Finite Group Databasebeta

← #21435117

Subgroups — #21435117

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
1C2×#14273723C211yesmaximal, characteristic, complemented
2#16837938C211yesmaximal, characteristic, complemented
3#16837788C212yesmaximal, complemented
4#16837765C212yesmaximal, complemented
5#16837600C212yesmaximal, complemented
6#16837596C212yesmaximal, complemented
7#16837552C211yesmaximal, characteristic, complemented
8#16837538C211yesmaximal, characteristic, complemented
9#16837536C211yesmaximal, characteristic, complemented
10#16837456C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59—21no—
14C22×C22⋊D59—21no—
15C2×#11745671C2212yescomplemented
16C2×D472C2212yescomplemented
17C2×#11745669C2212yescomplemented
18#14273723—21no—
19#14273723—21no—
20#14273723C2212yes—
21#14273723C2212yes—
22#14273722—21no—
23#14273718C2211yescharacteristic, complemented
24#14273716C2211yescharacteristic, complemented
25#14273711—21no—
26#14273701C2211yescharacteristic
27Q8×Dic59C2211yescharacteristic, complemented
28#14273689C2211yescharacteristic, complemented
29(C22⋊D59)⋊C4—21no—
30#14273685—21no—
31C2×C4×Dic59C2211yescharacteristic
32Dic59⋊C8C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273647C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37#14273637C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×SD16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C23×D59—21no—
45C2×Q8×C59C2311yescharacteristic
46C2×D4×C59—21no—
47C2×D4×C59—42no—
48C2×D4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—21no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—22no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—22no—
55C2×C22⋊D59—44no—
56C2×C22⋊D59—44no—
57C22×Dic59D411yescharacteristic
58C22×Dic59D411yescharacteristic
59C2×D236—21no—
60C2×D236—42no—
61C2×D236C2312yes—
62C2×D236C2311yescharacteristic
63C2×C4×D59—21no—
64SD16×C59—42no—
65C2×C8×C59C2312yes—
66C4⋊C4×C59—21no—
67C4⋊C4×C59C2311yescharacteristic
68C22⋊C4×C59—21no—
69#11745671—42no—
70#11745669—42no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59—21no—
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C4×Dic59C2311yescharacteristic
84C2×C59⋊C8C2312yes—
85D472—42no—
86C23×C59—22no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59C2×D411yescharacteristic
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—42no—
94C22×D59—42no—
95C22×D59—42no—
96Q8×C59—22no—
97D4×C59—22no—
98D4×C59—22no—
99D4×C59—42no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59C2×D411yescharacteristic
104C2×C4×C59C2411yescharacteristic
105C22⋊D59—44no—
106C22⋊D59—44no—
107C22⋊D59—44no—
108C22⋊D59—42no—
109C22⋊D59—44no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—21no—
113C2×Dic59C2×D411yescharacteristic
114C2×Dic59C2×D411yescharacteristic
115C2×Dic59C2×D411yescharacteristic
116C2×Dic59C2×D411yescharacteristic
117D236—22no—
118D236—22no—
119D236—22no—
120D236—42no—
121C4×D59—42no—
122C8×C59—22no—
123C59⋊C8—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—84no—
130C22×C59AGL(1,ℤ/8)12yes—
131C22×C59C22×D411yescharacteristic
132C2×D59—42no—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—42no—
136C2×D59—42no—
137C2×D59—41no—
138C2×D59—84no—
139C4×C59—21no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59H(5,𝔽2)12yes—
143Dic59—21no—
144Dic59—21no—
145Dic59—21no—
146Dic59—41no—
147Dic59AGL(1,ℤ/8)12yes—
148#3217—591nomaximal, Sylow
149C2×C59—22no—
150C2×C59—82no—
151C2×C59#53511yescharacteristic
152C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
154D59—81no—
155D59—82no—
156C2×#127—591no—
157#529—591no—
158#496—592no—
159#490—592no—
160#460—592no—
161#459—592no—
162#450—591no—
163#450D5911yescharacteristic, complemented
164#447—591no—
165#447—591no—
166#432—591no—
167C59#321711yescharacteristic, complemented, Sylow
168C22×D4—591no—
169C22×D4—1181no—
170C22×D4—1181no—
171C22×D4C2×D5911yescharacteristic
172C2×SD16—592no—
173C2×SD16C2×D5912yescomplemented
174C2×D8—592no—
175C2×D8—592no—
176#127—592no—
177#127—592no—
178#127—1181no—
179#127—1181no—
180#126—1181no—
181#125—591no—
182#124—591no—
183#123—1181no—
184C22⋊Q8C2×D5911yescharacteristic
185#121—591no—
186C4×Q8—591no—
187C4×D4—591no—
188C4×D4—1181no—
189#117—1181no—
190C2×C42—591no—
191C4⋊C8—592no—
192#102—592no—
193#102—592no—
194#102—592no—
195#102C2×D5912yescomplemented
196C22⋊C8—591no—
197C22⋊C8C2×D5911yescharacteristic
198#97—592no—
199C24—21no—
200C24—1181no—
201C2×Q8C22×D5911yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—592no—
205C2×D4—591no—
206C2×D4—1182no—
207C2×D4—1182no—
208C2×D4—1182no—
209C2×D4—1182no—
210C2×D4—1181no—
211C2×D4—1181no—
212C2×D4—2362no—
213C2×D4—2364no—
214C2×D4—2364no—
215C2×D4C22×D5912yes—
216C22×C4—591no—
217C22×C4—591no—
218C22×C4—1181no—
219C22×C4C22×D5911yescharacteristic
220SD16—42no—
221SD16—2362no—
222D8—2362no—
223D8—2362no—
224C2×C8—592no—
225C2×C8C22×D5912yes—
226C4⋊C4—21no—
227C4⋊C4—591no—
228C4⋊C4—1181no—
229C4⋊C4—1181no—
230C4⋊C4—1181no—
231C4⋊C4C22×D5911yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—1181no—
234C22⋊C4—1181no—
235C22⋊C4—1181no—
236C22⋊C4—1181no—
237C42—591no—
238C42—591no—
239C42—591no—
240C42—591no—
241C42—1181no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—1181no—
247C23—1182no—
248C23—2362no—
249C23—2362no—
250C23—2362no—
251C23D4×D5911yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—1182no—
257D4—1182no—
258D4—1182no—
259D4—2362no—
260D4—2364no—
261D4—2364no—
262D4—2364no—
263D4—2364no—
264D4—2362no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—591no—
269C2×C4—591no—
270C2×C4—591no—
271C2×C4—591no—
272C2×C4—1182no—
273C2×C4—1182no—
274C2×C4—1181no—
275C2×C4—2362no—
276C2×C4D4×D5911yescharacteristic
277C2×C4C23×D5911yescharacteristic
278C8—22no—
279C8—1182no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—4724no—
287C22—2361no—
288C22—2362no—
289C22—2362no—
290C22—2361no—
291C22—2362no—
292C22—2361no—
293C22#1427375412yes—
294C22C2×D4×D5911yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—592no—
299C4—1181no—
300C4—1181no—
301C4—1181no—
302C4—2361no—
303C4#1427377212yes—
304C2—22no—
305C2—82no—
306C2—4722no—
307C2—4721no—
308C2#1683796711yescharacteristic
309C2AGL(1,ℤ/8)×D5911yescharacteristic
310C2C2×#1427375411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3217
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#14273723C2semidirect
2. #16837938C2semidirect
3. #16837788C2semidirect
4. #16837765C2semidirect
5. #16837600C2semidirect
6. #16837596C2semidirect
7. #16837552C2semidirect
8. #16837538C2semidirect
9. #16837536C2semidirect
10. #16837456C2semidirect
11. C59×#450C2semidirect
15. C2×#11745671C22semidirect
16. C2×D472C22semidirect
17. C2×#11745669C22semidirect
23. #14273718C22semidirect
24. #14273716C22semidirect
27. Q8×Dic59C22semidirect
28. #14273689C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273647C22semidirect
37. #14273637C22semidirect
39. C2×SD16×C59C22semidirect
41. #102×C59C22semidirect
163. #450D59semidirect
167. C59#3217semidirect
173. C2×SD16C2×D59semidirect
195. #102C2×D59semidirect