Finite Group Databasebeta

← #20278103

Subgroups — #20278103

Order 6784.

786 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133198C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic
3C2×#13133173C211yesmaximal, characteristic, complemented
4C22×#11719921C211yesmaximal, characteristic, complemented
5#16810547C212yesmaximal, complemented
6#16810538C212yesmaximal, complemented
7#16810523C212yesmaximal, complemented
8#16810512C212yesmaximal, complemented
9#16810496C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic, complemented
11C53×#386C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C23×Dic53C2211yescharacteristic
15#13133198—24no—
16#13133185—24no—
17#13133173—24no—
18C2×#11719919C2211yescharacteristic, complemented
19C2×#11719919C2212yescomplemented
20C2×#11719919C2211yescharacteristic, complemented
21C2×#11719919C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic, complemented
25C2×C22⋊Dic53C2211yescharacteristic, complemented
26C2×#11719921—24no—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31C2×#11719921C2212yes—
32C2×C4×Dic53C2212yescomplemented
33#13133124C2212yescomplemented
34C22×D4×C53C2211yescharacteristic
35C23×C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yescomplemented
38#95×C53C2212yescomplemented
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C23×D53—22no—
42C23×D53—24no—
43C23×D53—22no—
44C23×D53—22no—
45C23×D53—42no—
46C23×D53C2312yes—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C2×D4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—42no—
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2312yes—
54C22×C4×C53C2312yes—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—42no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53D414yescomplemented
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2312yes—
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22⋊C4×C53—22no—
67C22⋊C4×C53—42no—
68C22⋊Dic53—22no—
69C22⋊Dic53—22no—
70C22⋊Dic53—22no—
71C22⋊Dic53D414yes—
72#11719921—24no—
73#11719921—24no—
74#11719921—24no—
75#11719921—24no—
76#11719921—42no—
77#11719919—24no—
78#11719919—22no—
79#11719919—22no—
80#11719919—42no—
81C4×Dic53—24no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—22no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53C2×D412yes—
90C23×C53C2×D412yes—
91C23×C53#4112yes—
92C23×C53C2411yescharacteristic
93C22×D53—28no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—48no—
101C22×D53—48no—
102C22×D53—44no—
103C22×D53—44no—
104C22×D53#4114yes—
105D4×C53—44no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—44no—
116C22⋊D53—48no—
117C2×Dic53—28no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53C2×D412yes—
129C2×Dic53C2×D412yes—
130C2×Dic53C2×D414yes—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—24no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53#12114yes—
141C22×C53C22×D411yescharacteristic
142C22×C53C22×D411yescharacteristic
143C22×C53C2×#4112yes—
144C22×C53C2×#4111yescharacteristic
145C22×C53H(5,𝔽2)12yes—
146C2×D53—24no—
147C2×D53—24no—
148C2×D53—24no—
149C2×D53—416no—
150C2×D53—42no—
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—42no—
154C4×C53—41no—
155C4×C53—42no—
156C4×C53—42no—
157Dic53—22no—
158Dic53—22no—
159Dic53—24no—
160Dic53—41no—
161Dic53—41no—
162#2411—531nomaximal, Sylow
163C2×C53—24no—
164C2×C53—81no—
165C2×C53C2×#12111yescharacteristic
166C2×C53#52511yescharacteristic
167C2×C53#53511yescharacteristic
168C2×C53#54612yes—
169C2×C53#55112yes—
170D53—44no—
171C2×#123—531no—
172C2×#120—531no—
173C2×#117—531no—
174C22×C22⋊C4—531no—
175#396—532no—
176#394—532no—
177#390—532no—
178#388—532no—
179#386D5311yescharacteristic, complemented
180#385—531no—
181#379—531no—
182C53#241111yescharacteristic, complemented, Sylow
183C25—531no—
184C22×D4—532no—
185C22×D4C2×D5311yescharacteristic
186C23×C4—531no—
187C23×C4C2×D5311yescharacteristic
188#123—1064no—
189#120—1064no—
190#117—1064no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—532no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—532no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4—532no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—1064no—
205C2×C22⋊C4C2×D5311yescharacteristic
206C2×C22⋊C4C2×D5312yescomplemented
207C2×C42—532no—
208#95—532no—
209#95C2×D5312yescomplemented
210C24—532no—
211C24—1064no—
212C24—1062no—
213C24—1062no—
214C24—1062no—
215C24—2122no—
216C24C22×D5311yescharacteristic
217C24C22×D5311yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—1064no—
222C2×D4—1064no—
223C2×D4—2122no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—534no—
227C22×C4—532no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4C22×D5312yes—
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5312yes—
237C4⋊C4—1064no—
238C4⋊C4—1062no—
239C4⋊C4—1062no—
240C4⋊C4—2122no—
241C22⋊C4—22no—
242C22⋊C4—42no—
243C22⋊C4—534no—
244C22⋊C4—1064no—
245C22⋊C4—1064no—
246C22⋊C4—1064no—
247C22⋊C4—1064no—
248C22⋊C4—1062no—
249C22⋊C4—1062no—
250C22⋊C4—1062no—
251C22⋊C4—2122no—
252C42—1064no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—534no—
261C23—1068no—
262C23—1062no—
263C23—1062no—
264C23—1062no—
265C23—1062no—
266C23—1062no—
267C23—1062no—
268C23—2128no—
269C23—2128no—
270C23—2124no—
271C23—2124no—
272C23#1171994512yes—
273C23D4×D5312yes—
274C23D4×D5312yes—
275C23C23×D5311yescharacteristic
276D4—44no—
277D4—2128no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—534no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—1068no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1062no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1062no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1062no—
301C2×C4—1062no—
302C22—24no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—1064no—
312C22—1064no—
313C22—1064no—
314C22—21216no—
315C22—2122no—
316C22—2122no—
317C22—2122no—
318C22—2122no—
319C22#1313319014yes—
320C22C2×D4×D5311yescharacteristic
321C22C2×D4×D5311yescharacteristic
322C22#41×D5312yes—
323C22C2×#1171994511yescharacteristic
324C22#1313326212yes—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—1062no—
329C4—1062no—
330C4—1064no—
331C4—2121no—
332C4—2121no—
333C2—24no—
334C2—81no—
335C2—2124no—
336C2C2×#1313319011yescharacteristic
337C2#1681113111yescharacteristic
338C2#1681118811yescharacteristic
339C2#1681126212yes—
340C2#1681130212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2411
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
3. C2×#13133173C2semidirect
4. C22×#11719921C2semidirect
5. #16810547C2semidirect
6. #16810538C2semidirect
7. #16810523C2semidirect
8. #16810512C2semidirect
9. #16810496C2semidirect
10. #16810478C2semidirect
11. C53×#386C2semidirect
18. C2×#11719919C22semidirect
19. C2×#11719919C22semidirect
20. C2×#11719919C22semidirect
21. C2×#11719919C22semidirect
24. C2×C22⋊Dic53C22semidirect
25. C2×C22⋊Dic53C22semidirect
32. C2×C4×Dic53C22semidirect
33. #13133124C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. #95×C53C22semidirect
60. C22×Dic53D4semidirect
179. #386D53semidirect
182. C53#2411semidirect
206. C2×C22⋊C4C2×D53semidirect
209. #95C2×D53semidirect