Finite Group Databasebeta

← #21435118

Subgroups — #21435118

Order 7552.

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

normal subgroups only

This group has 490 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
1#127×D59C211yesmaximal, characteristic, complemented
2#16837939C211yesmaximal, characteristic, complemented
3#16837788C212yesmaximal, complemented
4#16837765C212yesmaximal, complemented
5#16837602C211yesmaximal, characteristic, complemented
6#16837598C211yesmaximal, characteristic, complemented
7#16837594C211yesmaximal, characteristic, complemented
8#16837551C212yesmaximal, complemented
9#16837537C212yesmaximal, complemented
10#16837458C211yesmaximal, characteristic, complemented
11C59×#460C211yesmaximal, characteristic, complemented
12C2×D4×D59—21no—
13C2×D4×D59—21no—
14C2×D4×D59C2212yes—
15C2×#11745671C2212yescomplemented
16C2×D472C2212yescomplemented
17C2×#11745669C2212yescomplemented
18#14273724C2211yescharacteristic
19#14273723—21no—
20#14273723—21no—
21#14273723C2212yes—
22#14273722—21no—
23#14273718C2211yescharacteristic, complemented
24#14273716C2211yescharacteristic, complemented
25#14273712—21no—
26#14273703C2211yescharacteristic
27#14273702C2211yescharacteristic
28C4×D236—21no—
29C4×D236C2211yescharacteristic, complemented
30#14273684—21no—
31C42×D59C2211yescharacteristic
32C4⋊(C59⋊C8)C2211yescharacteristic, complemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273647C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37#14273637C2212yescomplemented
38C2×SD16×C59C2212yescomplemented
39#127×C59C2211yescharacteristic
40C4×Q8×C59C2211yescharacteristic, complemented
41C4⋊C8×C59C2211yescharacteristic, complemented
42#102×C59C2212yescomplemented
43C23×D59—22no—
44C2×Q8×C59C2311yescharacteristic
45C2×D4×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59C2312yes—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51D4×D59—42no—
52D4×D59—42no—
53D4×D59—44no—
54D4×D59—44no—
55C2×D236—21no—
56C2×D236—21no—
57C2×D236—21no—
58C2×D236C2312yes—
59C2×D236C2311yescharacteristic
60C2×C4×D59—21no—
61C2×C4×D59D411yescharacteristic
62C2×C4×D59D411yescharacteristic
63C2×C4×D59C2311yescharacteristic
64SD16×C59—42no—
65C2×C8×C59C2312yes—
66C4⋊C4×C59—21no—
67C4⋊C4×C59C2311yescharacteristic
68C42×C59—21no—
69C42×C59C2311yescharacteristic
70#11745671—42no—
71#11745669—42no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77C4⋊Dic59—21no—
78C4⋊Dic59C2311yescharacteristic
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C4×Dic59D411yescharacteristic
82C4×Dic59D411yescharacteristic
83C4×Dic59C2311yescharacteristic
84C2×C59⋊C8C2312yes—
85D472—42no—
86C23×C59—22no—
87C22×D59—21no—
88C22×D59—22no—
89C22×D59—44no—
90C22×D59—44no—
91C22×D59—44no—
92C22×D59C2×D411yescharacteristic
93Q8×C59—21no—
94Q8×C59—21no—
95D4×C59—22no—
96D4×C59—22no—
97D4×C59—42no—
98D4×C59—42no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—44no—
105C22⋊D59—44no—
106C22⋊D59—44no—
107C2×Dic59—21no—
108C2×Dic59C2×D411yescharacteristic
109C2×Dic59C2×D411yescharacteristic
110C2×Dic59C2×D411yescharacteristic
111D236—21no—
112D236—22no—
113D236—21no—
114D236—22no—
115D236—42no—
116D236—42no—
117D236—42no—
118C4×D59—22no—
119C4×D59—22no—
120C4×D59—21no—
121C4×D59—21no—
122C4×D59—42no—
123C8×C59—22no—
124C59⋊C8—22no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59C22×D411yescharacteristic
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—42no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—42no—
135C2×D59—42no—
136C2×D59—41no—
137C2×D59—88no—
138C2×D59AGL(1,ℤ/8)12yes—
139C4×C59—21no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59AGL(1,ℤ/8)12yes—
143C4×C59H(5,𝔽2)11yescharacteristic
144C4×C59H(5,𝔽2)11yescharacteristic
145Dic59—21no—
146Dic59—21no—
147Dic59—21no—
148Dic59—41no—
149#3217—591nomaximal, Sylow
150C2×C59—82no—
151C2×C59#53511yescharacteristic
152C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
154D59—22no—
155D59—81no—
156D59—82no—
157C2×#127—591no—
158#529—591no—
159#496—592no—
160#490—592no—
161#460—591no—
162#460D5911yescharacteristic, complemented
163#459—591no—
164#459—591no—
165#450—592no—
166#447—592no—
167#432—591no—
168C59#321711yescharacteristic, complemented, Sylow
169C22×D4—592no—
170C22×D4—1181no—
171C22×D4—1181no—
172C2×SD16—592no—
173C2×SD16C2×D5912yescomplemented
174C2×D8—592no—
175C2×D8—592no—
176#127—592no—
177#127—591no—
178#127—1181no—
179#127—1181no—
180#127C2×D5911yescharacteristic
181#126—1181no—
182#125—591no—
183#124—591no—
184#123—1181no—
185C22⋊Q8—591no—
186#121—591no—
187C4×Q8C2×D5911yescharacteristic, complemented
188C4×D4—591no—
189C4×D4—1181no—
190#117—1181no—
191C2×C42—591no—
192C4⋊C8—591no—
193C4⋊C8C2×D5911yescharacteristic, complemented
194#102—592no—
195#102—592no—
196#102—592no—
197#102C2×D5912yescomplemented
198C22⋊C8—592no—
199#97—592no—
200C24—1182no—
201C2×Q8C22×D5911yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—592no—
205C2×D4—591no—
206C2×D4—1181no—
207C2×D4—1182no—
208C2×D4—1182no—
209C2×D4—1182no—
210C2×D4—1181no—
211C2×D4—1181no—
212C2×D4—2362no—
213C2×D4—2362no—
214C2×D4—2364no—
215C2×D4—2364no—
216C2×D4C22×D5912yes—
217C22×C4—591no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—1181no—
221SD16—42no—
222SD16—2362no—
223D8—2362no—
224D8—2362no—
225C2×C8—592no—
226C2×C8C22×D5912yes—
227C4⋊C4—21no—
228C4⋊C4—591no—
229C4⋊C4—1181no—
230C4⋊C4—1181no—
231C4⋊C4—1181no—
232C4⋊C4C22×D5911yescharacteristic
233C22⋊C4—1181no—
234C22⋊C4—1181no—
235C22⋊C4—1181no—
236C22⋊C4—1181no—
237C22⋊C4—1181no—
238C42—21no—
239C42—591no—
240C42—591no—
241C42—591no—
242C42C22×D5911yescharacteristic
243C23—22no—
244C23—591no—
245C23—1181no—
246C23—1182no—
247C23—2364no—
248C23—2364no—
249C23—2364no—
250Q8—21no—
251Q8—21no—
252D4—22no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—1181no—
257D4—1182no—
258D4—1181no—
259D4—1182no—
260D4—2362no—
261D4—2362no—
262D4—2364no—
263D4—2364no—
264D4—2364no—
265D4—2362no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—591no—
269C2×C4—591no—
270C2×C4—591no—
271C2×C4—1182no—
272C2×C4—1182no—
273C2×C4—1181no—
274C2×C4—1181no—
275C2×C4—1181no—
276C2×C4—2362no—
277C2×C4D4×D5911yescharacteristic
278C2×C4D4×D5911yescharacteristic
279C2×C4C23×D5911yescharacteristic
280C8—22no—
281C8—1182no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—592no—
286C22—1181no—
287C22—1181no—
288C22—4728no—
289C22—2362no—
290C22—2361no—
291C22—2361no—
292C22—2362no—
293C22—2362no—
294C22—2361no—
295C22C2×D4×D5911yescharacteristic
296C4—21no—
297C4—41no—
298C4—41no—
299C4—1181no—
300C4—1181no—
301C4—1181no—
302C4—2361no—
303C4#1427375412yes—
304C4#1427377211yescharacteristic
305C4#1427377311yescharacteristic
306C2—82no—
307C2—1182no—
308C2—4722no—
309C2—4721no—
310C2#1683796811yescharacteristic
311C2AGL(1,ℤ/8)×D5911yescharacteristic
312C2C2×#1427375411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#460C2
socleC22×C59C22×D4
hypercenter#460D59
hypocenterC59#3217
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #127×D59C2semidirect
2. #16837939C2semidirect
3. #16837788C2semidirect
4. #16837765C2semidirect
5. #16837602C2semidirect
6. #16837598C2semidirect
7. #16837594C2semidirect
8. #16837551C2semidirect
9. #16837537C2semidirect
10. #16837458C2semidirect
11. C59×#460C2semidirect
15. C2×#11745671C22semidirect
16. C2×D472C22semidirect
17. C2×#11745669C22semidirect
23. #14273718C22semidirect
24. #14273716C22semidirect
29. C4×D236C22semidirect
32. C4⋊(C59⋊C8)C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273647C22semidirect
37. #14273637C22semidirect
38. C2×SD16×C59C22semidirect
40. C4×Q8×C59C22semidirect
41. C4⋊C8×C59C22semidirect
42. #102×C59C22semidirect
162. #460D59semidirect
168. C59#3217semidirect
173. C2×SD16C2×D59semidirect
187. C4×Q8C2×D59semidirect
193. C4⋊C8C2×D59semidirect
197. #102C2×D59semidirect