Finite Group Databasebeta

← #20280493

Subgroups — #20280493

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×#13133187C211yesmaximal, characteristic
2C22×#11719921C211yesmaximal, characteristic, complemented
3#16810552C212yesmaximal, complemented
4#16810549C212yesmaximal, complemented
5#16810541C212yesmaximal, complemented
6#16810540C212yesmaximal, complemented
7#16810502C211yesmaximal, characteristic, complemented
8#16810500C211yesmaximal, characteristic, complemented
9#16810495C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic, complemented
11C53×#385C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×D212C2211yescharacteristic
15C22×C22⋊D53C2211yescharacteristic
16C23×Dic53C2211yescharacteristic
17#13133187—24no—
18C2×#11719919C2212yescomplemented
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921—24no—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yescomplemented
28C2×#11719921C2212yescomplemented
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32C23×C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2212yescomplemented
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—22no—
40C23×D53—42no—
41C23×D53—42no—
42C23×D53D414yes—
43C23×D53C2311yescharacteristic
44C23×D53C2311yescharacteristic
45C23×D53C2311yescharacteristic
46C22×C4×C53—22no—
47C22×C4×C53—42no—
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C22×Dic53—22no—
56C22×Dic53—42no—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2312yes—
59C22×Dic53C2312yes—
60C2×D212—22no—
61C2×D212—22no—
62C2×D212—42no—
63C2×D212—42no—
64C4⋊C4×C53—42no—
65C22⋊C4×C53—22no—
66C22⋊C4×C53—22no—
67C22⋊Dic53—22no—
68#11719921—24no—
69#11719921—24no—
70#11719921—22no—
71#11719921—22no—
72#11719921—42no—
73#11719921—42no—
74#11719921—42no—
75#11719921—42no—
76#11719919—42no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53C2×D412yes—
80C23×C53#4112yes—
81C23×C53#4112yes—
82C23×C53C2411yescharacteristic
83C22×D53—28no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—44no—
98C22×D53—44no—
99C22×D53—48no—
100C22×D53—48no—
101C22×D53C2×D412yes—
102C22×D53C2×D412yes—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
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—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—44no—
125D212—44no—
126D212—44no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53#12314yes—
132C22×C53C22×D411yescharacteristic
133C22×C53C2×#4111yescharacteristic
134C22×C53C2×#4111yescharacteristic
135C22×C53H(5,𝔽2)12yes—
136C22×C53H(5,𝔽2)12yes—
137C2×D53—22no—
138C2×D53—22no—
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—22no—
142C2×D53—22no—
143C2×D53—416no—
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53—41no—
149C2×D53—41no—
150C2×D53—41no—
151C2×D53—42no—
152C2×D53—42no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—42no—
156C4×C53—42no—
157Dic53—41no—
158Dic53—42no—
159Dic53—42no—
160#2622—531nomaximal, Sylow
161C2×C53—24no—
162C2×C53C2×#12311yescharacteristic
163C2×C53#53411yescharacteristic
164C2×C53#53411yescharacteristic
165C2×C53#53812yes—
166C2×C53#53812yes—
167D53—42no—
168D53—42no—
169D53—81no—
170C2×#120—531no—
171C22×C22⋊C4—531no—
172#397—532no—
173#397—532no—
174#394—532no—
175#394—532no—
176#386—531no—
177#386—531no—
178#385—531no—
179#385D5311yescharacteristic, complemented
180#379—531no—
181C53#262211yescharacteristic, complemented, Sylow
182C25—531no—
183C22×D4—531no—
184C22×D4—531no—
185C22×D4—531no—
186C23×C4—531no—
187C23×C4C2×D5311yescharacteristic
188#120—1064no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4C2×D5312yescomplemented
191C2×C22⋊C4—531no—
192C2×C22⋊C4—532no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4—531no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—1064no—
201C2×C22⋊C4C2×D5311yescharacteristic
202C2×C22⋊C4C2×D5311yescharacteristic
203#95—532no—
204#95—532no—
205#95—532no—
206#95C2×D5312yescomplemented
207C24—531no—
208C24—531no—
209C24—534no—
210C24—531no—
211C24—1062no—
212C24—1062no—
213C24—2122no—
214C24—2122no—
215C24C22×D5311yescharacteristic
216C2×D4—1062no—
217C2×D4—1062no—
218C2×D4—1062no—
219C2×D4—1062no—
220C2×D4—1062no—
221C2×D4—2122no—
222C2×D4—2122no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—531no—
226C22×C4—532no—
227C22×C4—532no—
228C22×C4—1062no—
229C22×C4—2122no—
230C22×C4C22×D5311yescharacteristic
231C22×C4C22×D5312yes—
232C22×C4C22×D5312yes—
233C22×C4C22×D5311yescharacteristic
234C4⋊C4—42no—
235C4⋊C4—2122no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—1064no—
239C22⋊C4—1064no—
240C22⋊C4—1062no—
241C22⋊C4—1062no—
242C22⋊C4—1062no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C23—22no—
248C23—22no—
249C23—532no—
250C23—532no—
251C23—1068no—
252C23—1061no—
253C23—1061no—
254C23—1061no—
255C23—1061no—
256C23—1062no—
257C23—1062no—
258C23—1062no—
259C23—1062no—
260C23—1061no—
261C23—1061no—
262C23—1061no—
263C23—1062no—
264C23—1062no—
265C23—2124no—
266C23—2124no—
267C23—2128no—
268C23—2128no—
269C23#1171994512yes—
270C23D4×D5312yes—
271C23#1171994912yes—
272C23C23×D5311yescharacteristic
273D4—2124no—
274D4—2124no—
275D4—2124no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—1062no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1062no—
291C2×C4—1061no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2124no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—1062no—
302C22—1062no—
303C22—1062no—
304C22—1062no—
305C22—1062no—
306C22—1062no—
307C22—21216no—
308C22—2121no—
309C22—2121no—
310C22—2121no—
311C22—2122no—
312C22—2122no—
313C22—2122no—
314C22—2122no—
315C22—2122no—
316C22—2122no—
317C22#1313320214yes—
318C22C2×D4×D5311yescharacteristic
319C22C2×#1171994511yescharacteristic
320C22C2×#1171994911yescharacteristic
321C22#1313326212yes—
322C22#1313326312yes—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—42no—
327C4—2121no—
328C4—2122no—
329C4—2122no—
330C2—24no—
331C2—4241no—
332C2—2122no—
333C2—2122no—
334C2C2×#1313320211yescharacteristic
335C2#1681118311yescharacteristic
336C2#1681118411yescharacteristic
337C2#1681120612yes—
338C2#1681121312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2622
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×#11719921C2semidirect
3. #16810552C2semidirect
4. #16810549C2semidirect
5. #16810541C2semidirect
6. #16810540C2semidirect
7. #16810502C2semidirect
8. #16810500C2semidirect
9. #16810495C2semidirect
10. #16810478C2semidirect
11. C53×#385C2semidirect
18. C2×#11719919C22semidirect
27. C2×#11719921C22semidirect
28. C2×#11719921C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
179. #385D53semidirect
181. C53#2622semidirect
190. C2×C4⋊C4C2×D53semidirect
206. #95C2×D53semidirect