Finite Group Databasebeta

← #20279608

Subgroups — #20279608

Order 6784.

800 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133186C211yesmaximal, characteristic
3C2×#13133177C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
5#16810546C212yesmaximal, complemented
6#16810538C212yesmaximal, complemented
7#16810516C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810484C211yesmaximal, characteristic, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C53×#379C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16C22×C4×D53C2211yescharacteristic, complemented
17#13133201—24no—
18#13133186—24no—
19#13133177—24no—
20C2×C4⋊Dic53C2212yescomplemented
21C2×#11719919C2211yescharacteristic, complemented
22C2×#11719919C2212yescomplemented
23C2×C22⋊Dic53—24no—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C22⋊Dic53C2212yes—
27C2×C22⋊Dic53C2211yescharacteristic
28C2×#11719921C2212yescomplemented
29C2×#11719921C2211yescharacteristic, complemented
30C2×C4×Dic53C2211yescharacteristic, complemented
31#13133124C2212yescomplemented
32#13133124C2212yescomplemented
33#13133124C2212yescomplemented
34C25×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2211yescharacteristic
38C2×C22⋊C4×C53C2212yes—
39C24×C53—22no—
40C24×C53—24no—
41C24×C53—22no—
42C24×C53—22no—
43C24×C53—42no—
44C24×C53C2312yes—
45C24×C53C2311yescharacteristic
46C23×D53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53D414yes—
56C22×Dic53—22no—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—42no—
60C22×Dic53D414yescomplemented
61C22×Dic53C2312yes—
62C22×Dic53C2312yes—
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C2×C4×D53—22no—
67C2×C4×D53—42no—
68C22⋊C4×C53—22no—
69C22⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—42no—
72C22⋊Dic53—22no—
73C22⋊Dic53—24no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76#11719921—24no—
77#11719921—22no—
78C4⋊Dic53—24no—
79#11719919—22no—
80#11719919—42no—
81C4×Dic53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—28no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—48no—
92C23×C53—48no—
93C23×C53—44no—
94C23×C53—44no—
95C23×C53C2×D412yes—
96C23×C53C2×D412yes—
97C23×C53#4112yes—
98C23×C53#4114yes—
99C23×C53C2411yescharacteristic
100C22×D53—21no—
101C22×D53—22no—
102C22×D53—21no—
103C22×D53—22no—
104C22×D53C2×D412yes—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C22⋊D53—28no—
113C22⋊D53—48no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126C2×Dic53C2×D412yes—
127C2×Dic53C2×D412yes—
128C2×Dic53C2×D412yes—
129C4×D53—44no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—24no—
133C22×C53—24no—
134C22×C53—24no—
135C22×C53—24no—
136C22×C53—24no—
137C22×C53—416no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53#12114yes—
143C22×C53C22×D411yescharacteristic
144C22×C53C22×D411yescharacteristic
145C22×C53C2×#4112yes—
146C22×C53C2×#4111yescharacteristic
147C22×C53H(5,𝔽2)12yes—
148C2×D53—22no—
149C2×D53—24no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—42no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—42no—
156Dic53—22no—
157Dic53—22no—
158Dic53—22no—
159Dic53—42no—
160Dic53—42no—
161#2534—531nomaximal, Sylow
162C2×C53—24no—
163C2×C53—44no—
164C2×C53C2×#12111yescharacteristic
165C2×C53#52511yescharacteristic
166C2×C53#53411yescharacteristic
167C2×C53#54612yes—
168C2×C53#55112yes—
169D53—42no—
170C2×#123—531no—
171C2×#120—531no—
172C2×C4×D4—531no—
173C22×C22⋊C4—531no—
174#396—532no—
175#394—532no—
176#388—532no—
177#386—532no—
178#381—531no—
179#380—531no—
180#379D5311yescharacteristic, complemented
181C53#253411yescharacteristic, complemented, Sylow
182C25C2×D5311yescharacteristic
183C22×D4—531no—
184C22×D4—532no—
185C23×C4—531no—
186C23×C4—531no—
187C23×C4—531no—
188#123—1064no—
189#120—1064no—
190C4×D4—1064no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—532no—
193C2×C4⋊C4—532no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—532no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—1064no—
201C2×C22⋊C4C2×D5311yescharacteristic
202C2×C22⋊C4C2×D5311yescharacteristic
203C2×C22⋊C4C2×D5312yes—
204C2×C22⋊C4C2×D5312yes—
205C2×C42—531no—
206#95—532no—
207#95—532no—
208#95—532no—
209C24—22no—
210C24—24no—
211C24—22no—
212C24—22no—
213C24—42no—
214C24—531no—
215C24C22×D5312yes—
216C24C22×D5311yescharacteristic
217C2×D4—534no—
218C2×D4—1062no—
219C2×D4—1062no—
220C2×D4—1064no—
221C2×D4—1064no—
222C2×D4—2122no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—534no—
227C22×C4—532no—
228C22×C4—532no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—2122no—
234C22×C4—2122no—
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C22×C4C22×D5312yes—
238C4⋊C4—1062no—
239C4⋊C4—1064no—
240C4⋊C4—2122no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—42no—
245C22⋊C4—1062no—
246C22⋊C4—1064no—
247C22⋊C4—1064no—
248C22⋊C4—1064no—
249C22⋊C4—1064no—
250C22⋊C4—1062no—
251C42—1062no—
252C23—22no—
253C23—22no—
254C23—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—48no—
262C23—48no—
263C23—44no—
264C23—44no—
265C23—532no—
266C23—1061no—
267C23—1062no—
268C23—1061no—
269C23—1062no—
270C23#1171994512yes—
271C23D4×D5312yes—
272C23D4×D5312yes—
273C23#1171994714yes—
274C23C23×D5311yescharacteristic
275D4—1068no—
276D4—2128no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—532no—
285C2×C4—532no—
286C2×C4—532no—
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—1062no—
297C2×C4—2124no—
298C2×C4—2124no—
299C2×C4—2124no—
300C22—24no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—416no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—1062no—
313C22—1064no—
314C22—2122no—
315C22—2121no—
316C22—2121no—
317C22#1313319014yes—
318C22C2×D4×D5311yescharacteristic
319C22C2×D4×D5311yescharacteristic
320C22C2×#1171994712yes—
321C22C2×#1171994511yescharacteristic
322C22#1313326212yes—
323C4—41no—
324C4—41no—
325C4—42no—
326C4—1062no—
327C4—1062no—
328C4—1062no—
329C4—2122no—
330C4—2122no—
331C2—24no—
332C2—44no—
333C2—2122no—
334C2C2×#1313319011yescharacteristic
335C2#1681112911yescharacteristic
336C2#1681118311yescharacteristic
337C2#1681126012yes—
338C2#1681129912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2534
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
3. C2×#13133177C2semidirect
4. C22×C22⋊Dic53C2semidirect
5. #16810546C2semidirect
6. #16810538C2semidirect
7. #16810516C2semidirect
8. #16810499C2semidirect
9. #16810484C2semidirect
10. #16810481C2semidirect
11. C53×#379C2semidirect
16. C22×C4×D53C22semidirect
20. C2×C4⋊Dic53C22semidirect
21. C2×#11719919C22semidirect
22. C2×#11719919C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. C2×C4×Dic53C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
33. #13133124C22semidirect
60. C22×Dic53D4semidirect
180. #379D53semidirect
181. C53#2534semidirect