Finite Group Databasebeta

← #20278098

Subgroups — #20278098

Order 6784.

786 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 161 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×#13133201C211yesmaximal, characteristic, complemented
2C2×#13133187C211yesmaximal, characteristic
3C2×#13133173C211yesmaximal, characteristic, complemented
4C22×#11719921C211yesmaximal, characteristic, complemented
5#16810545C212yesmaximal, complemented
6#16810541C212yesmaximal, complemented
7#16810524C212yesmaximal, complemented
8#16810515C212yesmaximal, complemented
9#16810502C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic, complemented
11C53×#385C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2212yes—
14C22×C22⋊D53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133201—24no—
17#13133187—24no—
18#13133173—24no—
19C2×C4⋊Dic53C2211yescharacteristic, complemented
20C2×C4⋊Dic53C2211yescharacteristic, complemented
21C2×#11719919C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×#11719921—24no—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2211yescharacteristic, complemented
30C2×#11719921C2211yescharacteristic, complemented
31C2×#11719921C2212yescomplemented
32C2×C4×Dic53C2212yescomplemented
33#13133124C2212yescomplemented
34C23×C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2212yescomplemented
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2211yescharacteristic
38#95×C53C2212yescomplemented
39C24×C53C2311yescharacteristic
40C23×D53—22no—
41C23×D53—24no—
42C23×D53—22no—
43C23×D53—22no—
44C23×D53—42no—
45C23×D53C2311yescharacteristic
46C23×D53C2312yes—
47C22×C4×C53—22no—
48C22×C4×C53—42no—
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C22×C4×C53C2312yes—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53D414yescomplemented
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C2×D212—24no—
64C2×D212—24no—
65C2×D212—42no—
66C4⋊C4×C53—42no—
67C22⋊C4×C53—22no—
68C22⋊C4×C53D414yes—
69C22⋊Dic53—22no—
70#11719921—24no—
71#11719921—24no—
72#11719921—24no—
73#11719921—24no—
74#11719921—22no—
75#11719921—22no—
76#11719921—42no—
77#11719921—42no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80#11719919—24no—
81C4×Dic53—24no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53C2×D412yes—
85C23×C53C2×D412yes—
86C23×C53#4112yes—
87C23×C53C2411yescharacteristic
88C22×D53—28no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—22no—
98C22×D53—21no—
99C22×D53—22no—
100C22×D53—48no—
101C22×D53—48no—
102C22×D53—44no—
103C22×D53—44no—
104C22×D53#4114yes—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—44no—
115C2×C4×C53C2×D414yes—
116C22⋊D53—44no—
117C2×Dic53—28no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—22no—
125C2×Dic53—21no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53C2×D412yes—
129C2×Dic53C2×D412yes—
130D212—48no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—24no—
135C22×C53#12114yes—
136C22×C53C22×D411yescharacteristic
137C22×C53C22×D411yescharacteristic
138C22×C53C2×#4112yes—
139C22×C53C2×#4111yescharacteristic
140C22×C53H(5,𝔽2)12yes—
141C2×D53—24no—
142C2×D53—24no—
143C2×D53—24no—
144C2×D53—416no—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53—42no—
149C2×D53—41no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—42no—
153C2×D53—42no—
154C4×C53—24no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—42no—
158Dic53—22no—
159Dic53—22no—
160Dic53—41no—
161Dic53—42no—
162#2411—531nomaximal, Sylow
163C2×C53—24no—
164C2×C53C2×#12111yescharacteristic
165C2×C53#52511yescharacteristic
166C2×C53#53511yescharacteristic
167C2×C53#54612yes—
168C2×C53#55112yes—
169D53—44no—
170D53—81no—
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#386—531no—
180#385D5311yescharacteristic, complemented
181#379—531no—
182C53#241111yescharacteristic, complemented, Sylow
183C25—531no—
184C22×D4—531no—
185C22×D4—532no—
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⋊C4C2×D5312yescomplemented
195C2×C22⋊C4—531no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—531no—
200C2×C22⋊C4—531no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4—532no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—1064no—
205C2×C22⋊C4C2×D5311yescharacteristic
206C2×C22⋊C4C2×D5311yescharacteristic
207C2×C42—532no—
208#95—532no—
209#95C2×D5312yescomplemented
210C24—532no—
211C24—531no—
212C24—1064no—
213C24—1062no—
214C24—1062no—
215C24—1062no—
216C24—2122no—
217C24C22×D5311yescharacteristic
218C2×D4—1064no—
219C2×D4—1062no—
220C2×D4—1062no—
221C2×D4—1062no—
222C2×D4—1064no—
223C2×D4—2122no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—534no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—532no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4C22×D5312yes—
234C22×C4C22×D5311yescharacteristic
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5312yes—
237C4⋊C4—42no—
238C4⋊C4—1062no—
239C4⋊C4—1062no—
240C4⋊C4—1064no—
241C22⋊C4—22no—
242C22⋊C4—1064no—
243C22⋊C4—1064no—
244C22⋊C4—1062no—
245C22⋊C4—1062no—
246C22⋊C4—1064no—
247C22⋊C4—1062no—
248C22⋊C4—1064no—
249C22⋊C4—2122no—
250C22⋊C4—2122no—
251C22⋊C4D21214yes—
252C42—1064no—
253C23—22no—
254C23—22no—
255C23—534no—
256C23—1068no—
257C23—1062no—
258C23—1062no—
259C23—1062no—
260C23—1062no—
261C23—1062no—
262C23—1062no—
263C23—1061no—
264C23—1061no—
265C23—1062no—
266C23—1061no—
267C23—1062no—
268C23—2128no—
269C23—2128no—
270C23—2124no—
271C23—2124no—
272C23C2×D21212yes—
273C23D4×D5312yes—
274C23#1171994912yes—
275C23C23×D5311yescharacteristic
276D4—2128no—
277D4—2124no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—532no—
289C2×C4—532no—
290C2×C4—1068no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1062no—
297C2×C4—1061no—
298C2×C4—1062no—
299C2×C4—1061no—
300C2×C4—1062no—
301C2×C4C2×D21214yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—1064no—
307C22—1064no—
308C22—1064no—
309C22—21216no—
310C22—2122no—
311C22—2122no—
312C22—2121no—
313C22—2121no—
314C22—2121no—
315C22—2122no—
316C22—2122no—
317C22—2122no—
318C22—2122no—
319C22#1313319214yes—
320C22C2×D4×D5311yescharacteristic
321C22C22×D21211yescharacteristic
322C22#41×D5312yes—
323C22C2×#1171994911yescharacteristic
324C22#1313326212yes—
325C4—24no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—1062no—
330C4—1062no—
331C4—2122no—
332C4—2121no—
333C2—24no—
334C2—4241no—
335C2—2124no—
336C2C2×#1313319211yescharacteristic
337C2#1681113111yescharacteristic
338C2#1681118811yescharacteristic
339C2#1681126812yes—
340C2#1681130012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2411
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
3. C2×#13133173C2semidirect
4. C22×#11719921C2semidirect
5. #16810545C2semidirect
6. #16810541C2semidirect
7. #16810524C2semidirect
8. #16810515C2semidirect
9. #16810502C2semidirect
10. #16810478C2semidirect
11. C53×#385C2semidirect
19. C2×C4⋊Dic53C22semidirect
20. C2×C4⋊Dic53C22semidirect
21. C2×#11719919C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
32. C2×C4×Dic53C22semidirect
33. #13133124C22semidirect
35. C2×C4⋊C4×C53C22semidirect
38. #95×C53C22semidirect
58. C22×Dic53D4semidirect
180. #385D53semidirect
182. C53#2411semidirect
194. C2×C4⋊C4C2×D53semidirect
209. #95C2×D53semidirect