Finite Group Databasebeta

← #21438877

Subgroups — #21438877

Order 7552.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#16838238C211yesmaximal, characteristic, complemented
2C2×#14273773C211yesmaximal, characteristic, complemented
3#16838214C218yesmaximal, complemented
4#16838207C218yesmaximal, complemented
5#16838202C216yesmaximal, complemented
6C2×#14273755C216yesmaximal, complemented
7C59×#568C211yesmaximal, characteristic, complemented
8#14273775—21no—
9#14273774C2214yescomplemented
10#14273773—24no—
11#14273773C2218yescomplemented
12#14273773C2214yescomplemented
13#14273772—23no—
14C2×#11745689—21no—
15C2×#11745685C2213yescomplemented
16C2×#11745685C2216yescomplemented
17#41×D59—24no—
18C22×D236C2213yescomplemented
19C2×D4×D59—23no—
20#14273760C2214yescomplemented
21#14273760C22112yescomplemented
22#14273755C22112yescomplemented
23#14273755C22124yescomplemented
24#14273755C22112yescomplemented
25#14273749C22124yescomplemented
26C2×#11745660C22112yescomplemented
27C2×D472C22112yescomplemented
28C2×#41×C59C2211yescharacteristic
29#131×C59C2218yescomplemented
30C2×M16×C59C2216yescomplemented
31C23×D59—23no—
32#41×C59D414yes—
33#41×C59C2314yes—
34C2×Q8×C59D411yescharacteristic
35C2×D4×C59D413yes—
36C22×C4×C59C2313yes—
37C2×C22⋊D59—23no—
38C2×C22⋊D59—26no—
39#11745689—24no—
40#11745689—28no—
41Q8×D59—24no—
42#11745687—212no—
43D4×D59—224no—
44D4×D59—212no—
45#11745685—26no—
46#11745685—26no—
47#11745685—26no—
48#11745685C23112yes—
49#11745685C2314yes—
50#11745685C23112yescomplemented
51C2×D236—23no—
52C2×D236—26no—
53C2×D236C23112yes—
54C2×D236C2313yes—
55C2×D236C2316yescomplemented
56C2×C4×D59—23no—
57C2×C4×D59—26no—
58C2×Dic118C2311yescharacteristic
59C2×Dic118C2316yescomplemented
60M16×C59C2314yes—
61M16×C59C23112yescomplemented
62C2×C8×C59C23112yescomplemented
63Dic236C2318yescomplemented
64D472C23124yescomplemented
65#11745660C23124yescomplemented
66#11745660C2318yescomplemented
67C23×C59C2×D413yes—
68C22×D59—23no—
69C22×D59—26no—
70C22×D59—212no—
71C22×D59—86no—
72Q8×C59C2×D414yes—
73D4×C59C2×D4112yes—
74C2×C4×C59C2×D413yes—
75C2×C4×C59C2×D416yes—
76C2×C4×C59C2416yes—
77C2×C4×C59C2411yescharacteristic
78C22⋊D59—24no—
79C22⋊D59—212no—
80C22⋊D59—212no—
81C2×Dic59—21no—
82C2×Dic59—26no—
83D236—23no—
84D236—212no—
85D236—26no—
86D236C2416yes—
87D236C2414yes—
88C4×D59—24no—
89C4×D59—212no—
90C4×D59—212no—
91Dic118—21no—
92Dic118—26no—
93Dic118C2414yes—
94Dic118C2412yes—
95C8×C59C2418yes—
96C22×C59—43no—
97C22×C59C22×D411yescharacteristic
98C22×C59C22×D416yes—
99C2×D59—24no—
100C2×D59—26no—
101C2×D59—46no—
102C2×D59—86no—
103C2×D59—812no—
104C4×C59C22×D411yescharacteristic
105C4×C59C22×D416yes—
106C4×C59C2511yescharacteristic
107Dic59—24no—
108Dic59—22no—
109#3585—591nomaximal, Sylow
110C2×C59—21no—
111C2×C59—26no—
112C2×C59C23×D411yescharacteristic
113D59—46no—
114D59—44no—
115#585—591no—
116C2×H(5,𝔽2)—591no—
117#577—598no—
118#576—598no—
119#575—596no—
120C2×AGL(1,ℤ/8)—596no—
121#568D5911yescharacteristic, complemented
122C59#358511yescharacteristic, complemented, Sylow
1232−1+4—594no—
1242−1+4—1181no—
125H(5,𝔽2)—594no—
126H(5,𝔽2)—598no—
127H(5,𝔽2)—1184no—
128H(5,𝔽2)—1183no—
129C2×#41—596no—
130C2×#41—593no—
131C2×#41—1184no—
132C2×#41—1181no—
133C2×#41C2×D5911yescharacteristic
134C22×D4—593no—
135C22×D4—1183no—
136#137—594no—
137#137—5912no—
138AGL(1,ℤ/8)—5912no—
139AGL(1,ℤ/8)—5924no—
140AGL(1,ℤ/8)—5912no—
141#135—5924no—
142C2×SD16—5912no—
143C2×D8—5912no—
144#131C2×D5918yescomplemented
145C2×M16C2×D5916yescomplemented
146C24—1183no—
147#41—5912no—
148#41—5912no—
149#41—594no—
150#41—1188no—
151#41—1184no—
152#41—11812no—
153#41—1186no—
154#41—1186no—
155#41—1186no—
156#41D23614yes—
157#41C22×D5914yes—
158C2×Q8—596no—
159C2×Q8—591no—
160C2×Q8—1184no—
161C2×Q8D23611yescharacteristic
162C2×D4—596no—
163C2×D4—5912no—
164C2×D4—593no—
165C2×D4—1183no—
166C2×D4—1186no—
167C2×D4—11824no—
168C2×D4—11812no—
169C2×D4—1183no—
170C2×D4—1186no—
171C2×D4D23613yes—
172C22×C4—1183no—
173C22×C4—1186no—
174C22×C4C22×D5913yes—
175Q16—598no—
176SD16—5924no—
177SD16—598no—
178D8—5924no—
179M16C22×D5914yes—
180M16C22×D59112yescomplemented
181C2×C8C22×D59112yescomplemented
182C23—1186no—
183C23—11812no—
184C23—1183no—
185C23—4726no—
186C23C2×D23613yes—
187Q8—594no—
188Q8—592no—
189Q8—1186no—
190Q8—1181no—
191Q8C2×D23614yes—
192D4—594no—
193D4—596no—
194D4—11812no—
195D4—1184no—
196D4—11812no—
197D4—1183no—
198D4—11812no—
199D4—1186no—
200D4C2×D236112yes—
201C2×C4—1181no—
202C2×C4—1186no—
203C2×C4—1184no—
204C2×C4—11812no—
205C2×C4—11812no—
206C2×C4C2×D23613yes—
207C2×C4C2×D23616yes—
208C2×C4C23×D5916yes—
209C2×C4C23×D5911yescharacteristic
210C8C23×D5918yes—
211C22—43no—
212C22—1184no—
213C22—1186no—
214C22—47212no—
215C22—4726no—
216C22—2366no—
217C22C22×D23611yescharacteristic
218C22C22×D23616yes—
219C4—1182no—
220C4—1184no—
221C4C22×D23611yescharacteristic
222C4C22×D23616yes—
223C4C24×D5911yescharacteristic
224C2—21no—
225C2—26no—
226C2—2366no—
227C2—2364no—
228C2C23×D23611yescharacteristic
subgroupG/H
centerC2C23×D236
commutatorC4×C59C25
FrattiniC4C24×D59
FittingC59×#568C2
socleC2×C59C23×D4
hypercenter#568D59
hypocenterC59#3585
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #16838238C2semidirect
2. C2×#14273773C2semidirect
3. #16838214C2semidirect
4. #16838207C2semidirect
5. #16838202C2semidirect
6. C2×#14273755C2semidirect
7. C59×#568C2semidirect
9. #14273774C22semidirect
11. #14273773C22semidirect
12. #14273773C22semidirect
15. C2×#11745685C22semidirect
16. C2×#11745685C22semidirect
18. C22×D236C22semidirect
20. #14273760C22semidirect
21. #14273760C22semidirect
22. #14273755C22semidirect
23. #14273755C22semidirect
24. #14273755C22semidirect
25. #14273749C22semidirect
26. C2×#11745660C22semidirect
27. C2×D472C22semidirect
29. #131×C59C22semidirect
30. C2×M16×C59C22semidirect
50. #11745685C23semidirect
55. C2×D236C23semidirect
59. C2×Dic118C23semidirect
61. M16×C59C23semidirect
62. C2×C8×C59C23semidirect
63. Dic236C23semidirect
64. D472C23semidirect
65. #11745660C23semidirect
66. #11745660C23semidirect
121. #568D59semidirect
122. C59#3585semidirect
144. #131C2×D59semidirect
145. C2×M16C2×D59semidirect
180. M16C22×D59semidirect
181. C2×C8C22×D59semidirect