Finite Group Databasebeta

← #21434851

Subgroups — #21434851

Order 7552.

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

normal subgroups only

This group has 468 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
1C2×#14273703C211yesmaximal, characteristic, complemented
2#121×D59C211yesmaximal, characteristic
3#16837731C212yesmaximal, complemented
4#16837721C211yesmaximal, characteristic, complemented
5#16837718C211yesmaximal, characteristic, complemented
6#16837634C212yesmaximal, complemented
7#16837628C212yesmaximal, complemented
8#16837551C212yesmaximal, complemented
9#16837541C211yesmaximal, characteristic, complemented
10C22⋊C8×D59C211yesmaximal, characteristic, complemented
11C59×#448C211yesmaximal, characteristic, complemented
12C2×D4×D59—21no—
13C2×D4×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C2×#11745671C2212yescomplemented
16C2×C8×D59C2212yescomplemented
17#14273703—21no—
18#14273703—21no—
19#14273703C2212yes—
20#14273703C2212yescomplemented
21#14273702C2211yescharacteristic
22#14273701C2211yescharacteristic
23#14273700—21no—
24#14273699—21no—
25#14273698C2211yescharacteristic
26C2×C4⋊Dic59C2211yescharacteristic, complemented
27C2×Dic59⋊C4—21no—
28C4⋊C4×D59C2211yescharacteristic
29C2×(C2×D59)⋊C4—21no—
30C22⋊C4×D59—21no—
31#14273659C2212yescomplemented
32#14273658C2212yescomplemented
33#14273651C2212yescomplemented
34#14273649C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic, complemented
37(C2×D59)⋊C8C2212yes—
38C22×Q8×C59C2211yescharacteristic, complemented
39C2×SD16×C59C2212yescomplemented
40#121×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic, complemented
43C23×D59—21no—
44C23×D59D411yescharacteristic
45C2×Q8×C59—21no—
46C2×Q8×C59—42no—
47C2×Q8×C59C2312yes—
48C2×D4×C59—21no—
49C2×D4×C59C2311yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—21no—
54C2×C22⋊D59—21no—
55C22×Dic59—21no—
56C22×Dic59D411yescharacteristic
57D4×D59—42no—
58D4×D59—44no—
59C2×D236—21no—
60C2×D236C2311yescharacteristic
61C2×C4×D59—21no—
62C2×C4×D59—42no—
63C2×C4×D59D411yescharacteristic
64C2×C4×D59D411yescharacteristic
65C2×C4×D59C2311yescharacteristic
66C2×C4×D59C2311yescharacteristic
67SD16×C59—42no—
68C2×C8×C59C2312yes—
69C4⋊C4×C59C2311yescharacteristic
70C22⋊C4×C59—21no—
71C22⋊Dic59—21no—
72#11745671—42no—
73(C2×D59)⋊C4—22no—
74(C2×D59)⋊C4—22no—
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77C4⋊Dic59—21no—
78C4⋊Dic59C2312yes—
79C4⋊Dic59C2311yescharacteristic
80Dic59⋊C4—21no—
81Dic59⋊C4—22no—
82Dic59⋊C4—22no—
83C2×C59⋊C8C2312yes—
84C8×D59—42no—
85C23×C59—21no—
86C23×C59C2×D411yescharacteristic
87C22×D59—21no—
88C22×D59—41no—
89C22×D59—41no—
90C22×D59—42no—
91C22×D59—42no—
92C22×D59—42no—
93C22×D59SD1614yescomplemented
94C22×D59C2×D411yescharacteristic
95C22×D59C2×D411yescharacteristic
96Q8×C59—22no—
97Q8×C59—22no—
98Q8×C59—42no—
99D4×C59—22no—
100D4×C59—42no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—42no—
105C2×C4×C59C2×D411yescharacteristic
106C2×C4×C59C2411yescharacteristic
107C22⋊D59—42no—
108C22⋊D59—42no—
109C22⋊D59—42no—
110C2×Dic59—22no—
111C2×Dic59—21no—
112C2×Dic59—41no—
113C2×Dic59—42no—
114C2×Dic59C2×D411yescharacteristic
115C2×Dic59C2×D411yescharacteristic
116D236—22no—
117D236—42no—
118C4×D59—22no—
119C4×D59—22no—
120C4×D59—41no—
121C4×D59—41no—
122C4×D59—42no—
123C8×C59—22no—
124C59⋊C8—22no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59C2×SD1612yes—
131C22×C59C22×D411yescharacteristic
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—21no—
136C2×D59—44no—
137C2×D59—41no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—84no—
141C2×D59C2×SD1612yes—
142C2×D59C2×SD1612yes—
143C4×C59—21no—
144C4×C59—42no—
145C4×C59—41no—
146C4×C59H(5,𝔽2)12yes—
147Dic59—21no—
148Dic59—21no—
149Dic59—41no—
150Dic59—42no—
151#3187—591nomaximal, Sylow
152C2×C59—22no—
153C2×C59—81no—
154C2×C59#53411yescharacteristic
155C2×C59C22×SD1611yescharacteristic
156C2×C59#57511yescharacteristic
157D59—22no—
158D59—22no—
159D59—81no—
160C2×C22⋊Q8—591no—
161C2×#121—591no—
162#483—592no—
163#481—591no—
164#481—591no—
165#465—592no—
166#465—592no—
167#450—592no—
168#448—591no—
169#448D5911yescharacteristic, complemented
170C2×C22⋊C8—591no—
171C59#318711yescharacteristic, complemented, Sylow
172C22×Q8C2×D5911yescharacteristic, complemented
173C22×D4—591no—
174C22×D4—1181no—
175C23×C4—591no—
176C2×SD16—592no—
177C2×SD16C2×D5912yescomplemented
178C22×C8—592no—
179C22⋊Q8—592no—
180C22⋊Q8—592no—
181C22⋊Q8—1181no—
182C22⋊Q8—1181no—
183#121—591no—
184#121—591no—
185#121—591no—
186#121—1181no—
187#121—1181no—
188#121C2×D5911yescharacteristic
189C2×C4⋊C4—591no—
190C2×C4⋊C4—591no—
191C2×C4⋊C4—1181no—
192C2×C22⋊C4—1181no—
193C2×C22⋊C4—1181no—
194#106—592no—
195#106—592no—
196#103—592no—
197#103C2×D5912yescomplemented
198#102—592no—
199#102—592no—
200C22⋊C8—591no—
201C22⋊C8—592no—
202C22⋊C8C2×D5911yescharacteristic, complemented
203C24—591no—
204C24—1181no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D5912yes—
208C2×D4—21no—
209C2×D4—591no—
210C2×D4—1181no—
211C2×D4—1181no—
212C2×D4—1181no—
213C2×D4—1181no—
214C2×D4—2362no—
215C2×D4—2364no—
216C2×D4C22×D5911yescharacteristic
217C22×C4—21no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—591no—
222C22×C4—591no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4—2362no—
226C22×C4C22×D5911yescharacteristic
227SD16—42no—
228SD16—2362no—
229C2×C8—592no—
230C2×C8—2362no—
231C2×C8C22×D5912yes—
232C4⋊C4—592no—
233C4⋊C4—591no—
234C4⋊C4—1181no—
235C4⋊C4—1182no—
236C4⋊C4—1182no—
237C4⋊C4—1181no—
238C4⋊C4C22×D5911yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1181no—
241C22⋊C4—1182no—
242C22⋊C4—1182no—
243C22⋊C4—1181no—
244C22⋊C4—1181no—
245C23—21no—
246C23—591no—
247C23—591no—
248C23—594no—
249C23—1181no—
250C23—2361no—
251C23—2361no—
252C23—2362no—
253C23—2362no—
254C23—2362no—
255C23D4×D5911yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—1182no—
262D4—2362no—
263D4—2362no—
264D4—2362no—
265D4—2362no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—591no—
271C2×C4—591no—
272C2×C4—1182no—
273C2×C4—1182no—
274C2×C4—1182no—
275C2×C4—1181no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2362no—
279C2×C4—2361no—
280C2×C4—2362no—
281C2×C4D4×D5911yescharacteristic
282C2×C4C23×D5911yescharacteristic
283C8—22no—
284C8—1182no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—592no—
291C22—592no—
292C22—1181no—
293C22—1181no—
294C22—1181no—
295C22—1181no—
296C22—4724no—
297C22—2364no—
298C22—2361no—
299C22—2361no—
300C22—2361no—
301C22SD16×D5912yes—
302C22C2×D4×D5911yescharacteristic
303C4—21no—
304C4—41no—
305C4—42no—
306C4—1181no—
307C4—1181no—
308C4—2361no—
309C4—2362no—
310C4#1427377212yes—
311C2—22no—
312C2—81no—
313C2—1182no—
314C2—1182no—
315C2—4721no—
316C2#1683796211yescharacteristic
317C2C2×SD16×D5911yescharacteristic
318C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#448C2
socleC22×C59C22×D4
hypercenter#448D59
hypocenterC59#3187
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#14273703C2semidirect
3. #16837731C2semidirect
4. #16837721C2semidirect
5. #16837718C2semidirect
6. #16837634C2semidirect
7. #16837628C2semidirect
8. #16837551C2semidirect
9. #16837541C2semidirect
10. C22⋊C8×D59C2semidirect
11. C59×#448C2semidirect
15. C2×#11745671C22semidirect
16. C2×C8×D59C22semidirect
20. #14273703C22semidirect
26. C2×C4⋊Dic59C22semidirect
31. #14273659C22semidirect
32. #14273658C22semidirect
33. #14273651C22semidirect
34. #14273649C22semidirect
35. #14273648C22semidirect
36. C22⋊(C59⋊C8)C22semidirect
38. C22×Q8×C59C22semidirect
39. C2×SD16×C59C22semidirect
41. #103×C59C22semidirect
42. C22⋊C8×C59C22semidirect
93. C22×D59SD16semidirect
169. #448D59semidirect
171. C59#3187semidirect
172. C22×Q8C2×D59semidirect
177. C2×SD16C2×D59semidirect
197. #103C2×D59semidirect
202. C22⋊C8C2×D59semidirect