Finite Group Databasebeta

← #21434894

Subgroups — #21434894

Order 7552.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14273698C211yesmaximal, characteristic, complemented
2#121×D59C211yesmaximal, characteristic, complemented
3#16837721C211yesmaximal, characteristic, complemented
4#16837719C212yesmaximal, complemented
5#16837718C211yesmaximal, characteristic, complemented
6#16837654C212yesmaximal, complemented
7#16837652C212yesmaximal, complemented
8#16837536C211yesmaximal, characteristic, complemented
9#16837534C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#447C211yesmaximal, characteristic, complemented
12C22×C22⋊D59—21no—
13C2×D4×D59—21no—
14C2×D4×D59C2211yescharacteristic, complemented
15C22×C4×D59C2211yescharacteristic
16C2×#11745669C2212yescomplemented
17C2×#11745659C2212yescomplemented
18#14273702C2211yescharacteristic, complemented
19#14273701C2211yescharacteristic
20#14273700—21no—
21#14273699—21no—
22#14273698—21no—
23#14273698—21no—
24#14273698C2212yes—
25#14273698C2211yescharacteristic, complemented
26#14273698C2212yescomplemented
27C2×C4⋊Dic59C2211yescharacteristic, complemented
28C4⋊C4×D59C2211yescharacteristic, complemented
29C2×C22⋊Dic59—21no—
30C22⋊C4×D59—21no—
31#14273659C2212yescomplemented
32#14273658C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic, complemented
37(C2×D59)⋊C8C2212yescomplemented
38C22×D4×C59C2211yescharacteristic, complemented
39C2×D8×C59C2212yescomplemented
40#121×C59C2211yescharacteristic
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic, complemented
43C24×C59—21no—
44C23×D59—21no—
45C23×D59D411yescharacteristic
46C2×D4×C59—21no—
47C2×D4×C59—21no—
48C2×D4×C59—42no—
49C2×D4×C59C2312yes—
50C2×D4×C59C2311yescharacteristic
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—22no—
55C2×C22⋊D59—21no—
56C2×C22⋊D59—21no—
57C2×C22⋊D59—44no—
58C22×Dic59—21no—
59C22×Dic59D411yescharacteristic
60D4×D59—42no—
61D4×D59—44no—
62C2×D236—21no—
63C2×D236C2311yescharacteristic
64C2×C4×D59—21no—
65C2×C4×D59—42no—
66C2×C4×D59D411yescharacteristic
67C2×C4×D59D411yescharacteristic
68C2×C4×D59C2311yescharacteristic
69C2×C4×D59C2311yescharacteristic
70D8×C59—42no—
71C2×C8×C59C2312yes—
72C4⋊C4×C59C2311yescharacteristic
73C22⋊C4×C59—21no—
74C22⋊Dic59—21no—
75C22⋊Dic59—22no—
76C22⋊Dic59—22no—
77#11745669—42no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80C4⋊Dic59—21no—
81C4⋊Dic59C2311yescharacteristic
82C4⋊Dic59C2312yes—
83Dic59⋊C4—21no—
84C2×C59⋊C8C2312yes—
85#11745659—42no—
86C23×C59—22no—
87C23×C59—21no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59C2×D411yescharacteristic
92C22×D59—22no—
93C22×D59—22no—
94C22×D59—21no—
95C22×D59—41no—
96C22×D59—42no—
97C22×D59—42no—
98C22×D59—42no—
99C22×D59C2×D411yescharacteristic
100C22×D59C2×D411yescharacteristic
101D4×C59—22no—
102D4×C59—22no—
103D4×C59—22no—
104D4×C59—42no—
105D4×C59—42no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59C2×D411yescharacteristic
109C2×C4×C59C2411yescharacteristic
110C22⋊D59—42no—
111C22⋊D59—42no—
112C22⋊D59—44no—
113C22⋊D59—42no—
114C22⋊D59—44no—
115C2×Dic59—21no—
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—42no—
119C2×Dic59C2×D411yescharacteristic
120C2×Dic59C2×D411yescharacteristic
121D236—22no—
122D236—42no—
123C4×D59—22no—
124C4×D59—22no—
125C4×D59—41no—
126C4×D59—41no—
127C4×D59—42no—
128C8×C59—22no—
129C59⋊C8—22no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—41no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—42no—
138C22×C59—84no—
139C22×C59AGL(1,ℤ/8)12yes—
140C22×C59C22×D411yescharacteristic
141C2×D59—21no—
142C2×D59—21no—
143C2×D59—21no—
144C2×D59—21no—
145C2×D59—21no—
146C2×D59—44no—
147C2×D59—41no—
148C2×D59—41no—
149C2×D59—41no—
150C2×D59—84no—
151C2×D59AGL(1,ℤ/8)12yes—
152C4×C59—21no—
153C4×C59—41no—
154C4×C59H(5,𝔽2)12yes—
155Dic59—21no—
156Dic59—21no—
157Dic59—41no—
158Dic59—42no—
159#3192—591nomaximal, Sylow
160C2×C59—22no—
161C2×C59—81no—
162C2×C59—82no—
163C2×C59#53411yescharacteristic
164C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
166D59—22no—
167D59—41no—
168D59—81no—
169C2×#121—591no—
170C2×#121—591no—
171#481—591no—
172#481—592no—
173#481—591no—
174#469—592no—
175#469—592no—
176#447—591no—
177#447—592no—
178#447D5911yescharacteristic, complemented
179#407—591no—
180C59#319211yescharacteristic, complemented, Sylow
181C22×D4—591no—
182C22×D4—1181no—
183C22×D4—1181no—
184C22×D4C2×D5911yescharacteristic, complemented
185C23×C4—591no—
186C2×D8—592no—
187C2×D8C2×D5912yescomplemented
188C2×M16—592no—
189#121—592no—
190#121—591no—
191#121—592no—
192#121—591no—
193#121—591no—
194#121—1181no—
195#121—1181no—
196#121—1181no—
197#121—1181no—
198#121C2×D5911yescharacteristic
199C2×C4⋊C4—591no—
200C2×C4⋊C4—591no—
201C2×C22⋊C4—1181no—
202C2×C22⋊C4—1181no—
203#106—592no—
204#106—592no—
205#102—592no—
206#102—592no—
207#102—592no—
208#102C2×D5912yescomplemented
209C22⋊C8—591no—
210C22⋊C8—592no—
211C22⋊C8C2×D5911yescharacteristic, complemented
212C24—21no—
213C24—591no—
214C24—1181no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—42no—
218C2×D4—591no—
219C2×D4—1181no—
220C2×D4—1181no—
221C2×D4—1182no—
222C2×D4—1181no—
223C2×D4—1182no—
224C2×D4—1181no—
225C2×D4—2362no—
226C2×D4—2364no—
227C2×D4—2364no—
228C2×D4C22×D5911yescharacteristic
229C2×D4C22×D5912yes—
230C22×C4—591no—
231C22×C4—591no—
232C22×C4—591no—
233C22×C4—591no—
234C22×C4—591no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4—2362no—
238C22×C4C22×D5911yescharacteristic
239D8—42no—
240D8—2362no—
241M16—2362no—
242C2×C8—592no—
243C2×C8C22×D5912yes—
244C4⋊C4—591no—
245C4⋊C4—592no—
246C4⋊C4—1181no—
247C4⋊C4—1181no—
248C4⋊C4C22×D5911yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—1181no—
251C22⋊C4—1181no—
252C22⋊C4—1182no—
253C22⋊C4—1182no—
254C22⋊C4—1181no—
255C23—21no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—591no—
261C23—591no—
262C23—1182no—
263C23—1182no—
264C23—1181no—
265C23—2361no—
266C23—2362no—
267C23—2362no—
268C23—2362no—
269C23D4×D5911yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—1182no—
276D4—2362no—
277D4—2362no—
278D4—2364no—
279D4—2364no—
280D4—2362no—
281D4—2362no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—591no—
285C2×C4—591no—
286C2×C4—1182no—
287C2×C4—1182no—
288C2×C4—1182no—
289C2×C4—1181no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2362no—
293C2×C4—2361no—
294C2×C4—2362no—
295C2×C4D4×D5911yescharacteristic
296C2×C4C23×D5911yescharacteristic
297C8—22no—
298C8—1182no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—84no—
308C22—592no—
309C22—1181no—
310C22—1181no—
311C22—1181no—
312C22—1181no—
313C22—1181no—
314C22—4724no—
315C22—2364no—
316C22—2361no—
317C22—2361no—
318C22—2361no—
319C22#1427375112yes—
320C22C2×D4×D5911yescharacteristic
321C4—21no—
322C4—41no—
323C4—1181no—
324C4—1181no—
325C4—2362no—
326C4—2361no—
327C4#1427377212yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—1182no—
332C2—4721no—
333C2—2361no—
334C2#1683796211yescharacteristic
335C2AGL(1,ℤ/8)×D5911yescharacteristic
336C2C2×#1427375111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#447C2
socleC22×C59C22×D4
hypercenter#447D59
hypocenterC59#3192
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. C2×#14273698C2semidirect
2. #121×D59C2semidirect
3. #16837721C2semidirect
4. #16837719C2semidirect
5. #16837718C2semidirect
6. #16837654C2semidirect
7. #16837652C2semidirect
8. #16837536C2semidirect
9. #16837534C2semidirect
10. #16837364C2semidirect
11. C59×#447C2semidirect
14. C2×D4×D59C22semidirect
16. C2×#11745669C22semidirect
17. C2×#11745659C22semidirect
18. #14273702C22semidirect
25. #14273698C22semidirect
26. #14273698C22semidirect
27. C2×C4⋊Dic59C22semidirect
28. C4⋊C4×D59C22semidirect
31. #14273659C22semidirect
32. #14273658C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273648C22semidirect
36. C22⋊(C59⋊C8)C22semidirect
37. (C2×D59)⋊C8C22semidirect
38. C22×D4×C59C22semidirect
39. C2×D8×C59C22semidirect
41. #102×C59C22semidirect
42. C22⋊C8×C59C22semidirect
178. #447D59semidirect
180. C59#3192semidirect
184. C22×D4C2×D59semidirect
187. C2×D8C2×D59semidirect
208. #102C2×D59semidirect
211. C22⋊C8C2×D59semidirect