Finite Group Databasebeta

← #20280448

Subgroups — #20280448

Order 6784.

824 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133188C211yesmaximal, characteristic, complemented
2C2×#13133185C212yesmaximal, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810538C212yesmaximal, complemented
7#16810537C212yesmaximal, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810499C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15C22×C4×D53C2211yescharacteristic
16C22×C4×D53C2211yescharacteristic
17#13133188—24no—
18#13133185—28no—
19C2×C4⋊Dic53C2211yescharacteristic, complemented
20C2×#11719919C2212yescomplemented
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yescomplemented
24C2×C22⋊Dic53C2212yescomplemented
25C2×#11719921C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C22⋊C4×D53—24no—
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32#13133124C2211yescharacteristic, complemented
33C22×D4×C53C2212yes—
34C22×D4×C53C2211yescharacteristic, complemented
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yescomplemented
37#95×C53C2211yescharacteristic, complemented
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—42no—
43C23×D53—42no—
44C23×D53D414yescomplemented
45C23×D53C2311yescharacteristic
46C23×D53C2311yescharacteristic
47C2×D4×C53—24no—
48C2×D4×C53—22no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C22×C4×C53C2312yes—
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—42no—
60C2×C22⋊D53—42no—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2312yes—
63C22×Dic53C2312yes—
64C22×Dic53C2311yescharacteristic
65C2×C4×D53—22no—
66C2×C4×D53—42no—
67C2×C4×D53—42no—
68C2×C4×D53D414yescomplemented
69C22⋊C4×C53—22no—
70C22⋊C4×C53—42no—
71C22⋊Dic53—24no—
72C22⋊Dic53—22no—
73C22⋊Dic53—24no—
74C22⋊Dic53—42no—
75#11719921—24no—
76#11719921—24no—
77#11719921—22no—
78#11719921—22no—
79C4⋊Dic53—22no—
80#11719919—42no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×D412yes—
93C23×C53C2411yescharacteristic
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—28no—
101C22×D53—48no—
102C22×D53—44no—
103C22×D53—44no—
104C22×D53—48no—
105C22×D53C2×D412yes—
106C22×D53C2×D412yes—
107C22×D53C2×D412yes—
108C22×D53#4112yes—
109D4×C53—44no—
110D4×C53—48no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53C2×D412yes—
118C22⋊D53—48no—
119C22⋊D53—48no—
120C2×Dic53—22no—
121C2×Dic53—21no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53C2×D412yes—
130C4×D53—44no—
131C4×D53—44no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53C22×D411yescharacteristic
145C22×C53C22×D411yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53H(5,𝔽2)12yes—
148C22×C53H(5,𝔽2)12yes—
149C2×D53—22no—
150C2×D53—24no—
151C2×D53—22no—
152C2×D53—24no—
153C2×D53—22no—
154C2×D53—24no—
155C2×D53—416no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—42no—
159C2×D53#12114yes—
160C4×C53—22no—
161C4×C53—42no—
162C4×C53—41no—
163Dic53—22no—
164Dic53—42no—
165Dic53—42no—
166Dic53—41no—
167#2617—531nomaximal, Sylow
168C2×C53—42no—
169C2×C53—82no—
170C2×C53C2×#12111yescharacteristic
171C2×C53#53411yescharacteristic
172C2×C53#53412yes—
173C2×C53#53511yescharacteristic
174C2×C53#53812yes—
175D53—24no—
176D53—42no—
177C2×#121—531no—
178C2×#120—532no—
179C22×C22⋊C4—531no—
180#397—531no—
181#397—531no—
182#394—532no—
183#394—532no—
184#392—531no—
185#392D5311yescharacteristic, complemented
186#386—532no—
187#380—531no—
188C53#261711yescharacteristic, complemented, Sylow
189C25—531no—
190C22×D4—532no—
191C22×D4—532no—
192C22×D4C2×D5312yes—
193C22×D4C2×D5311yescharacteristic, complemented
194C23×C4—531no—
195C23×C4—531no—
196#121—1064no—
197#120—1068no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—532no—
200C2×C22⋊C4—531no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4—532no—
203C2×C22⋊C4—531no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4—532no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—1064no—
209C2×C22⋊C4C2×D5311yescharacteristic
210C2×C22⋊C4C2×D5312yescomplemented
211#95—531no—
212#95—532no—
213#95—532no—
214#95C2×D5311yescharacteristic, complemented
215C24—531no—
216C24—531no—
217C24—534no—
218C24—1062no—
219C24—1062no—
220C24—2122no—
221C24—2122no—
222C24C22×D5312yes—
223C24C22×D5311yescharacteristic
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—1064no—
230C2×D4—1064no—
231C2×D4—1064no—
232C2×D4—1064no—
233C2×D4—2122no—
234C2×D4—2122no—
235C22×C4—531no—
236C22×C4—534no—
237C22×C4—531no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—1062no—
241C22×C4—2122no—
242C22×C4—2122no—
243C22×C4C22×D5311yescharacteristic
244C22×C4C22×D5312yes—
245C22×C4C22×D5311yescharacteristic
246C4⋊C4—1062no—
247C4⋊C4—2122no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—1062no—
251C22⋊C4—1064no—
252C22⋊C4—1064no—
253C22⋊C4—1062no—
254C22⋊C4—1064no—
255C22⋊C4—1064no—
256C22⋊C4—1062no—
257C22⋊C4—2122no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—532no—
270C23—532no—
271C23—532no—
272C23—532no—
273C23—1068no—
274C23—1062no—
275C23—1062no—
276C23—1061no—
277C23—1061no—
278C23—1062no—
279C23—1062no—
280C23—2128no—
281C23—2124no—
282C23—2124no—
283C23—2128no—
284C23D4×D5312yes—
285C23C23×D5311yescharacteristic
286D4—44no—
287D4—48no—
288D4—2128no—
289D4—2128no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—532no—
297C2×C4—1062no—
298C2×C4—1061no—
299C2×C4—1062no—
300C2×C4—1062no—
301C2×C4—1062no—
302C2×C4—1062no—
303C2×C4—1062no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4D4×D5312yes—
309C22—22no—
310C22—24no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—534no—
322C22—1062no—
323C22—1064no—
324C22—1062no—
325C22—1064no—
326C22—1062no—
327C22—1064no—
328C22—21216no—
329C22—2121no—
330C22—2121no—
331C22—2122no—
332C22C2×D4×D5311yescharacteristic
333C22C2×D4×D5311yescharacteristic
334C22#41×D5311yescharacteristic
335C22#1313326212yes—
336C22#1313326212yes—
337C4—22no—
338C4—42no—
339C4—41no—
340C4—1062no—
341C4—2122no—
342C4—2122no—
343C4—2121no—
344C2—42no—
345C2—82no—
346C2—1064no—
347C2—2122no—
348C2#121×D5311yescharacteristic
349C2#1681118311yescharacteristic
350C2#1681118312yes—
351C2#1681118611yescharacteristic
352C2#1681120512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2617
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133188C2semidirect
2. C2×#13133185C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810550C2semidirect
5. #16810548C2semidirect
6. #16810538C2semidirect
7. #16810537C2semidirect
8. #16810531C2semidirect
9. #16810499C2semidirect
10. #16810482C2semidirect
11. C53×#392C2semidirect
19. C2×C4⋊Dic53C22semidirect
20. C2×#11719919C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×C22⋊Dic53C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
34. C22×D4×C53C22semidirect
36. C2×C22⋊C4×C53C22semidirect
37. #95×C53C22semidirect
44. C23×D53D4semidirect
68. C2×C4×D53D4semidirect
185. #392D53semidirect
188. C53#2617semidirect
193. C22×D4C2×D53semidirect
210. C2×C22⋊C4C2×D53semidirect
214. #95C2×D53semidirect