Finite Group Databasebeta

← #20280932

Subgroups — #20280932

Order 6784.

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

normal subgroups only

This group has 932 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×#13133213C211yesmaximal, characteristic, complemented
2C2×#13133189C212yesmaximal, complemented
3C2×#13133186C212yesmaximal, complemented
4#16810538C212yesmaximal, complemented
5#16810532C212yesmaximal, complemented
6#16810531C212yesmaximal, complemented
7#16810511C212yesmaximal, complemented
8#16810503C211yesmaximal, characteristic, complemented
9C2×#120×C53C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic, complemented
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yescomplemented
15C23×Dic53C2211yescharacteristic, complemented
16#13133213—24no—
17#13133189—28no—
18#13133186—28no—
19C2×#11719919C2212yescomplemented
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2211yescharacteristic, complemented
24C2×#11719921C2212yescomplemented
25C2×#11719921C2212yescomplemented
26C2×C4×Dic53C2211yescharacteristic, complemented
27#13133124C2212yescomplemented
28#13133124C2212yescomplemented
29C25×C53C2211yescharacteristic
30C22×D4×C53C2212yes—
31C22×D4×C53C2211yescharacteristic
32#120×C53—24no—
33C2×C22⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C24×C53—22no—
36C24×C53—24no—
37C24×C53—42no—
38C24×C53—41no—
39C24×C53—41no—
40C24×C53C2311yescharacteristic
41C24×C53C2312yes—
42C24×C53C2311yescharacteristic
43C23×D53C2312yes—
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53—42no—
49C2×D4×C53D414yescomplemented
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—24no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—42no—
60C2×C22⋊D53—42no—
61C2×C22⋊D53—42no—
62C2×C22⋊D53—42no—
63C2×C22⋊D53—42no—
64C22×Dic53—24no—
65C22×Dic53—41no—
66C22×Dic53C2312yes—
67C22×Dic53C2312yes—
68C22×Dic53C2312yes—
69C2×D212—22no—
70C2×D212—41no—
71C2×D212—41no—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—24no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76C22⋊Dic53—41no—
77C22⋊Dic53D418yescomplemented
78#11719921—24no—
79#11719921—42no—
80#11719919—24no—
81C4×Dic53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53—21no—
96C23×C53—44no—
97C23×C53—48no—
98C23×C53—44no—
99C23×C53—48no—
100C23×C53—44no—
101C23×C53C2×D414yes—
102C23×C53C2×D412yes—
103C23×C53C2×D412yes—
104C23×C53C2411yescharacteristic
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—22no—
112D4×C53—44no—
113D4×C53—48no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53C2×D412yes—
120C22⋊D53—48no—
121C22⋊D53—48no—
122C22⋊D53—48no—
123C22⋊D53—48no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—44no—
132C2×Dic53C2×D414yes—
133C2×Dic53C2×D414yes—
134D212—44no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—24no—
138C22×C53—24no—
139C22×C53—24no—
140C22×C53—24no—
141C22×C53—24no—
142C22×C53—416no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—42no—
153C22×C53C22×D412yes—
154C22×C53C22×D411yescharacteristic
155C22×C53H(5,𝔽2)11yescharacteristic
156C22×C53H(5,𝔽2)11yescharacteristic
157C22×C53H(5,𝔽2)12yes—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—42no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—42no—
165C4×C53—22no—
166C4×C53—42no—
167Dic53—24no—
168Dic53—24no—
169Dic53—42no—
170#2669—531nomaximal, Sylow
171C2×C53—42no—
172C2×C53—42no—
173C2×C53—44no—
174C2×C53#53411yescharacteristic
175C2×C53#53512yes—
176C2×C53D4211yescharacteristic
177C2×C53D4212yes—
178C2×C53#56011yescharacteristic
179D53—82no—
180C2×#127—531no—
181C2×#121—532no—
182C2×#120—532no—
183C2×#120D5311yescharacteristic, complemented
184#394—532no—
185#392—532no—
186#392—532no—
187#388—532no—
188#386—531no—
189C53#266911yescharacteristic, complemented, Sylow
190C25C2×D5311yescharacteristic
191C22×D4—532no—
192C22×D4—532no—
193C22×D4—532no—
194C22×D4—532no—
195C22×D4—531no—
196C22×D4C2×D5312yes—
197C22×D4C2×D5311yescharacteristic
198C23×C4—531no—
199#127—1064no—
200#121—1068no—
201#120—24no—
202#120—1068no—
203C2×C4⋊C4—532no—
204C2×C22⋊C4—532no—
205C2×C22⋊C4—532no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—531no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4C2×D5312yes—
211C2×C22⋊C4C2×D5311yescharacteristic
212C2×C42—531no—
213#95—532no—
214#95—532no—
215C24—24no—
216C24—22no—
217C24—41no—
218C24—42no—
219C24—41no—
220C24—532no—
221C24C22×D5312yes—
222C24C22×D5311yescharacteristic
223C24C22×D5311yescharacteristic
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—1064no—
230C2×D4—1064no—
231C2×D4—1064no—
232C2×D4—1064no—
233C2×D4—1064no—
234C2×D4—1064no—
235C2×D4—1064no—
236C2×D4—1062no—
237C2×D4—2121no—
238C2×D4—2121no—
239C2×D4—2122no—
240C2×D4—2122no—
241C2×D4—2122no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4C22⋊D5314yescomplemented
245C22×C4—532no—
246C22×C4—532no—
247C22×C4—532no—
248C22×C4—1064no—
249C22×C4—2121no—
250C22×C4C22×D5312yes—
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—1064no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—538no—
256C22⋊C4—1064no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—2122no—
260C22⋊C4—2121no—
261C42—1062no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—44no—
277C23—48no—
278C23—44no—
279C23—48no—
280C23—44no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—1062no—
285C23—1062no—
286C23—1062no—
287C23—1062no—
288C23D4×D5314yes—
289C23C2×C22⋊D5312yes—
290C23C2×C22⋊D5312yes—
291C23C23×D5311yescharacteristic
292D4—44no—
293D4—48no—
294D4—2124no—
295D4—2128no—
296D4—2128no—
297D4—2128no—
298D4—2128no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—534no—
305C2×C4—534no—
306C2×C4—1062no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4—2124no—
314C2×C4C2×C22⋊D5312yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—416no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—2122no—
334C22—2122no—
335C22—2122no—
336C22—2122no—
337C22—2122no—
338C22—2122no—
339C22—2122no—
340C22C2×D4×D5312yes—
341C22C22×C22⋊D5311yescharacteristic
342C22#1313326211yescharacteristic
343C22#1313326212yes—
344C22#1313326211yescharacteristic
345C4—22no—
346C4—42no—
347C4—1064no—
348C4—1064no—
349C4—2122no—
350C2—42no—
351C2—44no—
352C2—42no—
353C2—4242no—
354C2#1681118211yescharacteristic
355C2#1681118812yes—
356C2#1681125511yescharacteristic
357C2D4×C22⋊D5312yes—
358C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2669
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133189C2semidirect
3. C2×#13133186C2semidirect
4. #16810538C2semidirect
5. #16810532C2semidirect
6. #16810531C2semidirect
7. #16810511C2semidirect
8. #16810503C2semidirect
9. C2×#120×C53C2semidirect
10. C22×D212C22semidirect
14. C22×C22⋊D53C22semidirect
15. C23×Dic53C22semidirect
19. C2×#11719919C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. C2×C4×Dic53C22semidirect
27. #13133124C22semidirect
28. #13133124C22semidirect
49. C2×D4×C53D4semidirect
77. C22⋊Dic53D4semidirect
183. C2×#120D53semidirect
189. C53#2669semidirect
244. C2×D4C22⋊D53semidirect