Finite Group Databasebeta

← #21427445

Subgroups — #21427445

Order 7552.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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×D4×D59C211yesmaximal, characteristic, complemented
2C2×#14273699C212yesmaximal, complemented
3C2×#14273695C212yesmaximal, complemented
4C2×(C22⋊D59)⋊C4C211yesmaximal, characteristic, complemented
5#16837320C211yesmaximal, characteristic, complemented
6#16837316C211yesmaximal, characteristic, complemented
7#16837311C211yesmaximal, characteristic, complemented
8#16837302C212yesmaximal, complemented
9#16837261C211yesmaximal, characteristic, complemented
10#16837257C212yesmaximal, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C24×D59C2212yes—
13C22×D236C2211yescharacteristic
14C22×C22⋊D59C2212yes—
15C22×C22⋊D59C2211yescharacteristic
16C22×C22⋊D59C2212yescomplemented
17C2×D4×D59—28no—
18C2×D4×D59—24no—
19C23×Dic59C2211yescharacteristic, complemented
20C22×C4×D59C2211yescharacteristic
21#14273699—28no—
22#14273695—28no—
23(C22⋊D59)⋊C4—24no—
24C2×Dic59⋊C4C2212yescomplemented
25C2×Dic59⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic59C2212yescomplemented
27C2×C22⋊Dic59C2212yescomplemented
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×(C2×D59)⋊C4C2212yes—
30C2×(C2×D59)⋊C4C2212yescomplemented
31C2×(C2×D59)⋊C4C2212yescomplemented
32C2×(C2×D59)⋊C4C2212yescomplemented
33C2×C4×Dic59C2211yescharacteristic, complemented
34C59⋊#95C2211yescharacteristic, complemented
35C22×D4×C59C2211yescharacteristic
36C22×D4×C59C2212yescomplemented
37C2×C22⋊C4×C59C2211yescharacteristic, complemented
38C2×C22⋊C4×C59C2212yescomplemented
39#95×C59C2211yescharacteristic, complemented
40C24×C59C2312yes—
41C24×C59C2311yescharacteristic
42C23×D59—24no—
43C23×D59—24no—
44C23×D59—24no—
45C23×D59—24no—
46C23×D59—24no—
47C23×D59—42no—
48C23×D59—42no—
49C23×D59C2312yes—
50C23×D59C2311yescharacteristic
51C2×D4×C59—24no—
52C2×D4×C59—24no—
53C2×D4×C59—24no—
54C2×D4×C59—22no—
55C2×D4×C59—42no—
56C22×C4×C59C2312yes—
57C22×C4×C59C2311yescharacteristic
58C22×C4×C59C2311yescharacteristic
59C2×C22⋊D59—22no—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—24no—
62C2×C22⋊D59—24no—
63C2×C22⋊D59—24no—
64C2×C22⋊D59—24no—
65C2×C22⋊D59—42no—
66C2×C22⋊D59D418yescomplemented
67C2×C22⋊D59D414yescomplemented
68C22×Dic59—22no—
69C22×Dic59—24no—
70C22×Dic59C2312yes—
71C22×Dic59C2311yescharacteristic
72C22×Dic59C2311yescharacteristic
73C22×Dic59C2311yescharacteristic
74D4×D59—232no—
75C2×D236—24no—
76C2×D236—22no—
77C2×C4×D59—24no—
78C2×C4×D59D414yescomplemented
79C22⋊C4×C59—22no—
80C22⋊C4×C59—42no—
81C22⋊Dic59—24no—
82C22⋊Dic59—24no—
83(C2×D59)⋊C4—22no—
84(C2×D59)⋊C4—24no—
85(C2×D59)⋊C4—24no—
86(C2×D59)⋊C4—24no—
87(C2×D59)⋊C4—42no—
88Dic59⋊C4—22no—
89Dic59⋊C4—24no—
90C4×Dic59—22no—
91C23×C59—22no—
92C23×C59—21no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—22no—
96C23×C59—21no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—22no—
101C23×C59C2×D414yes—
102C23×C59C2×D412yescomplemented
103C23×C59C2411yescharacteristic
104C22×D59—22no—
105C22×D59—22no—
106C22×D59—216no—
107C22×D59—22no—
108C22×D59—22no—
109C22×D59—22no—
110C22×D59—22no—
111C22×D59—22no—
112C22×D59—22no—
113C22×D59—48no—
114C22×D59—48no—
115C22×D59—48no—
116C22×D59—48no—
117C22×D59—48no—
118C22×D59—48no—
119C22×D59C2×D414yes—
120C22×D59C2×D412yes—
121C22×D59C2×D412yes—
122C22×D59#4112yes—
123D4×C59—28no—
124D4×C59—48no—
125C2×C4×C59—22no—
126C2×C4×C59—21no—
127C2×C4×C59—22no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59C2×D412yes—
132C22⋊D59—28no—
133C22⋊D59—216no—
134C22⋊D59—48no—
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—21no—
138C2×Dic59—22no—
139C2×Dic59—22no—
140C2×Dic59—22no—
141C2×Dic59—22no—
142C2×Dic59—44no—
143C2×Dic59C2×D412yes—
144C2×Dic59C2×D414yes—
145C2×Dic59C2×D412yescomplemented
146D236—28no—
147C4×D59—28no—
148C22×C59—24no—
149C22×C59—22no—
150C22×C59—24no—
151C22×C59—24no—
152C22×C59—24no—
153C22×C59—41no—
154C22×C59—42no—
155C22×C59—42no—
156C22×C59—41no—
157C22×C59—42no—
158C22×C59—42no—
159C22×C59—42no—
160C22×C59C22×D411yescharacteristic
161C22×C59C22×D412yes—
162C22×C59C22×D411yescharacteristic
163C22×C59C2×#4111yescharacteristic
164C22×C59H(5,𝔽2)12yes—
165C2×D59—22no—
166C2×D59—22no—
167C2×D59—24no—
168C2×D59—24no—
169C2×D59—24no—
170C2×D59—24no—
171C2×D59—24no—
172C2×D59—432no—
173C2×D59—42no—
174C2×D59—42no—
175C2×D59—42no—
176C2×D59—42no—
177C2×D59#12114yescomplemented
178C4×C59—22no—
179C4×C59—41no—
180C4×C59—42no—
181Dic59—22no—
182Dic59—41no—
183Dic59—42no—
184Dic59#12014yescomplemented
185#2408—591nomaximal, Sylow
186C2×C59—44no—
187C2×C59—42no—
188C2×C59C2×#12011yescharacteristic
189C2×C59C2×#12111yescharacteristic
190C2×C59#53511yescharacteristic
191C2×C59D4212yes—
192C2×C59#54612yes—
193D59—24no—
194D59—44no—
195C23×D4—591no—
196C2×#121—592no—
197C2×#120—592no—
198C2×C4×D4—591no—
199#394—591no—
200#394—591no—
201#392—591no—
202#392D5911yescharacteristic, complemented
203#390—592no—
204#380—591no—
205#379—592no—
206C59#240811yescharacteristic, complemented, Sylow
207C25—592no—
208C22×D4—591no—
209C22×D4—592no—
210C22×D4—592no—
211C22×D4—591no—
212C22×D4—1184no—
213C22×D4—1188no—
214C22×D4C2×D5911yescharacteristic
215C22×D4C2×D5912yescomplemented
216C23×C4—591no—
217C23×C4—591no—
218#121—1188no—
219#120—1188no—
220C4×D4—1184no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—592no—
223C2×C22⋊C4—592no—
224C2×C22⋊C4—592no—
225C2×C22⋊C4—591no—
226C2×C22⋊C4—592no—
227C2×C22⋊C4—592no—
228C2×C22⋊C4—592no—
229C2×C22⋊C4—592no—
230C2×C22⋊C4C2×D5911yescharacteristic, complemented
231C2×C22⋊C4C2×D5912yescomplemented
232C2×C42—591no—
233#95—591no—
234#95C2×D5911yescharacteristic, complemented
235C24—591no—
236C24—592no—
237C24—1184no—
238C24—1184no—
239C24—1184no—
240C24—1184no—
241C24—1184no—
242C24—2362no—
243C24—2362no—
244C24C22×D5912yes—
245C24C22×D5911yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—598no—
252C2×D4—594no—
253C2×D4—1184no—
254C2×D4—1184no—
255C2×D4—1184no—
256C2×D4—11832no—
257C2×D4—1182no—
258C2×D4—1182no—
259C2×D4—1184no—
260C2×D4—1184no—
261C2×D4—1182no—
262C2×D4—2362no—
263C22×C4—591no—
264C22×C4—591no—
265C22×C4—594no—
266C22×C4—592no—
267C22×C4—591no—
268C22×C4—1184no—
269C22×C4—1182no—
270C22×C4—1184no—
271C22×C4C22×D5912yes—
272C22×C4C22×D5911yescharacteristic
273C22×C4C22×D5911yescharacteristic
274C4⋊C4—1184no—
275C4⋊C4—1182no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—1184no—
279C22⋊C4—1182no—
280C22⋊C4—1184no—
281C22⋊C4—1184no—
282C22⋊C4—1184no—
283C22⋊C4—1184no—
284C22⋊C4—2362no—
285C42—1182no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—592no—
297C23—592no—
298C23—592no—
299C23—594no—
300C23—11816no—
301C23—1182no—
302C23—1182no—
303C23—1182no—
304C23—1182no—
305C23—1182no—
306C23—1182no—
307C23—1182no—
308C23—1182no—
309C23—2368no—
310C23—2368no—
311C23—2368no—
312C23—2368no—
313C23—2368no—
314C23—2368no—
315C23D4×D5914yes—
316C23D4×D5912yescomplemented
317C23C23×D5911yescharacteristic
318D4—28no—
319D4—48no—
320D4—1188no—
321D4—11816no—
322D4—1188no—
323D4—2368no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—592no—
331C2×C4—594no—
332C2×C4—592no—
333C2×C4—1188no—
334C2×C4—1181no—
335C2×C4—1181no—
336C2×C4—1182no—
337C2×C4—1182no—
338C2×C4—1181no—
339C2×C4—1182no—
340C2×C4—1182no—
341C2×C4—2364no—
342C2×C4D4×D5912yes—
343C22—22no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—42no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—594no—
356C22—1182no—
357C22—1184no—
358C22—1184no—
359C22—1182no—
360C22—1184no—
361C22—1184no—
362C22—1184no—
363C22—23632no—
364C22—2362no—
365C22—2362no—
366C22—2362no—
367C22—2362no—
368C22C2×D4×D5911yescharacteristic
369C22C2×D4×D5911yescharacteristic
370C22C2×D4×D5912yes—
371C22#41×D5911yescharacteristic
372C22#1427377212yes—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—594no—
377C4—1182no—
378C4—2362no—
379C4—2361no—
380C2—44no—
381C2—42no—
382C2—1184no—
383C2—2364no—
384C2#120×D5911yescharacteristic
385C2#121×D5911yescharacteristic
386C2#1683796511yescharacteristic
387C2#1683803412yes—
388C2#1683804112yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2408
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D59C2semidirect
2. C2×#14273699C2semidirect
3. C2×#14273695C2semidirect
4. C2×(C22⋊D59)⋊C4C2semidirect
5. #16837320C2semidirect
6. #16837316C2semidirect
7. #16837311C2semidirect
8. #16837302C2semidirect
9. #16837261C2semidirect
10. #16837257C2semidirect
11. C59×#392C2semidirect
16. C22×C22⋊D59C22semidirect
19. C23×Dic59C22semidirect
24. C2×Dic59⋊C4C22semidirect
25. C2×Dic59⋊C4C22semidirect
26. C2×C22⋊Dic59C22semidirect
27. C2×C22⋊Dic59C22semidirect
30. C2×(C2×D59)⋊C4C22semidirect
31. C2×(C2×D59)⋊C4C22semidirect
32. C2×(C2×D59)⋊C4C22semidirect
33. C2×C4×Dic59C22semidirect
34. C59⋊#95C22semidirect
36. C22×D4×C59C22semidirect
37. C2×C22⋊C4×C59C22semidirect
38. C2×C22⋊C4×C59C22semidirect
39. #95×C59C22semidirect
66. C2×C22⋊D59D4semidirect
67. C2×C22⋊D59D4semidirect
78. C2×C4×D59D4semidirect
102. C23×C59C2×D4semidirect
145. C2×Dic59C2×D4semidirect
177. C2×D59#121semidirect
184. Dic59#120semidirect
202. #392D59semidirect
206. C59#2408semidirect
215. C22×D4C2×D59semidirect
230. C2×C22⋊C4C2×D59semidirect
231. C2×C22⋊C4C2×D59semidirect
234. #95C2×D59semidirect
316. C23D4×D59semidirect