Finite Group Databasebeta

← #21435791

Subgroups — #21435791

Order 7552.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D59C212yesmaximal, complemented
2#16838067C211yesmaximal, characteristic, complemented
3#16838037C211yesmaximal, characteristic
4#16837628C212yesmaximal, complemented
5#16837604C211yesmaximal, characteristic, complemented
6#16837599C211yesmaximal, characteristic, complemented
7#16837551C212yesmaximal, complemented
8#16837539C212yesmaximal, complemented
9C4×#11745660C211yesmaximal, characteristic, complemented
10C8×D236C211yesmaximal, characteristic, complemented
11C59×#492C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2212yescomplemented
13C2×D4×D59—21no—
14C2×D4×D59C2212yes—
15C2×#11745660C2211yescharacteristic
16C2×#11745670C2212yescomplemented
17C2×#11745671C2212yes—
18SD16×D59—28no—
19C2×C8×D59C2212yescomplemented
20#14273726—21no—
21#14273705—22no—
22#14273703C2212yescomplemented
23#14273698—21no—
24#14273698C2212yes—
25#14273695—22no—
26C4×Dic118C2211yescharacteristic, complemented
27C4×D236C2211yescharacteristic
28C4×D236C2211yescharacteristic
29#14273689—21no—
30#14273659C2211yescharacteristic, complemented
31C4⋊(C59⋊C8)C2211yescharacteristic, complemented
32#14273652C2211yescharacteristic, complemented
33#14273651C2212yescomplemented
34#14273649C2211yescharacteristic
35#14273648C2212yescomplemented
36(C2×D59)⋊C8C2212yescomplemented
37C2×SD16×C59C2212yescomplemented
38C2×SD16×C59C2212yescomplemented
39C4⋊Q8×C59C2211yescharacteristic, complemented
40#127×C59C2211yescharacteristic
41C4×C8×C59C2211yescharacteristic, complemented
42C23×D59—22no—
43C2×Q8×C59C2312yes—
44C2×D4×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59C2312yes—
47C2×C22⋊D59—22no—
48C2×C22⋊D59—22no—
49Q8×D59—24no—
50Q8×D59—42no—
51D4×D59—24no—
52D4×D59—42no—
53D4×D59—44no—
54C2×D236D411yescharacteristic
55C2×D236C2311yescharacteristic
56C2×C4×D59—22no—
57C2×C4×D59—21no—
58C2×C4×D59C2312yes—
59C2×Dic118—22no—
60C2×Dic118C2311yescharacteristic
61SD16×C59—24no—
62SD16×C59—42no—
63C2×C8×C59C2311yescharacteristic
64C2×C8×C59C2311yescharacteristic
65C4⋊C4×C59—21no—
66C4⋊C4×C59—21no—
67C42×C59C2311yescharacteristic
68C22⋊Dic59—22no—
69#11745671—24no—
70#11745670—24no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4D412yes—
74C4⋊Dic59D411yescharacteristic
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—21no—
78Dic59⋊C4—22no—
79C4×Dic59—21no—
80C2×C59⋊C8C2312yes—
81#11745660D414yescomplemented
82C8×D59—24no—
83C23×C59—22no—
84C22×D59—21no—
85C22×D59—44no—
86C22×D59—44no—
87C22×D59—82no—
88C22×D59C2×D412yes—
89Q8×C59—22no—
90Q8×C59—22no—
91D4×C59—22no—
92D4×C59—22no—
93D4×C59—42no—
94D4×C59—81no—
95C2×C4×C59—22no—
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—44no—
100C22⋊D59—44no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103C2×Dic59—21no—
104C2×Dic59C2×D412yes—
105D236—21no—
106D236SD1614yescomplemented
107D236C2×D412yes—
108C4×D59—22no—
109C4×D59—22no—
110C4×D59—44no—
111C4×D59—42no—
112Dic118—21no—
113Dic118—44no—
114Dic118C2×D412yescomplemented
115C8×C59—21no—
116C8×C59C2×D412yescomplemented
117C59⋊C8—22no—
118C22×C59—42no—
119C22×C59—42no—
120C22×C59—42no—
121C22×C59C22×D411yescharacteristic
122C2×D59—22no—
123C2×D59—22no—
124C2×D59—22no—
125C2×D59—41no—
126C2×D59—41no—
127C2×D59—88no—
128C2×D59C2×SD1614yescomplemented
129C4×C59—21no—
130C4×C59—42no—
131C4×C59C2×SD1612yescomplemented
132C4×C59C22×D411yescharacteristic
133C4×C59H(5,𝔽2)11yescharacteristic
134Dic59—22no—
135Dic59—22no—
136Dic59—41no—
137Dic59—41no—
138D4×SD16—591nomaximal, Sylow
139C2×C59—82no—
140C2×C59D4211yescharacteristic
141C2×C59C22×SD1611yescharacteristic
142C2×C59#57711yescharacteristic
143D59—24no—
144D59—42no—
145C22×SD16—592no—
146D4×Q8—591no—
147D42—591no—
148#492D5911yescharacteristic, complemented
149#465—592no—
150#461—591no—
151#460—591no—
152#450—592no—
153#448—592no—
154C4×SD16—591no—
155C8×D4—591no—
156C59D4×SD1611yescharacteristic, complemented, Sylow
157C22×Q8—592no—
158C22×D4—592no—
159C22×D4—1181no—
160C2×SD16—592no—
161C2×SD16—592no—
162C2×SD16—591no—
163C2×SD16—1188no—
164C2×SD16C2×D5912yescomplemented
165C2×SD16C2×D5912yescomplemented
166C22×C8—592no—
167C4⋊Q8—1181no—
168C4⋊Q8C2×D5911yescharacteristic, complemented
169#127C2×D5911yescharacteristic
170C22⋊Q8—592no—
171C22⋊Q8—1182no—
172#121—592no—
173#121—1181no—
174#120—1182no—
175C4×Q8—591no—
176C4×D4—591no—
177C4×D4—591no—
178C4×D4—1181no—
179#106—591no—
180C4⋊C8—591no—
181#103—592no—
182#103—591no—
183#102—592no—
184#102—591no—
185C22⋊C8—592no—
186C4×C8C2×D5911yescharacteristic, complemented
187C24—1182no—
188C2×Q8—591no—
189C2×Q8—1182no—
190C2×Q8—1184no—
191C2×Q8—2362no—
192C2×Q8C22×D5912yes—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—591no—
196C2×D4—591no—
197C2×D4—1182no—
198C2×D4—1184no—
199C2×D4—1182no—
200C2×D4—2362no—
201C2×D4—2364no—
202C2×D4C22×D5912yes—
203C22×C4—592no—
204C22×C4—1182no—
205C22×C4—1181no—
206SD16—24no—
207SD16—42no—
208SD16—594no—
209SD16—1184no—
210SD16—1184no—
211C2×C8—592no—
212C2×C8—1184no—
213C2×C8C22×D5911yescharacteristic
214C2×C8C22×D5911yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—591no—
218C4⋊C4—591no—
219C4⋊C4—591no—
220C4⋊C4—1182no—
221C4⋊C4—1181no—
222C22⋊C4—592no—
223C22⋊C4—1182no—
224C22⋊C4—1182no—
225C22⋊C4—1181no—
226C42—1181no—
227C42C22×D5911yescharacteristic
228C23—22no—
229C23—592no—
230C23—1181no—
231C23—4722no—
232C23—2364no—
233C23—2364no—
234Q8—22no—
235Q8—22no—
236Q8—592no—
237Q8—1181no—
238Q8—2364no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—594no—
244D4—592no—
245D4—1181no—
246D4—2364no—
247D4—2364no—
248C2×C4—22no—
249C2×C4—592no—
250C2×C4—1181no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—1181no—
254C2×C4—1181no—
255C2×C4—2364no—
256C2×C4—2362no—
257C2×C4D4×D5911yescharacteristic
258C2×C4D4×D5911yescharacteristic
259C2×C4C23×D5911yescharacteristic
260C8—21no—
261C8—1182no—
262C8D4×D5912yescomplemented
263C22—42no—
264C22—42no—
265C22—42no—
266C22—594no—
267C22—1182no—
268C22—1182no—
269C22—1182no—
270C22—4728no—
271C22—2361no—
272C22—2361no—
273C22C2×D4×D5911yescharacteristic
274C4—21no—
275C4—42no—
276C4—1182no—
277C4—1182no—
278C4—2361no—
279C4—2361no—
280C4SD16×D5912yescomplemented
281C4C2×D4×D5911yescharacteristic
282C4#1427377211yescharacteristic
283C2—82no—
284C2—1184no—
285C2—2362no—
286C2#1683803711yescharacteristic
287C2C2×SD16×D5911yescharacteristic
288C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#492C2
socleC22×C59C22×D4
hypercenter#492D59
hypocenterC59D4×SD16
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D59C2semidirect
2. #16838067C2semidirect
4. #16837628C2semidirect
5. #16837604C2semidirect
6. #16837599C2semidirect
7. #16837551C2semidirect
8. #16837539C2semidirect
9. C4×#11745660C2semidirect
10. C8×D236C2semidirect
11. C59×#492C2semidirect
12. C2×Q8×D59C22semidirect
16. C2×#11745670C22semidirect
19. C2×C8×D59C22semidirect
22. #14273703C22semidirect
26. C4×Dic118C22semidirect
30. #14273659C22semidirect
31. C4⋊(C59⋊C8)C22semidirect
32. #14273652C22semidirect
33. #14273651C22semidirect
35. #14273648C22semidirect
36. (C2×D59)⋊C8C22semidirect
37. C2×SD16×C59C22semidirect
38. C2×SD16×C59C22semidirect
39. C4⋊Q8×C59C22semidirect
41. C4×C8×C59C22semidirect
81. #11745660D4semidirect
106. D236SD16semidirect
114. Dic118C2×D4semidirect
116. C8×C59C2×D4semidirect
128. C2×D59C2×SD16semidirect
131. C4×C59C2×SD16semidirect
148. #492D59semidirect
156. C59D4×SD16semidirect
164. C2×SD16C2×D59semidirect
165. C2×SD16C2×D59semidirect
168. C4⋊Q8C2×D59semidirect
186. C4×C8C2×D59semidirect
262. C8D4×D59semidirect
280. C4SD16×D59semidirect