Finite Group Databasebeta

← #21435864

Subgroups — #21435864

Order 7552.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#16838210C212yesmaximal, complemented
2#16838206C212yesmaximal, complemented
3#16837792C211yesmaximal, characteristic, complemented
4#16837670C212yesmaximal, complemented
5#16837668C212yesmaximal, complemented
6#16837579C211yesmaximal, characteristic, complemented
7#16837571C212yesmaximal, complemented
8#16837521C212yesmaximal, complemented
9C59×#454C211yesmaximal, characteristic, complemented
10#14273773C2212yescomplemented
11#14273772C2211yescharacteristic
12#14273758—22no—
13#14273758C2212yescomplemented
14#14273755C2212yes—
15#14273754—22no—
16#14273752—22no—
17#14273752C2212yescomplemented
18#14273751—22no—
19#14273748—22no—
20#14273747—22no—
21#14273747C2212yes—
22C2×#11745671—22no—
23C2×#11745671C2212yescomplemented
24C2×#11745669C2212yescomplemented
25D8×D59—22no—
26#14273735C2212yescomplemented
27#14273734C2212yescomplemented
28#14273724C2211yescharacteristic, complemented
29#14273662C2212yescomplemented
30#14273655C2212yescomplemented
31#14273644C2211yescharacteristic, complemented
32#14273643C2212yescomplemented
33#14273638C2211yescharacteristic, complemented
342−1+4×C59C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C59C2212yescomplemented
36#124×C59C2211yescharacteristic, complemented
37#104×C59C2212yescomplemented
38#101×C59C2211yescharacteristic, complemented
39#41×C59—22no—
40#41×C59—41no—
41#41×C59C2312yes—
42C2×Q8×C59—22no—
43C2×Q8×C59C2311yescharacteristic
44C2×D4×C59C2311yescharacteristic
45C2×C22⋊D59—41no—
46#11745689—22no—
47#11745687—22no—
48D4×D59—22no—
49D4×D59—22no—
50D4×D59—42no—
51#11745685—22no—
52#11745685C2312yes—
53C2×D236—22no—
54C2×D236—41no—
55C2×D236C2312yes—
56SD16×C59—22no—
57D8×C59—22no—
58M16×C59C2312yes—
59C22⋊C4×C59—41no—
60C42×C59C2311yescharacteristic
61#11745672—22no—
62#11745672D412yescomplemented
63#11745671—22no—
64#11745671—22no—
65#11745671—22no—
66#11745671—22no—
67#11745671D412yescomplemented
68#11745670D412yescomplemented
69#11745669—22no—
70#11745669—22no—
71#11745669—22no—
72#11745669D412yescomplemented
73#11745664—22no—
74#11745664C2312yes—
75C2×C59⋊C8—22no—
76C2×C59⋊C8C2312yes—
77D472—22no—
78#11745660—22no—
79#11745659—22no—
80C8×D59—22no—
81C23×C59—21no—
82C22×D59—22no—
83C22×D59—41no—
84C22×D59—42no—
85Q8×C59—22no—
86Q8×C59—41no—
87Q8×C59C2×D412yes—
88D4×C59—22no—
89D4×C59—22no—
90D4×C59—41no—
91D4×C59C2×D412yes—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—42no—
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—22no—
98C22⋊D59—42no—
99C22⋊D59—42no—
100C2×Dic59—41no—
101D236—22no—
102D236—22no—
103D236—42no—
104D236—82no—
105D236C2×D412yes—
106C4×D59—22no—
107C4×D59—42no—
108Dic118C2×D412yes—
109C8×C59—22no—
110C59⋊C8—22no—
111C59⋊C8C2×D412yes—
112C59⋊C8C2×D412yes—
113C22×C59—22no—
114C22×C59—41no—
115C22×C59—81no—
116C22×C59H(5,𝔽2)11yescharacteristic
117C2×D59—22no—
118C2×D59—42no—
119C2×D59—82no—
120C2×D59—161no—
121C2×D59—162no—
122C4×C59—22no—
123C4×C59—41no—
124C4×C59—41no—
125C4×C59C22×D412yes—
126Dic59—22no—
127#3288—591nomaximal, Sylow
128C2×C59—21no—
129C2×C59—42no—
130C2×C59—81no—
131C2×C59D4211yescharacteristic
132D59—42no—
133D59—82no—
134#577—592no—
135#576—592no—
136#496—591no—
137#472—592no—
138#471—592no—
139#455—591no—
140#454D5911yescharacteristic, complemented
141#453—592no—
142#444—592no—
143C59#328811yescharacteristic, complemented, Sylow
1442−1+4C2×D5911yescharacteristic, complemented
145H(5,𝔽2)—592no—
146H(5,𝔽2)—591no—
147#137—592no—
148#137—1182no—
149AGL(1,ℤ/8)—592no—
150AGL(1,ℤ/8)—592no—
151AGL(1,ℤ/8)—1182no—
152AGL(1,ℤ/8)—1182no—
153AGL(1,ℤ/8)—1182no—
154AGL(1,ℤ/8)C2×D5912yescomplemented
155#135—592no—
156#135—1182no—
157#135—1182no—
158C2×SD16—592no—
159C2×SD16—1182no—
160C2×D8—592no—
161C2×D8—1182no—
162#131—592no—
163#131—592no—
164#127—591no—
165#124C2×D5911yescharacteristic, complemented
166#108—592no—
167#104—592no—
168#104C2×D5912yescomplemented
169#101C2×D5911yescharacteristic, complemented
170#100—591no—
171#100—592no—
172#97—591no—
173#41—22no—
174#41—41no—
175#41—592no—
176#41—1182no—
177#41—1182no—
178#41—1182no—
179#41C22×D5912yes—
180C2×Q8—22no—
181C2×Q8C22×D5911yescharacteristic
182C2×D4—592no—
183C2×D4—1182no—
184C2×D4—1182no—
185C2×D4—1182no—
186C2×D4—2361no—
187C2×D4—2362no—
188C2×D4—2361no—
189C2×D4C22×D5911yescharacteristic
190Q16—592no—
191Q16—1182no—
192SD16—22no—
193SD16—592no—
194SD16—592no—
195SD16—1182no—
196SD16—1182no—
197SD16—1182no—
198SD16—1182no—
199SD16—1182no—
200D8—22no—
201D8—592no—
202D8—1182no—
203D8—1182no—
204D8—1182no—
205D8—1182no—
206M16—592no—
207M16—1182no—
208M16—1182no—
209M16C22×D5912yes—
210C2×C8—592no—
211C2×C8—1182no—
212C2×C8—1182no—
213C22⋊C4—41no—
214C42C22×D5911yescharacteristic
215C23—21no—
216C23—1182no—
217C23—2362no—
218C23—2361no—
219Q8—22no—
220Q8—41no—
221Q8—592no—
222Q8D4×D5912yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—592no—
227D4—1182no—
228D4—1182no—
229D4—1182no—
230D4—4722no—
231D4—2362no—
232D4—2362no—
233D4—2362no—
234D4D4×D5912yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—1182no—
240C2×C4—2362no—
241C2×C4—2361no—
242C2×C4C23×D5911yescharacteristic
243C8—22no—
244C8—592no—
245C8—592no—
246C8—1182no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—1182no—
251C22—9442no—
252C22—9441no—
253C22—4722no—
254C22—2362no—
255C22#1427377211yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—1182no—
260C4C2×D4×D5912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—4722no—
265C2—2362no—
266C2#1683803411yescharacteristic
subgroupG/H
centerC2#16838034
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#454C2
socleC2×C59D42
hypercenter#454D59
hypocenterC59#3288
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #16838210C2semidirect
2. #16838206C2semidirect
3. #16837792C2semidirect
4. #16837670C2semidirect
5. #16837668C2semidirect
6. #16837579C2semidirect
7. #16837571C2semidirect
8. #16837521C2semidirect
9. C59×#454C2semidirect
10. #14273773C22semidirect
13. #14273758C22semidirect
17. #14273752C22semidirect
23. C2×#11745671C22semidirect
24. C2×#11745669C22semidirect
26. #14273735C22semidirect
27. #14273734C22semidirect
28. #14273724C22semidirect
29. #14273662C22semidirect
30. #14273655C22semidirect
31. #14273644C22semidirect
32. #14273643C22semidirect
33. #14273638C22semidirect
34. 2−1+4×C59C22semidirect
35. AGL(1,ℤ/8)×C59C22semidirect
36. #124×C59C22semidirect
37. #104×C59C22semidirect
38. #101×C59C22semidirect
62. #11745672D4semidirect
67. #11745671D4semidirect
68. #11745670D4semidirect
72. #11745669D4semidirect
140. #454D59semidirect
143. C59#3288semidirect
144. 2−1+4C2×D59semidirect
154. AGL(1,ℤ/8)C2×D59semidirect
165. #124C2×D59semidirect
168. #104C2×D59semidirect
169. #101C2×D59semidirect