Finite Group Databasebeta

← #21436326

Subgroups — #21436326

Order 7552.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 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
1#16838126C211yesmaximal, characteristic, complemented
2#16838037C211yesmaximal, characteristic
3#16837725C212yesmaximal, complemented
4#16837679C211yesmaximal, characteristic, complemented
5#16837647C212yesmaximal, complemented
6#16837599C211yesmaximal, characteristic, complemented
7#16837554C212yesmaximal, complemented
8C4×#11745660C211yesmaximal, characteristic, complemented
9#16837481C211yesmaximal, characteristic, complemented
10#16837405C212yesmaximal, complemented
11C59×#486C211yesmaximal, characteristic, complemented
12C2×D4×D59—21no—
13C2×D4×D59C2212yes—
14C2×#11745660C2211yescharacteristic
15C2×#11745670C2212yescomplemented
16C2×#11745659C2212yescomplemented
17#14273722—22no—
18#14273716—21no—
19#14273707C2212yescomplemented
20#14273698—21no—
21#14273698C2212yes—
22#14273695—22no—
23Q8×Dic59—21no—
24C4×Dic118C2211yescharacteristic, complemented
25C4×D236C2211yescharacteristic
26C4×D236C2211yescharacteristic
27#14273685C2212yescomplemented
28#14273660C2212yescomplemented
29#14273659C2211yescharacteristic, complemented
30C4⋊(C59⋊C8)C2211yescharacteristic, complemented
31#14273652C2211yescharacteristic, complemented
32#14273649C2211yescharacteristic
33#14273648C2212yescomplemented
34#14273647C2212yes—
35(C2×D59)⋊C8C2212yescomplemented
36C4⋊Q8×C59C2211yescharacteristic, complemented
37#127×C59C2211yescharacteristic
38#102×C59C2212yescomplemented
39#102×C59C2212yescomplemented
40C4×C8×C59C2211yescharacteristic, complemented
41C23×D59—22no—
42C2×Q8×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—21no—
45C2×D4×C59C2312yes—
46C2×C22⋊D59—22no—
47C2×C22⋊D59—22no—
48D4×D59—24no—
49D4×D59—42no—
50D4×D59—44no—
51C2×D236D411yescharacteristic
52C2×D236C2311yescharacteristic
53C2×C4×D59—21no—
54C2×C4×D59C2312yes—
55C2×Dic118C2311yescharacteristic
56C2×C8×C59C2311yescharacteristic
57C2×C8×C59C2311yescharacteristic
58C4⋊C4×C59—21no—
59C4⋊C4×C59C2312yes—
60C42×C59C2311yescharacteristic
61C22⋊Dic59—22no—
62#11745670—42no—
63(C2×D59)⋊C4—22no—
64(C2×D59)⋊C4—22no—
65(C2×D59)⋊C4—21no—
66(C2×D59)⋊C4D412yes—
67C4⋊Dic59—22no—
68C4⋊Dic59D411yescharacteristic
69C4⋊Dic59C2311yescharacteristic
70C4⋊Dic59C2311yescharacteristic
71Dic59⋊C4—21no—
72C4×Dic59—21no—
73C4×Dic59—22no—
74C2×C59⋊C8C2312yes—
75#11745660—22no—
76#11745659—42no—
77C23×C59—22no—
78C22×D59—21no—
79C22×D59—44no—
80C22×D59—44no—
81C22×D59—82no—
82C22×D59C2×D412yes—
83Q8×C59—42no—
84D4×C59—22no—
85D4×C59—22no—
86D4×C59—42no—
87D4×C59—81no—
88C2×C4×C59—22no—
89C2×C4×C59C2×D411yescharacteristic
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2411yescharacteristic
92C22⋊D59—44no—
93C22⋊D59—44no—
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—21no—
97C2×Dic59C2×D412yes—
98D236—22no—
99D236—21no—
100D236#4112yes—
101C4×D59—22no—
102C4×D59—22no—
103C4×D59—42no—
104Dic118—21no—
105Dic118—21no—
106C8×C59—21no—
107C8×C59—21no—
108C59⋊C8—22no—
109C22×C59—42no—
110C22×C59—42no—
111C22×C59—42no—
112C22×C59C22×D411yescharacteristic
113C2×D59—22no—
114C2×D59—22no—
115C2×D59—22no—
116C2×D59—22no—
117C2×D59—41no—
118C2×D59—41no—
119C2×D59—88no—
120C4×C59—21no—
121C4×C59—42no—
122C4×C59AGL(1,ℤ/8)12yes—
123C4×C59C2×#4111yescharacteristic
124C4×C59H(5,𝔽2)11yescharacteristic
125Dic59—22no—
126Dic59—41no—
127Dic59—41no—
128Dic59—41no—
129#3324—591nomaximal, Sylow
130C2×C59—82no—
131C2×C59#54611yescharacteristic
132C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
133C2×C59#57711yescharacteristic
134D59—42no—
135D59—42no—
136#556—591no—
137D42—591no—
138#486D5911yescharacteristic, complemented
139#482—592no—
140#474—591no—
141#468—592no—
142#460—591no—
143#450—592no—
144C4×SD16—591no—
145#436—591no—
146#418—592no—
147C59#332411yescharacteristic, complemented, Sylow
148C22×D4—592no—
149C22×D4—1181no—
150C2×SD16—592no—
151C2×SD16—591no—
152C2×M16—592no—
153C4⋊Q8C2×D5911yescharacteristic, complemented
154#127C2×D5911yescharacteristic
155#126—1182no—
156#124—1181no—
157C22⋊Q8—592no—
158#121—592no—
159#121—1181no—
160#120—1182no—
161C4×Q8—591no—
162C4×Q8—1181no—
163C4×D4—591no—
164C4×D4—591no—
165#117—592no—
166#107—592no—
167#106—591no—
168C4⋊C8—591no—
169#103—591no—
170#102—592no—
171#102—591no—
172#102—592no—
173#102C2×D5912yescomplemented
174#102C2×D5912yescomplemented
175C22⋊C8—592no—
176C4×C8C2×D5911yescharacteristic, complemented
177C24—1182no—
178C2×Q8—21no—
179C2×Q8—591no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—591no—
183C2×D4—591no—
184C2×D4—1182no—
185C2×D4—1184no—
186C2×D4—1182no—
187C2×D4—2362no—
188C2×D4—2364no—
189C2×D4C22×D5912yes—
190C22×C4—592no—
191C22×C4—1181no—
192SD16—1182no—
193SD16—2362no—
194M16—2362no—
195C2×C8—592no—
196C2×C8C22×D5911yescharacteristic
197C2×C8C22×D5911yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—591no—
200C4⋊C4—591no—
201C4⋊C4—591no—
202C4⋊C4—1182no—
203C4⋊C4—1181no—
204C4⋊C4C22×D5912yes—
205C22⋊C4—592no—
206C22⋊C4—1182no—
207C22⋊C4—1182no—
208C22⋊C4—1182no—
209C22⋊C4—1181no—
210C42—1182no—
211C42—1181no—
212C42C22×D5911yescharacteristic
213C23—22no—
214C23—592no—
215C23—1181no—
216C23—4722no—
217C23—2364no—
218C23—2364no—
219Q8—42no—
220Q8—1181no—
221Q8—1181no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—81no—
226D4—592no—
227D4—1182no—
228D4—1181no—
229D4—2364no—
230D4—2364no—
231C2×C4—22no—
232C2×C4—592no—
233C2×C4—1182no—
234C2×C4—1182no—
235C2×C4—1181no—
236C2×C4—1181no—
237C2×C4—1181no—
238C2×C4—2362no—
239C2×C4D4×D5911yescharacteristic
240C2×C4D4×D5911yescharacteristic
241C2×C4C23×D5911yescharacteristic
242C8—21no—
243C8—21no—
244C8—1182no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—1182no—
249C22—1182no—
250C22—1182no—
251C22—1182no—
252C22—4728no—
253C22—2361no—
254C22—2361no—
255C22C2×D4×D5911yescharacteristic
256C4—21no—
257C4—42no—
258C4—1182no—
259C4—2361no—
260C4—2361no—
261C4—2361no—
262C4#1427375112yes—
263C4#41×D5911yescharacteristic
264C4#1427377311yescharacteristic
265C2—82no—
266C2—2362no—
267C2—2362no—
268C2#1683804511yescharacteristic
269C2C2×#1427375111yescharacteristic
270C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#486C2
socleC22×C59C22×D4
hypercenter#486D59
hypocenterC59#3324
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16838126C2semidirect
3. #16837725C2semidirect
4. #16837679C2semidirect
5. #16837647C2semidirect
6. #16837599C2semidirect
7. #16837554C2semidirect
8. C4×#11745660C2semidirect
9. #16837481C2semidirect
10. #16837405C2semidirect
11. C59×#486C2semidirect
15. C2×#11745670C22semidirect
16. C2×#11745659C22semidirect
19. #14273707C22semidirect
24. C4×Dic118C22semidirect
27. #14273685C22semidirect
28. #14273660C22semidirect
29. #14273659C22semidirect
30. C4⋊(C59⋊C8)C22semidirect
31. #14273652C22semidirect
33. #14273648C22semidirect
35. (C2×D59)⋊C8C22semidirect
36. C4⋊Q8×C59C22semidirect
38. #102×C59C22semidirect
39. #102×C59C22semidirect
40. C4×C8×C59C22semidirect
138. #486D59semidirect
147. C59#3324semidirect
153. C4⋊Q8C2×D59semidirect
173. #102C2×D59semidirect
174. #102C2×D59semidirect
176. C4×C8C2×D59semidirect