Finite Group Databasebeta

← #21433343

Subgroups — #21433343

Order 7552.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 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
1C22×#11745687C211yesmaximal, characteristic, complemented
2C2×#14273751C212yesmaximal, complemented
3#121×D59C211yesmaximal, characteristic, complemented
4#16837548C212yesmaximal, complemented
5#16837544C212yesmaximal, complemented
6#16837543C211yesmaximal, characteristic, complemented
7#16837540C211yesmaximal, characteristic, complemented
8#16837536C211yesmaximal, characteristic, complemented
9#16837400C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#447C211yesmaximal, characteristic, complemented
12C2×#11745687—28no—
13C2×#11745687—21no—
14C2×#11745687—21no—
15C2×#11745687C2212yes—
16C2×#11745687C2212yescomplemented
17C22×Dic118C2211yescharacteristic, complemented
18C22×C22⋊D59—21no—
19C2×D4×D59—21no—
20C2×D4×D59C2211yescharacteristic, complemented
21C23×Dic59—21no—
22C22×C4×D59C2211yescharacteristic
23#14273751—28no—
24C2×#11745660C2212yescomplemented
25C2×#11745670C2212yescomplemented
26C2×#11745669C2212yescomplemented
27C2×#11745659C2212yescomplemented
28#14273702C2211yescharacteristic, complemented
29#14273701C2211yescharacteristic
30#14273700—21no—
31#14273699—21no—
32#14273698C2211yescharacteristic, complemented
33C4⋊C4×D59C2211yescharacteristic, complemented
34C22⋊C4×D59—21no—
35#14273652C2212yescomplemented
36#14273650C2212yescomplemented
37#14273648C2212yescomplemented
38C22⋊(C59⋊C8)C2211yescharacteristic, complemented
39(C2×D59)⋊C8C2212yescomplemented
40C22×D4×C59C2211yescharacteristic, complemented
41C2×D8×C59C2212yescomplemented
42#121×C59C2211yescharacteristic
43#102×C59C2212yescomplemented
44C22⋊C8×C59C2211yescharacteristic, complemented
45C24×C59—21no—
46C23×D59—21no—
47C23×D59D411yescharacteristic
48C2×D4×C59—24no—
49C2×D4×C59—21no—
50C2×D4×C59—21no—
51C2×D4×C59C2311yescharacteristic
52C2×D4×C59C2312yes—
53C22×C4×C59C2311yescharacteristic
54C2×C22⋊D59—28no—
55C2×C22⋊D59—22no—
56C2×C22⋊D59—22no—
57C2×C22⋊D59—21no—
58C2×C22⋊D59—21no—
59C2×C22⋊D59—21no—
60C22×Dic59—28no—
61C22×Dic59—21no—
62C22×Dic59—22no—
63C22×Dic59—22no—
64C22×Dic59D411yescharacteristic
65#11745687—28no—
66#11745687—28no—
67#11745687—24no—
68#11745687—24no—
69#11745687—24no—
70#11745687D418yescomplemented
71D4×D59—24no—
72D4×D59—44no—
73C2×D236—21no—
74C2×D236C2311yescharacteristic
75C2×C4×D59—24no—
76C2×C4×D59—21no—
77C2×C4×D59D411yescharacteristic
78C2×C4×D59D411yescharacteristic
79C2×C4×D59C2311yescharacteristic
80C2×C4×D59C2311yescharacteristic
81C2×Dic118—21no—
82C2×Dic118—24no—
83C2×Dic118C2312yes—
84D8×C59—24no—
85C2×C8×C59C2312yes—
86C4⋊C4×C59C2311yescharacteristic
87C22⋊C4×C59—21no—
88C22⋊Dic59—21no—
89#11745670—24no—
90#11745669—24no—
91(C2×D59)⋊C4—21no—
92(C2×D59)⋊C4—21no—
93C4⋊Dic59C2311yescharacteristic
94Dic59⋊C4—21no—
95C2×C59⋊C8C2312yes—
96#11745660—24no—
97#11745659—24no—
98C23×C59—24no—
99C23×C59—22no—
100C23×C59—21no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59C2×D411yescharacteristic
104C22×D59—22no—
105C22×D59—22no—
106C22×D59—21no—
107C22×D59—41no—
108C22×D59—42no—
109C22×D59—42no—
110C22×D59—42no—
111C22×D59C2×D411yescharacteristic
112C22×D59C2×D411yescharacteristic
113D4×C59—24no—
114D4×C59—22no—
115D4×C59—22no—
116D4×C59—42no—
117D4×C59C2×D414yescomplemented
118C2×C4×C59—22no—
119C2×C4×C59—21no—
120C2×C4×C59C2×D411yescharacteristic
121C2×C4×C59C2411yescharacteristic
122C22⋊D59—28no—
123C22⋊D59—28no—
124C22⋊D59—42no—
125C22⋊D59—42no—
126C22⋊D59—42no—
127C2×Dic59—24no—
128C2×Dic59—28no—
129C2×Dic59—22no—
130C2×Dic59—22no—
131C2×Dic59—28no—
132C2×Dic59—21no—
133C2×Dic59C2×D411yescharacteristic
134C2×Dic59C2×D411yescharacteristic
135D236—22no—
136D236—42no—
137C4×D59—22no—
138C4×D59—22no—
139C4×D59—22no—
140C4×D59—42no—
141C4×D59C2×D414yes—
142Dic118—24no—
143Dic118—22no—
144Dic118C2×D414yescomplemented
145C8×C59—22no—
146C59⋊C8—22no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—24no—
150C22×C59—48no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59AGL(1,ℤ/8)12yescomplemented
157C22×C59C22×D411yescharacteristic
158C2×D59—21no—
159C2×D59—21no—
160C2×D59—21no—
161C2×D59—21no—
162C2×D59—21no—
163C2×D59—44no—
164C2×D59—41no—
165C2×D59—41no—
166C2×D59—41no—
167C2×D59—84no—
168C2×D59#12012yes—
169C4×C59—21no—
170C4×C59—41no—
171C4×C59C22×D412yes—
172Dic59—24no—
173Dic59—21no—
174Dic59—41no—
175Dic59#12012yes—
176#3006—591nomaximal, Sylow
177C2×C59—22no—
178C2×C59—44no—
179C2×C59—81no—
180C2×C59C2×#12011yescharacteristic
181C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C59#57511yescharacteristic
183D59—22no—
184D59—41no—
185D59—81no—
186C22×#41—591no—
187C2×AGL(1,ℤ/8)—592no—
188C2×#121—591no—
189#449—592no—
190#449—592no—
191#448—591no—
192#448—591no—
193#447—591no—
194#447D5911yescharacteristic, complemented
195#417—592no—
196#407—591no—
197C59#300611yescharacteristic, complemented, Sylow
198C2×#41—592no—
199C2×#41—592no—
200C2×#41—1181no—
201C2×#41—1181no—
202C2×#41—1188no—
203C22×Q8—591no—
204C22×D4—591no—
205C22×D4—1181no—
206C22×D4—1181no—
207C22×D4C2×D5911yescharacteristic, complemented
208C23×C4—591no—
209C23×C4—1181no—
210AGL(1,ℤ/8)—1188no—
211C2×SD16—592no—
212C2×SD16—592no—
213C2×D8—592no—
214C2×D8C2×D5912yescomplemented
215C2×M16—592no—
216#121—591no—
217#121—591no—
218#121—591no—
219#121—1181no—
220#121—1181no—
221#121C2×D5911yescharacteristic
222C2×C4⋊C4—591no—
223C2×C22⋊C4—1181no—
224#103—592no—
225#103—592no—
226#102—592no—
227#102C2×D5912yescomplemented
228C22⋊C8—591no—
229C22⋊C8—592no—
230C22⋊C8C2×D5911yescharacteristic, complemented
231C24—21no—
232C24—591no—
233C24—1181no—
234#41—598no—
235#41—1184no—
236#41—1184no—
237#41—1188no—
238#41—1188no—
239#41—1184no—
240C2×Q8—592no—
241C2×Q8—1184no—
242C2×Q8—1181no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—591no—
247C2×D4—1181no—
248C2×D4—1184no—
249C2×D4—1181no—
250C2×D4—1182no—
251C2×D4—1182no—
252C2×D4—1188no—
253C2×D4—1181no—
254C2×D4—1181no—
255C2×D4—2364no—
256C2×D4C22×D5911yescharacteristic
257C2×D4C22×D5912yes—
258C22×C4—591no—
259C22×C4—591no—
260C22×C4—591no—
261C22×C4—591no—
262C22×C4—591no—
263C22×C4—1184no—
264C22×C4—1181no—
265C22×C4—1182no—
266C22×C4—1182no—
267C22×C4—1188no—
268C22×C4—1181no—
269C22×C4C22×D5911yescharacteristic
270SD16—1184no—
271SD16—1184no—
272D8—24no—
273D8—1184no—
274M16—1184no—
275C2×C8—592no—
276C2×C8C22×D5912yes—
277C4⋊C4—591no—
278C4⋊C4—1181no—
279C4⋊C4C22×D5911yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1181no—
282C22⋊C4—1181no—
283C22⋊C4—1181no—
284C23—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—591no—
290C23—591no—
291C23—1182no—
292C23—1182no—
293C23—1181no—
294C23—2361no—
295C23—2362no—
296C23—2362no—
297C23—2362no—
298C23D4×D5911yescharacteristic
299Q8—594no—
300Q8—1184no—
301Q8—1182no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—1188no—
307D4—1188no—
308D4—1182no—
309D4—2362no—
310D4—2362no—
311D4—2362no—
312D4—2362no—
313D4D4×D5914yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—591no—
317C2×C4—594no—
318C2×C4—591no—
319C2×C4—1182no—
320C2×C4—1182no—
321C2×C4—1182no—
322C2×C4—1188no—
323C2×C4—1184no—
324C2×C4—1188no—
325C2×C4—1182no—
326C2×C4—1182no—
327C2×C4—1181no—
328C2×C4—2362no—
329C2×C4D4×D5911yescharacteristic
330C2×C4C23×D5911yescharacteristic
331C8—22no—
332C8—1182no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—592no—
343C22—1181no—
344C22—1181no—
345C22—1181no—
346C22—1181no—
347C22—1181no—
348C22—4724no—
349C22—2364no—
350C22—2361no—
351C22—2361no—
352C22—2361no—
353C22#1427375112yescomplemented
354C22C2×D4×D5911yescharacteristic
355C4—21no—
356C4—41no—
357C4—592no—
358C4—1181no—
359C4—1184no—
360C4—2361no—
361C4C2×D4×D5912yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—1182no—
366C2—4721no—
367C2—2361no—
368C2#120×D5911yescharacteristic
369C2C2×#1427375111yescharacteristic
370C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#447C2
socleC22×C59C22×D4
hypercenter#447D59
hypocenterC59#3006
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#11745687C2semidirect
2. C2×#14273751C2semidirect
3. #121×D59C2semidirect
4. #16837548C2semidirect
5. #16837544C2semidirect
6. #16837543C2semidirect
7. #16837540C2semidirect
8. #16837536C2semidirect
9. #16837400C2semidirect
10. #16837364C2semidirect
11. C59×#447C2semidirect
16. C2×#11745687C22semidirect
17. C22×Dic118C22semidirect
20. C2×D4×D59C22semidirect
24. C2×#11745660C22semidirect
25. C2×#11745670C22semidirect
26. C2×#11745669C22semidirect
27. C2×#11745659C22semidirect
28. #14273702C22semidirect
32. #14273698C22semidirect
33. C4⋊C4×D59C22semidirect
35. #14273652C22semidirect
36. #14273650C22semidirect
37. #14273648C22semidirect
38. C22⋊(C59⋊C8)C22semidirect
39. (C2×D59)⋊C8C22semidirect
40. C22×D4×C59C22semidirect
41. C2×D8×C59C22semidirect
43. #102×C59C22semidirect
44. C22⋊C8×C59C22semidirect
70. #11745687D4semidirect
117. D4×C59C2×D4semidirect
144. Dic118C2×D4semidirect
156. C22×C59AGL(1,ℤ/8)semidirect
194. #447D59semidirect
197. C59#3006semidirect
207. C22×D4C2×D59semidirect
214. C2×D8C2×D59semidirect
227. #102C2×D59semidirect
230. C22⋊C8C2×D59semidirect
313. D4D4×D59semidirect
353. C22#14273751semidirect