Finite Group Databasebeta

← #21433453

Subgroups — #21433453

Order 7552.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#16838200C212yesmaximal, complemented
2#16837969C211yesmaximal, characteristic, complemented
3#16837580C214yesmaximal, complemented
4#16837570C214yesmaximal, complemented
5#16837419C212yesmaximal, complemented
6C2×#14273644C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C59C211yesmaximal, characteristic, complemented
8C2×#11745685C2211yescharacteristic, complemented
9#14273758—22no—
10#14273757C2214yescomplemented
11#14273753—22no—
12#14273752C2214yescomplemented
13#14273747—24no—
14C2×#11745664C2212yescomplemented
15#14273723C2212yescomplemented
16#14273714C2212yescomplemented
17#14273698—22no—
18#14273696—22no—
19#14273686C2211yescharacteristic, complemented
20#14273654C2218yescomplemented
21#14273644C2214yescomplemented
22H(5,𝔽2)×C59—24no—
23H(5,𝔽2)×C59—22no—
24H(5,𝔽2)×C59C2214yes—
25C2×#41×C59—22no—
26C2×#41×C59C2212yes—
27C22×D4×C59—22no—
28C22×D4×C59—22no—
29C22×D4×C59C2211yescharacteristic
30C24×C59—22no—
31C24×C59D412yes—
32#41×C59—28no—
33#41×C59—28no—
34#41×C59—22no—
35#41×C59—22no—
36#41×C59D414yescomplemented
37#41×C59C2314yes—
38C2×Q8×C59D412yescomplemented
39C2×D4×C59—24no—
40C2×D4×C59—28no—
41C2×D4×C59—28no—
42C2×D4×C59—22no—
43C2×D4×C59—28no—
44C2×D4×C59—28no—
45C2×D4×C59—22no—
46C2×D4×C59—22no—
47C2×D4×C59D412yes—
48C2×D4×C59D412yescomplemented
49C2×D4×C59C2312yes—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59C2311yescharacteristic
53C2×C22⋊D59—21no—
54C2×C22⋊D59—24no—
55#11745685—22no—
56#11745685—22no—
57C2×D236C2311yescharacteristic, complemented
58C2×C4×D59—21no—
59C2×Dic118C2311yescharacteristic, complemented
60C22⋊Dic59—21no—
61C22⋊Dic59—24no—
62#11745672—24no—
63#11745671—24no—
64#11745670—24no—
65#11745669—24no—
66C4⋊Dic59—21no—
67C4×Dic59C2312yescomplemented
68#11745664—22no—
69#11745664C2314yescomplemented
70C2×C59⋊C8—22no—
71C23×C59—28no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—82no—
76C23×C59—84no—
77C23×C59C2×D414yes—
78C23×C59C2×D411yescharacteristic
79C22×D59—21no—
80Q8×C59—22no—
81Q8×C59C2×D414yes—
82D4×C59—216no—
83D4×C59—24no—
84D4×C59—24no—
85D4×C59—24no—
86D4×C59—22no—
87D4×C59—24no—
88D4×C59C2×D414yes—
89C2×C4×C59—28no—
90C2×C4×C59—22no—
91C2×C4×C59—24no—
92C2×C4×C59—24no—
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2×D412yes—
95C2×C4×C59C2×D412yes—
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—41no—
98C22⋊D59—41no—
99C22⋊D59—84no—
100C2×Dic59—22no—
101C2×Dic59—21no—
102D236—22no—
103C4×D59—42no—
104Dic118—22no—
105C59⋊C8—24no—
106C22×C59—24no—
107C22×C59—24no—
108C22×C59—41no—
109C22×C59—44no—
110C22×C59—44no—
111C22×C59—44no—
112C22×C59—82no—
113C22×C59—88no—
114C22×C59#12011yescharacteristic
115C22×C59#12011yescharacteristic
116C22×C59C22×D411yescharacteristic
117C2×D59—42no—
118C2×D59—41no—
119C4×C59—24no—
120C4×C59#12012yes—
121C4×C59C22×D412yes—
122Dic59—42no—
123Dic59—41no—
124#3019—591nomaximal, Sylow
125C2×C59—21no—
126C2×C59—21no—
127C2×C59—21no—
128C2×C59—44no—
129C2×C59—44no—
130C2×C59C2×#12011yescharacteristic
131D59—81no—
132C2×H(5,𝔽2)D5911yescharacteristic, complemented
133#575—592no—
134#535—591no—
135#455—594no—
136#453—594no—
137#421—592no—
138C2×#100—591no—
139C59#301911yescharacteristic, complemented, Sylow
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D5914yes—
143C2×#41—22no—
144C2×#41—591no—
145C2×#41C2×D5912yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D5911yescharacteristic
149#137—594no—
150#137—1182no—
151AGL(1,ℤ/8)—594no—
152AGL(1,ℤ/8)—1182no—
153#135—1184no—
154C2×M16—592no—
155#127—592no—
156#124—592no—
157#121—1182no—
158#120—1182no—
159#117—591no—
160#104—598no—
161#100—594no—
162C24—22no—
163C24C22⋊D5912yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—1182no—
169#41—1182no—
170#41C22⋊D5914yescomplemented
171#41C22×D5914yes—
172C2×Q8—591no—
173C2×Q8C22⋊D5912yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—591no—
183C2×D4—1184no—
184C2×D4—1181no—
185C2×D4C22⋊D5912yes—
186C2×D4C22⋊D5912yescomplemented
187C2×D4C22×D5912yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1181no—
191C22×C4C22×D5911yescharacteristic
192Q16—1184no—
193SD16—1184no—
194SD16—1184no—
195D8—1184no—
196M16—594no—
197M16—1182no—
198C2×C8—1182no—
199C4⋊C4—1181no—
200C22⋊C4—1181no—
201C22⋊C4—1184no—
202C42—592no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—1181no—
210C23C2×C22⋊D5914yes—
211C23C2×C22⋊D5911yescharacteristic
212Q8—22no—
213Q8—1182no—
214Q8C2×C22⋊D5914yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—1182no—
222D4—4724no—
223D4—2361no—
224D4—2361no—
225D4C2×C22⋊D5914yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—1182no—
231C2×C4—1181no—
232C2×C4—2362no—
233C2×C4C2×C22⋊D5911yescharacteristic
234C2×C4C2×C22⋊D5912yes—
235C2×C4C2×C22⋊D5912yes—
236C2×C4C23×D5911yescharacteristic
237C8—1184no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—2362no—
247C22—2361no—
248C22#1427369611yescharacteristic
249C22#1427369611yescharacteristic
250C22C22×C22⋊D5911yescharacteristic
251C4—24no—
252C4—2362no—
253C4—2361no—
254C4#1427369612yes—
255C4C22×C22⋊D5912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—4721no—
262C2C2×#1427369611yescharacteristic
subgroupG/H
centerC2C2×#14273696
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×H(5,𝔽2)×C59C2
socleC2×C59C2×#120
hypercenterC2×H(5,𝔽2)D59
hypocenterC59#3019
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #16838200C2semidirect
2. #16837969C2semidirect
3. #16837580C2semidirect
4. #16837570C2semidirect
5. #16837419C2semidirect
6. C2×#14273644C2semidirect
7. C2×H(5,𝔽2)×C59C2semidirect
8. C2×#11745685C22semidirect
10. #14273757C22semidirect
12. #14273752C22semidirect
14. C2×#11745664C22semidirect
15. #14273723C22semidirect
16. #14273714C22semidirect
19. #14273686C22semidirect
20. #14273654C22semidirect
21. #14273644C22semidirect
36. #41×C59D4semidirect
38. C2×Q8×C59D4semidirect
48. C2×D4×C59D4semidirect
57. C2×D236C23semidirect
59. C2×Dic118C23semidirect
67. C4×Dic59C23semidirect
69. #11745664C23semidirect
132. C2×H(5,𝔽2)D59semidirect
139. C59#3019semidirect
170. #41C22⋊D59semidirect
173. C2×Q8C22⋊D59semidirect
186. C2×D4C22⋊D59semidirect