#397—532no—
173#397—532no—
174#394—532no—
175#394—532no—
176#386—531no—
177#386D5311yescharacteristic, complemented
178#385—531no—
179#385—531no—
180#379—531no—
181C53#262211yescharacteristic, complemented, Sylow
182C25—531no—
183C22×D4—531no—
184C22×D4—531no—
185C22×D4C2×D5311yescharacteristic
186C23×C4—531no—
187C23×C4C2×D5311yescharacteristic
188#120—1064no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—532no—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—1064no—
201C2×C22⋊C4C2×D5311yescharacteristic
202C2×C22⋊C4C2×D5312yescomplemented
203#95—532no—
204#95—532no—
205#95—532no—
206#95C2×D5312yescomplemented
207C24—531no—
208C24—531no—
209C24—534no—
210C24—1062no—
211C24—1062no—
212C24—2122no—
213C24—2122no—
214C24C22×D5311yescharacteristic
215C24C22×D5311yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—1062no—
220C2×D4—1062no—
221C2×D4—2122no—
222C2×D4—2122no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—532no—
226C22×C4—532no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—1062no—
230C22×C4—2122no—
231C22×C4C22×D5311yescharacteristic
232C22×C4C22×D5312yes—
233C22×C4C22×D5312yes—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—1064no—
239C22⋊C4—1064no—
240C22⋊C4—1062no—
241C22⋊C4—1062no—
242C22⋊C4—1062no—
243C22⋊C4—1062no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—532no—
255C23—532no—
256C23—1068no—
257C23—1061no—
258C23—1061no—
259C23—1061no—
260C23—1061no—
261C23—1062no—
262C23—1062no—
263C23—1062no—
264C23—1062no—
265C23—2124no—
266C23—2124no—
267C23—2128no—
268C23—2128no—
269C23#1171994512yes—
270C23D4×D5312yes—
271C23#1171994712yes—
272C23C23×D5311yescharacteristic
273D4—44no—
274D4—2124no—
275D4—2124no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—1062no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1062no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2124no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—1062no—
307C22—1062no—
308C22—1062no—
309C22—1062no—
310C22—1062no—
311C22—1062no—
312C22—21216no—
313C22—2122no—
314C22—2122no—
315C22—2122no—
316C22—2122no—
317C22#1313319914yes—
318C22C2×D4×D5311yescharacteristic
319C22C2×#1171994711yescharacteristic
320C22C2×#1171994511yescharacteristic
321C22#1313326212yes—
322C22#1313326312yes—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—2121no—
327C4—2122no—
328C4—2121no—
329C4—2122no—
330C2—24no—
331C2—81no—
332C2—2122no—
333C2—2122no—
334C2C2×#1313319911yescharacteristic
335C2#1681118311yescharacteristic
336C2#1681118411yescharacteristic
337C2#1681120612yes—
338C2#1681121012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2622
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#11719921C2semidirect
3. #16810552C2semidirect
4. #16810550C2semidirect
5. #16810539C2semidirect
6. #16810538C2semidirect
7. #16810503C2semidirect
8. #16810498C2semidirect
9. #16810496C2semidirect
10. #16810478C2semidirect
11. C53×#386C2semidirect
17. C2×C4⋊Dic53C22semidirect
18. C2×#11719919C22semidirect
21. C2×C22⋊Dic53C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
35. C2×C22⋊C4×C53C22semidirect
36. #95×C53C22semidirect
177. #386D53semidirect
181. C53#2622semidirect
202. C2×C22⋊C4C2×D53semidirect
206. #95C2×D53semidirect