Finite Group Databasebeta

← #21434913

Subgroups — #21434913

Order 7552.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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#16837915C211yesmaximal, characteristic, complemented
2C2×#14273698C211yesmaximal, characteristic, complemented
3#16837730C211yesmaximal, characteristic, complemented
4#16837724C212yesmaximal, complemented
5#16837716C211yesmaximal, characteristic, complemented
6#16837654C212yesmaximal, complemented
7#16837645C212yesmaximal, complemented
8#16837546C212yesmaximal, complemented
9#16837536C211yesmaximal, characteristic, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic, complemented
12C2×#11745689C2211yescharacteristic, complemented
13C22×C22⋊D59—21no—
14C22×C4×D59C2211yescharacteristic
15C2×#11745669C2212yescomplemented
16C2×#11745659C2212yescomplemented
17#14273712—21no—
18#14273703C2211yescharacteristic, complemented
19#14273702C2211yescharacteristic, complemented
20#14273701C2211yescharacteristic
21#14273698—21no—
22#14273698—21no—
23#14273698C2212yes—
24#14273698C2212yes—
25#14273697—21no—
26#14273689—21no—
27(C22⋊D59)⋊C4—21no—
28#14273685C2211yescharacteristic, complemented
29C2×C4⋊Dic59C2211yescharacteristic
30C2×C22⋊Dic59—21no—
31#14273661C2212yescomplemented
32#14273658C2212yescomplemented
33#14273651C2212yescomplemented
34#14273649C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×SD16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C23×D59D411yescharacteristic
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—44no—
54C22×Dic59—21no—
55C22×Dic59D411yescharacteristic
56#11745689—42no—
57C2×D236—21no—
58C2×D236—21no—
59C2×D236C2311yescharacteristic
60C2×C4×D59—21no—
61C2×C4×D59—42no—
62C2×C4×D59D411yescharacteristic
63C2×C4×D59D411yescharacteristic
64C2×C4×D59C2311yescharacteristic
65C2×C4×D59C2311yescharacteristic
66SD16×C59—42no—
67C2×C8×C59C2312yes—
68C4⋊C4×C59—21no—
69C4⋊C4×C59C2311yescharacteristic
70C22⋊C4×C59—21no—
71C22⋊Dic59—22no—
72C22⋊Dic59—22no—
73#11745669—42no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77(C2×D59)⋊C4—21no—
78C4⋊Dic59—21no—
79C4⋊Dic59C2311yescharacteristic
80C4⋊Dic59C2312yes—
81Dic59⋊C4—21no—
82C4×Dic59—21no—
83C2×C59⋊C8C2312yes—
84#11745659—42no—
85C23×C59—22no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59C2×D411yescharacteristic
90C22×D59—22no—
91C22×D59—21no—
92C22×D59—41no—
93C22×D59—41no—
94C22×D59C2×D411yescharacteristic
95C22×D59C2×D411yescharacteristic
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—42no—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—41no—
111C2×Dic59—42no—
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114D236—22no—
115D236—42no—
116D236—42no—
117C4×D59—22no—
118C4×D59—22no—
119C4×D59—41no—
120C4×D59—41no—
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—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—21no—
136C2×D59—21no—
137C2×D59—21no—
138C2×D59—44no—
139C2×D59—41no—
140C2×D59—41no—
141C2×D59—41no—
142C4×C59—21no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59H(5,𝔽2)12yes—
146Dic59—21no—
147Dic59—21no—
148Dic59—42no—
149Dic59—41no—
150#3194—591nomaximal, Sylow
151C2×C59—22no—
152C2×C59—82no—
153C2×C59#53411yescharacteristic
154C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C59#57511yescharacteristic
156D59—41no—
157D59—41no—
158D59—81no—
159#525—591no—
160C2×#121—591no—
161#483—591no—
162#482—592no—
163#480—591no—
164#469—592no—
165#468—592no—
166#450D5911yescharacteristic, complemented
167#449—592no—
168#447—591no—
169#407—591no—
170C59#319411yescharacteristic, complemented, Sylow
171C2×#41—591no—
172C22×D4—1181no—
173C22×D4C2×D5911yescharacteristic
174C23×C4—591no—
175C2×SD16C2×D5912yescomplemented
176C2×D8—592no—
177C2×M16—592no—
178#123—1181no—
179C22⋊Q8—591no—
180C22⋊Q8C2×D5911yescharacteristic
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#102C2×D5912yescomplemented
199C22⋊C8—591no—
200C22⋊C8—592no—
201C22⋊C8C2×D5911yescharacteristic
202C24—21no—
203C24—591no—
204#41—2362no—
205C2×Q8C22×D5911yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—591no—
209C2×D4—1181no—
210C2×D4—1181no—
211C2×D4—1182no—
212C2×D4—1182no—
213C2×D4—1181no—
214C2×D4—2364no—
215C2×D4C22×D5912yes—
216C22×C4—591no—
217C22×C4—591no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—1181no—
222C22×C4—1181no—
223C22×C4—2362no—
224C22×C4C22×D5911yescharacteristic
225SD16—42no—
226D8—2362no—
227M16—2362no—
228C2×C8—592no—
229C2×C8C22×D5912yes—
230C4⋊C4—21no—
231C4⋊C4—591no—
232C4⋊C4—592no—
233C4⋊C4—1181no—
234C4⋊C4—1181no—
235C4⋊C4C22×D5911yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1181no—
238C22⋊C4—1181no—
239C22⋊C4—1181no—
240C22⋊C4—1182no—
241C22⋊C4—1182no—
242C22⋊C4—1181no—
243C42—1181no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—591no—
249C23—591no—
250C23—1182no—
251C23—1181no—
252C23—2361no—
253C23—2361no—
254C23D4×D5911yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—1182no—
260D4—2362no—
261D4—2362no—
262D4—2364no—
263D4—2364no—
264D4—2362no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—591no—
269C2×C4—591no—
270C2×C4—1182no—
271C2×C4—1182no—
272C2×C4—1181no—
273C2×C4—1182no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2362no—
277C2×C4—2361no—
278C2×C4—2362no—
279C2×C4D4×D5911yescharacteristic
280C2×C4C23×D5911yescharacteristic
281C8—22no—
282C8—1182no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—1181no—
290C22—1181no—
291C22—1181no—
292C22—1181no—
293C22—1181no—
294C22—1181no—
295C22—2364no—
296C22—2361no—
297C22—2361no—
298C22—2361no—
299C22#1427375412yes—
300C22C2×D4×D5911yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—1181no—
305C4—1181no—
306C4—2362no—
307C4—2361no—
308C4#1427377212yes—
309C2—22no—
310C2—82no—
311C2—4721no—
312C2—2361no—
313C2—2361no—
314C2#1683796211yescharacteristic
315C2C2×#1427375411yescharacteristic
316C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3194
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. #16837915C2semidirect
2. C2×#14273698C2semidirect
3. #16837730C2semidirect
4. #16837724C2semidirect
5. #16837716C2semidirect
6. #16837654C2semidirect
7. #16837645C2semidirect
8. #16837546C2semidirect
9. #16837536C2semidirect
10. #16837364C2semidirect
11. C59×#450C2semidirect
12. C2×#11745689C22semidirect
15. C2×#11745669C22semidirect
16. C2×#11745659C22semidirect
18. #14273703C22semidirect
19. #14273702C22semidirect
28. #14273685C22semidirect
31. #14273661C22semidirect
32. #14273658C22semidirect
33. #14273651C22semidirect
34. #14273649C22semidirect
35. #14273648C22semidirect
37. (C2×D59)⋊C8C22semidirect
39. C2×SD16×C59C22semidirect
41. #102×C59C22semidirect
166. #450D59semidirect
170. C59#3194semidirect
175. C2×SD16C2×D59semidirect
198. #102C2×D59semidirect