Finite Group Databasebeta

← #21429807

Subgroups — #21429807

Order 7552.

1134 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14273709C211yesmaximal, characteristic, complemented
3C2×#14273695C212yesmaximal, complemented
4#16837320C211yesmaximal, characteristic, complemented
5#16837318C212yesmaximal, complemented
6#16837316C211yesmaximal, characteristic, complemented
7#16837311C211yesmaximal, characteristic, complemented
8#16837310C212yesmaximal, complemented
9#16837264C211yesmaximal, characteristic, complemented
10#16837257C212yesmaximal, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C24×D59C2212yes—
13C22×D236C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15C22×C22⋊D59C2212yes—
16C22×C22⋊D59C2212yescomplemented
17C2×D4×D59—28no—
18C2×D4×D59—24no—
19C22×C4×D59C2211yescharacteristic
20#14273709—24no—
21#14273695—28no—
22C2×C4⋊Dic59C2211yescharacteristic, complemented
23C2×Dic59⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic59C2211yescharacteristic, complemented
25C2×C22⋊Dic59C2212yescomplemented
26C2×C22⋊Dic59C2212yescomplemented
27C2×(C2×D59)⋊C4C2211yescharacteristic
28C2×(C2×D59)⋊C4C2212yes—
29C2×(C2×D59)⋊C4C2212yescomplemented
30C2×(C2×D59)⋊C4C2212yescomplemented
31C2×(C2×D59)⋊C4C2212yescomplemented
32C59⋊#95C2212yescomplemented
33C59⋊#95C2211yescharacteristic, complemented
34C22×D4×C59C2211yescharacteristic
35C22×D4×C59C2212yescomplemented
36C2×C22⋊C4×C59C2211yescharacteristic, complemented
37C2×C22⋊C4×C59C2212yescomplemented
38#95×C59C2211yescharacteristic, complemented
39C24×C59C2311yescharacteristic
40C24×C59C2312yes—
41C23×D59—24no—
42C23×D59—24no—
43C23×D59—42no—
44C23×D59—42no—
45C23×D59—42no—
46C23×D59—42no—
47C23×D59D418yescomplemented
48C23×D59C2312yes—
49C23×D59C2311yescharacteristic
50C2×D4×C59—24no—
51C2×D4×C59—24no—
52C2×D4×C59—22no—
53C2×D4×C59—42no—
54C2×D4×C59—42no—
55C22×C4×C59C2312yes—
56C22×C4×C59C2311yescharacteristic
57C22×C4×C59C2311yescharacteristic
58C2×C22⋊D59—24no—
59C2×C22⋊D59—24no—
60C2×C22⋊D59—24no—
61C2×C22⋊D59—24no—
62C2×C22⋊D59—22no—
63C2×C22⋊D59—24no—
64C2×C22⋊D59—42no—
65C2×C22⋊D59—42no—
66C22×Dic59C2312yes—
67C22×Dic59C2311yescharacteristic
68C22×Dic59C2311yescharacteristic
69C22×Dic59C2311yescharacteristic
70D4×D59—232no—
71C2×D236—24no—
72C2×D236—22no—
73C2×C4×D59—24no—
74C2×C4×D59D414yescomplemented
75C22⋊C4×C59—22no—
76C22⋊C4×C59—42no—
77C22⋊Dic59—24no—
78C22⋊Dic59—22no—
79C22⋊Dic59—42no—
80(C2×D59)⋊C4—24no—
81(C2×D59)⋊C4—22no—
82(C2×D59)⋊C4—24no—
83(C2×D59)⋊C4—42no—
84(C2×D59)⋊C4—42no—
85C4⋊Dic59—22no—
86Dic59⋊C4—22no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—21no—
94C23×C59—22no—
95C23×C59—21no—
96C23×C59—22no—
97C23×C59—21no—
98C23×C59C2×D414yes—
99C23×C59C2411yescharacteristic
100C22×D59—22no—
101C22×D59—21no—
102C22×D59—21no—
103C22×D59—22no—
104C22×D59—22no—
105C22×D59—22no—
106C22×D59—22no—
107C22×D59—22no—
108C22×D59—22no—
109C22×D59—216no—
110C22×D59—48no—
111C22×D59—48no—
112C22×D59—48no—
113C22×D59—48no—
114C22×D59—48no—
115C22×D59—48no—
116C22×D59C2×D414yes—
117C22×D59C2×D414yes—
118C22×D59C2×D412yes—
119D4×C59—28no—
120D4×C59—48no—
121C2×C4×C59—22no—
122C2×C4×C59—21no—
123C2×C4×C59—22no—
124C2×C4×C59—22no—
125C2×C4×C59—22no—
126C2×C4×C59—22no—
127C2×C4×C59C2×D412yes—
128C22⋊D59—216no—
129C22⋊D59—48no—
130C22⋊D59—44no—
131C2×Dic59—22no—
132C2×Dic59—21no—
133C2×Dic59—21no—
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59—21no—
137C2×Dic59—22no—
138C2×Dic59—21no—
139C2×Dic59—22no—
140C2×Dic59C2×D412yes—
141D236—28no—
142C4×D59—28no—
143C22×C59—24no—
144C22×C59—24no—
145C22×C59—24no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—41no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59C22×D412yes—
156C22×C59C22×D411yescharacteristic
157C22×C59H(5,𝔽2)12yes—
158C22×C59H(5,𝔽2)11yescharacteristic
159C22×C59H(5,𝔽2)11yescharacteristic
160C2×D59—24no—
161C2×D59—24no—
162C2×D59—22no—
163C2×D59—22no—
164C2×D59—24no—
165C2×D59—24no—
166C2×D59—24no—
167C2×D59—432no—
168C2×D59—42no—
169C2×D59—42no—
170C2×D59—42no—
171C2×D59—42no—
172C2×D59#12014yes—
173C4×C59—22no—
174C4×C59—42no—
175C4×C59—41no—
176Dic59—22no—
177Dic59—41no—
178Dic59—42no—
179Dic59—41no—
180#2613—591nomaximal, Sylow
181C2×C59—44no—
182C2×C59—81no—
183C2×C59C2×#12011yescharacteristic
184C2×C59#53412yes—
185C2×C59#53412yes—
186C2×C59#53511yescharacteristic
187C2×C59#53511yescharacteristic
188D59—24no—
189D59—44no—
190C23×D4—591no—
191C2×#123—591no—
192C2×#120—592no—
193#394—591no—
194#394—592no—
195#394—591no—
196#392—592no—
197#392—591no—
198#392D5911yescharacteristic, complemented
199#381—591no—
200#379—592no—
201C59#261311yescharacteristic, complemented, Sylow
202C25—592no—
203C22×D4—591no—
204C22×D4—592no—
205C22×D4—592no—
206C22×D4—591no—
207C22×D4—1184no—
208C22×D4—1188no—
209C22×D4C2×D5911yescharacteristic
210C22×D4C2×D5912yescomplemented
211C23×C4—591no—
212#123—1184no—
213#120—1188no—
214C2×C4⋊C4—591no—
215C2×C4⋊C4—591no—
216C2×C22⋊C4—592no—
217C2×C22⋊C4—591no—
218C2×C22⋊C4—592no—
219C2×C22⋊C4—592no—
220C2×C22⋊C4—591no—
221C2×C22⋊C4—592no—
222C2×C22⋊C4—592no—
223C2×C22⋊C4—592no—
224C2×C22⋊C4C2×D5911yescharacteristic, complemented
225C2×C22⋊C4C2×D5912yescomplemented
226#95—592no—
227#95—591no—
228#95C2×D5911yescharacteristic, complemented
229C24—591no—
230C24—592no—
231C24—598no—
232C24—1184no—
233C24—1184no—
234C24—2362no—
235C24—2362no—
236C24—2362no—
237C24—2362no—
238C24C22×D5911yescharacteristic
239C24C22×D5912yes—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—1184no—
246C2×D4—1184no—
247C2×D4—11832no—
248C2×D4—1184no—
249C2×D4—1184no—
250C2×D4—1184no—
251C2×D4—1184no—
252C2×D4—1182no—
253C2×D4—1182no—
254C2×D4—2362no—
255C2×D4—2362no—
256C22×C4—591no—
257C22×C4—594no—
258C22×C4—592no—
259C22×C4—591no—
260C22×C4—591no—
261C22×C4—1184no—
262C22×C4C22×D5912yes—
263C22×C4C22×D5911yescharacteristic
264C22×C4C22×D5911yescharacteristic
265C4⋊C4—1182no—
266C4⋊C4—1182no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—1184no—
270C22⋊C4—1184no—
271C22⋊C4—1182no—
272C22⋊C4—1182no—
273C22⋊C4—1184no—
274C22⋊C4—2362no—
275C22⋊C4—2362no—
276C22⋊C4—2362no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—21no—
288C23—592no—
289C23—594no—
290C23—594no—
291C23—11816no—
292C23—1182no—
293C23—1181no—
294C23—1181no—
295C23—1182no—
296C23—1182no—
297C23—1182no—
298C23—1182no—
299C23—1182no—
300C23—1182no—
301C23—2368no—
302C23—2368no—
303C23—2368no—
304C23—2368no—
305C23—2368no—
306C23—2368no—
307C23D4×D5914yes—
308C23C23×D5911yescharacteristic
309D4—28no—
310D4—48no—
311D4—1188no—
312D4—11816no—
313D4—2364no—
314D4—2368no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—592no—
322C2×C4—1188no—
323C2×C4—1182no—
324C2×C4—1181no—
325C2×C4—1182no—
326C2×C4—1182no—
327C2×C4—1181no—
328C2×C4—1181no—
329C2×C4—1182no—
330C2×C4—1181no—
331C2×C4—1182no—
332C2×C4D4×D5912yes—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—594no—
346C22—1184no—
347C22—1184no—
348C22—1182no—
349C22—1182no—
350C22—1184no—
351C22—1184no—
352C22—1184no—
353C22—23632no—
354C22—2362no—
355C22—2362no—
356C22—2362no—
357C22—2362no—
358C22C2×D4×D5912yes—
359C22C2×D4×D5911yescharacteristic
360C22#1427377212yes—
361C22#1427377211yescharacteristic
362C22#1427377311yescharacteristic
363C4—22no—
364C4—42no—
365C4—41no—
366C4—1182no—
367C4—2362no—
368C4—2361no—
369C4—2361no—
370C2—44no—
371C2—81no—
372C2—1184no—
373C2—2364no—
374C2#120×D5911yescharacteristic
375C2#1683796012yes—
376C2#1683796212yes—
377C2#1683796511yescharacteristic
378C2#1683796811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2613
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×#14273709C2semidirect
3. C2×#14273695C2semidirect
4. #16837320C2semidirect
5. #16837318C2semidirect
6. #16837316C2semidirect
7. #16837311C2semidirect
8. #16837310C2semidirect
9. #16837264C2semidirect
10. #16837257C2semidirect
11. C59×#392C2semidirect
16. C22×C22⋊D59C22semidirect
22. C2×C4⋊Dic59C22semidirect
23. C2×Dic59⋊C4C22semidirect
24. C2×C22⋊Dic59C22semidirect
25. C2×C22⋊Dic59C22semidirect
26. C2×C22⋊Dic59C22semidirect
29. C2×(C2×D59)⋊C4C22semidirect
30. C2×(C2×D59)⋊C4C22semidirect
31. C2×(C2×D59)⋊C4C22semidirect
32. C59⋊#95C22semidirect
33. C59⋊#95C22semidirect
35. C22×D4×C59C22semidirect
36. C2×C22⋊C4×C59C22semidirect
37. C2×C22⋊C4×C59C22semidirect
38. #95×C59C22semidirect
47. C23×D59D4semidirect
74. C2×C4×D59D4semidirect
198. #392D59semidirect
201. C59#2613semidirect
210. C22×D4C2×D59semidirect
224. C2×C22⋊C4C2×D59semidirect
225. C2×C22⋊C4C2×D59semidirect
228. #95C2×D59semidirect