Finite Group Databasebeta

← #21434912

Subgroups — #21434912

Order 7552.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 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#16837913C211yesmaximal, characteristic, complemented
2C2×#14273701C211yesmaximal, characteristic, complemented
3#16837733C211yesmaximal, characteristic, complemented
4#16837727C212yesmaximal, complemented
5#16837716C211yesmaximal, characteristic, complemented
6#16837655C212yesmaximal, complemented
7#16837648C212yesmaximal, complemented
8#16837555C211yesmaximal, characteristic, complemented
9#16837549C212yesmaximal, complemented
10#16837538C211yesmaximal, characteristic, complemented
11C59×#407C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic, complemented
13C22×D236C2211yescharacteristic
14C22×C22⋊D59—21no—
15C2×#11745660C2212yescomplemented
16C2×D472C2212yescomplemented
17#14273710—21no—
18#14273706C2211yescharacteristic
19#14273706C2211yescharacteristic, complemented
20#14273701—21no—
21#14273701—21no—
22#14273701C2212yes—
23#14273701C2212yes—
24#14273701C2211yescharacteristic
25#14273701C2211yescharacteristic, complemented
26#14273696—21no—
27C4×C22⋊D59—21no—
28C4×C22⋊D59—21no—
29#14273686C2211yescharacteristic, complemented
30C2×C4⋊Dic59C2211yescharacteristic
31C2×(C2×D59)⋊C4—21no—
32#14273661C2212yescomplemented
33#14273659C2212yescomplemented
34#14273652C2212yescomplemented
35#14273649C2212yescomplemented
36#14273649C2212yescomplemented
37#14273649C2212yescomplemented
38C23×C4×C59C2211yescharacteristic
39C2×M16×C59C2212yescomplemented
40C22⋊C8×C59C2211yescharacteristic
41C22⋊C8×C59C2211yescharacteristic
42C22⋊C8×C59C2212yescomplemented
43C24×C59D411yescharacteristic
44C23×D59—21no—
45C22×C4×C59—42no—
46C22×C4×C59D411yescharacteristic
47C22×C4×C59D411yescharacteristic
48C22×C4×C59D411yescharacteristic
49C22×C4×C59C2311yescharacteristic
50C22×C4×C59C2311yescharacteristic
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—21no—
54C2×C22⋊D59—21no—
55C2×C22⋊D59—22no—
56C2×C22⋊D59—22no—
57C2×C22⋊D59—44no—
58C22×Dic59—21no—
59#11745685—42no—
60C2×D236—21no—
61C2×D236—42no—
62C2×D236C2312yes—
63C2×D236C2311yescharacteristic
64C2×C4×D59—21no—
65C2×Dic118C2311yescharacteristic
66M16×C59—42no—
67C2×C8×C59C2312yes—
68C2×C8×C59C2312yes—
69C22⋊Dic59—21no—
70C22⋊Dic59—21no—
71C22⋊Dic59—21no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—22no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76C4⋊Dic59—21no—
77C4⋊Dic59C2311yescharacteristic
78C4⋊Dic59C2311yescharacteristic
79C4⋊Dic59C2312yes—
80Dic59⋊C4—21no—
81Dic59⋊C4—21no—
82C4×Dic59—21no—
83D472—42no—
84#11745660—42no—
85C23×C59—22no—
86C23×C59—41no—
87C23×C59—41no—
88C23×C59C2×D411yescharacteristic
89C23×C59C2×D411yescharacteristic
90C23×C59C2×D411yescharacteristic
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—42no—
94C22×D59—42no—
95C22×D59—42no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2×D411yescharacteristic
104C2×C4×C59C2×D411yescharacteristic
105C2×C4×C59C2411yescharacteristic
106C22⋊D59—42no—
107C22⋊D59—42no—
108C22⋊D59—44no—
109C22⋊D59—44no—
110C22⋊D59—42no—
111C2×Dic59—21no—
112C2×Dic59—21no—
113C2×Dic59—21no—
114C2×Dic59—22no—
115C2×Dic59—42no—
116D236—22no—
117D236—22no—
118D236—22no—
119D236—42no—
120C4×D59—42no—
121Dic118—22no—
122C8×C59—22no—
123C8×C59—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—44no—
133C22×C59AGL(1,ℤ/8)12yes—
134C22×C59C22×D411yescharacteristic
135C2×D59—41no—
136C2×D59—42no—
137C2×D59—42no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—42no—
141C2×D59—84no—
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—21no—
145C4×C59H(5,𝔽2)12yes—
146Dic59—41no—
147Dic59—42no—
148Dic59—41no—
149Dic59—41no—
150#3194—591nomaximal, Sylow
151C2×C59—22no—
152C2×C59—41no—
153C2×C59—41no—
154C2×C59#53411yescharacteristic
155C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C59#57511yescharacteristic
157D59—81no—
158D59—82no—
159#525—591no—
160C2×#121—591no—
161#483—591no—
162#482—592no—
163#480—591no—
164#469—592no—
165#468—592no—
166#450—591no—
167#449—592no—
168#447—591no—
169#407D5911yescharacteristic, complemented
170C59#319411yescharacteristic, complemented, Sylow
171C2×#41—591no—
172C22×D4—591no—
173C22×D4—1181no—
174C23×C4C2×D5911yescharacteristic
175C2×SD16—592no—
176C2×D8—592no—
177C2×M16C2×D5912yescomplemented
178#123—1181no—
179C22⋊Q8—591no—
180C22⋊Q8—591no—
181#121—592no—
182#121—592no—
183#121—591no—
184#121—591no—
185#121—1181no—
186#121—1181no—
187#120—1181no—
188C4×D4—1181no—
189C4×D4—1181no—
190#117—591no—
191C2×C4⋊C4—591no—
192C2×C22⋊C4—1181no—
193#107—592no—
194#106—592no—
195#103—592no—
196#102—592no—
197#102—592no—
198#102—592no—
199C22⋊C8C2×D5911yescharacteristic
200C22⋊C8C2×D5911yescharacteristic
201C22⋊C8C2×D5912yescomplemented
202C24—1181no—
203C24D23611yescharacteristic
204#41—2362no—
205C2×Q8—591no—
206C2×D4—591no—
207C2×D4—592no—
208C2×D4—1181no—
209C2×D4—1181no—
210C2×D4—1181no—
211C2×D4—1182no—
212C2×D4—1182no—
213C2×D4—1181no—
214C2×D4—2362no—
215C2×D4—2364no—
216C22×C4—42no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4D23611yescharacteristic
220C22×C4D23611yescharacteristic
221C22×C4D23611yescharacteristic
222C22×C4C22×D5911yescharacteristic
223C22×C4C22×D5911yescharacteristic
224C22×C4C22×D5911yescharacteristic
225SD16—2362no—
226D8—2362no—
227M16—42no—
228C2×C8C22×D5912yes—
229C2×C8C22×D5912yes—
230C4⋊C4—591no—
231C4⋊C4—591no—
232C4⋊C4—592no—
233C4⋊C4—1181no—
234C4⋊C4—1181no—
235C4⋊C4—1181no—
236C22⋊C4—1181no—
237C22⋊C4—1181no—
238C22⋊C4—1181no—
239C22⋊C4—1182no—
240C22⋊C4—1182no—
241C22⋊C4—1181no—
242C22⋊C4—1181no—
243C42—1181no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—1181no—
248C23—1182no—
249C23—2362no—
250C23—2362no—
251C23—2362no—
252C23C2×D23611yescharacteristic
253C23C2×D23611yescharacteristic
254C23C2×D23611yescharacteristic
255Q8—1182no—
256D4—1182no—
257D4—1182no—
258D4—1182no—
259D4—2362no—
260D4—2362no—
261D4—2364no—
262D4—2364no—
263D4—2362no—
264D4—2362no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1181no—
272C2×C4—1181no—
273C2×C4—1182no—
274C2×C4—1181no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4C2×D23611yescharacteristic
278C2×C4C2×D23611yescharacteristic
279C2×C4C2×D23611yescharacteristic
280C2×C4C23×D5911yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—4724no—
293C22—2362no—
294C22—2361no—
295C22—2361no—
296C22—2362no—
297C22—2361no—
298C22—2362no—
299C22#1427375512yes—
300C22C22×D23611yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—2362no—
305C4—2361no—
306C4—2361no—
307C4—2361no—
308C4#1427377212yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—4722no—
313C2—4721no—
314C2#1683796411yescharacteristic
315C2C2×#1427375511yescharacteristic
316C2#1683820211yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#407C2
socleC22×C59C22×D4
hypercenter#407D59
hypocenterC59#3194
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16837913C2semidirect
2. C2×#14273701C2semidirect
3. #16837733C2semidirect
4. #16837727C2semidirect
5. #16837716C2semidirect
6. #16837655C2semidirect
7. #16837648C2semidirect
8. #16837555C2semidirect
9. #16837549C2semidirect
10. #16837538C2semidirect
11. C59×#407C2semidirect
12. C2×#11745685C22semidirect
15. C2×#11745660C22semidirect
16. C2×D472C22semidirect
19. #14273706C22semidirect
25. #14273701C22semidirect
29. #14273686C22semidirect
32. #14273661C22semidirect
33. #14273659C22semidirect
34. #14273652C22semidirect
35. #14273649C22semidirect
36. #14273649C22semidirect
37. #14273649C22semidirect
39. C2×M16×C59C22semidirect
42. C22⋊C8×C59C22semidirect
169. #407D59semidirect
170. C59#3194semidirect
177. C2×M16C2×D59semidirect
201. C22⋊C8C2×D59semidirect