Finite Group Databasebeta

← #30653509

Subgroups — #30653509

Order 16256.

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
1C2×#16861255C211yesmaximal, characteristic, complemented
2#22550684C218yesmaximal, complemented
3#22550679C218yesmaximal, complemented
4#22550674C216yesmaximal, complemented
5C2×#16861234C216yesmaximal, complemented
6#22550643C211yesmaximal, characteristic, complemented
7#585×C127C211yesmaximal, characteristic, complemented
8#16861255—24no—
9#16861255C2218yescomplemented
10C2×#14523249—21no—
11C2×#14523248C2213yescomplemented
12C22×D508C2213yescomplemented
13C2×D4×D127—23no—
14#16861240C22112yescomplemented
15#16861239C2214yescomplemented
16#16861235C22112yescomplemented
17#16861234C22112yescomplemented
18#16861234C22124yescomplemented
19#16861229C22124yescomplemented
20C2×#14523235C22112yescomplemented
21C2×#14523233C22112yescomplemented
22#16861216C2218yescomplemented
23C2×#14523232C2216yescomplemented
242−1+4×C127—21no—
252−1+4×C127C2214yescomplemented
26H(5,𝔽2)×C127—23no—
27H(5,𝔽2)×C127C2214yescomplemented
28C2×#41×C127—24no—
29C2×#41×C127C2211yescharacteristic
30C2×#41×C127C2216yescomplemented
31C23×D127—23no—
32#14523249—28no—
33#14523248—26no—
34#14523248C23112yes—
35C2×Dic254C2311yescharacteristic
36C2×D508—23no—
37C2×D508—26no—
38C2×D508C23112yes—
39C2×D508C2313yes—
40C2×C22⋊D127—23no—
41D4×D127—224no—
42C2×C4×D127—23no—
43#14523238C2318yescomplemented
44#14523236C2318yescomplemented
45#14523235C23124yescomplemented
46#14523233C23124yescomplemented
47#14523232C2314yes—
48#14523232C23112yescomplemented
49C2×C127⋊C8C23112yescomplemented
50#41×C127—24no—
51#41×C127—212no—
52#41×C127—26no—
53#41×C127—26no—
54#41×C127D414yes—
55#41×C127C2314yes—
56#41×C127C2314yes—
57#41×C127C23112yescomplemented
58C2×Q8×C127—24no—
59C2×Q8×C127D411yescharacteristic
60C2×Q8×C127C2316yescomplemented
61C2×D4×C127—26no—
62C2×D4×C127—212no—
63C2×D4×C127D413yes—
64C2×D4×C127C2316yescomplemented
65C22×C4×C127—26no—
66C22×C4×C127C2313yes—
67C22×D127—23no—
68C22×D127—212no—
69C22×D127—86no—
70Dic254—21no—
71Dic254C2412yes—
72D508—212no—
73D508—23no—
74D508C2416yes—
75C22⋊D127—212no—
76C2×Dic127—21no—
77C4×D127—212no—
78C127⋊C8C2418yes—
79C23×C127—26no—
80C23×C127C2×D413yes—
81Q8×C127—26no—
82Q8×C127C2×D414yes—
83Q8×C127C2414yes—
84D4×C127—24no—
85D4×C127—212no—
86D4×C127—26no—
87D4×C127C2×D4112yes—
88D4×C127C2414yes—
89C2×C4×C127—26no—
90C2×C4×C127—24no—
91C2×C4×C127—212no—
92C2×C4×C127C2×D416yes—
93C2×C4×C127C2×D413yes—
94C2×C4×C127C2411yescharacteristic
95C2×C4×C127C2416yes—
96C22×C127—24no—
97C22×C127—43no—
98C22×C127—86no—
99C22×C127C22×D416yes—
100C22×C127C22×D411yescharacteristic
101C2×D127—26no—
102C2×D127—46no—
103C2×D127—812no—
104C4×C127—24no—
105C4×C127C22×D411yescharacteristic
106C4×C127C22×D416yes—
107C4×C127C2511yescharacteristic
108Dic127—22no—
109C2×C127—21no—
110C2×C127—26no—
111C2×C127—44no—
112C2×C127C23×D411yescharacteristic
113D127—46no—
114#3585—1271nomaximal, Sylow
115C127#358511yescharacteristic, complemented, Sylow
116#585D12711yescharacteristic, complemented
117C2×H(5,𝔽2)—1271no—
118#577—1278no—
119#576—1278no—
120#575—1276no—
121C2×AGL(1,ℤ/8)—1276no—
122#568—1271no—
1232−1+4—21no—
1242−1+4C2×D12714yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—1278no—
127H(5,𝔽2)—2544no—
128H(5,𝔽2)C2×D12714yescomplemented
129C2×#41—24no—
130C2×#41—1273no—
131C2×#41—2541no—
132C2×#41C2×D12711yescharacteristic
133C2×#41C2×D12716yescomplemented
134C22×D4—1273no—
135C22×D4—2543no—
136#137—1274no—
137#137—12712no—
138AGL(1,ℤ/8)—12712no—
139AGL(1,ℤ/8)—12724no—
140AGL(1,ℤ/8)—12712no—
141#135—12724no—
142C2×SD16—12712no—
143C2×D8—12712no—
144#131—1278no—
145C2×M16—1276no—
146C24—2543no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—12712no—
152#41—2548no—
153#41—2546no—
154#41C22⋊D12714yes—
155#41C22×D12714yes—
156#41C22×D12714yes—
157#41C22×D127112yescomplemented
158C2×Q8—24no—
159C2×Q8—1271no—
160C2×Q8C22⋊D12711yescharacteristic
161C2×Q8C22×D12716yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—12712no—
165C2×D4—1273no—
166C2×D4—2543no—
167C2×D4—25424no—
168C2×D4—2543no—
169C2×D4—2546no—
170C2×D4C22⋊D12713yes—
171C2×D4C22×D12716yescomplemented
172C22×C4—26no—
173C22×C4—2543no—
174C22×C4C22×D12713yes—
175Q16—1278no—
176SD16—12724no—
177SD16—1278no—
178D8—12724no—
179M16—1274no—
180M16—12712no—
181C2×C8—12712no—
182C23—26no—
183C23—10166no—
184C23—25412no—
185C23—2543no—
186C23C2×C22⋊D12713yes—
187Q8—26no—
188Q8—1272no—
189Q8—2541no—
190Q8C2×C22⋊D12714yes—
191Q8C23×D12714yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—1276no—
196D4—25412no—
197D4—2543no—
198D4—25412no—
199D4C2×C22⋊D127112yes—
200D4C23×D12714yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—2541no—
205C2×C4—25412no—
206C2×C4C2×C22⋊D12716yes—
207C2×C4C2×C22⋊D12713yes—
208C2×C4C23×D12716yes—
209C2×C4C23×D12711yescharacteristic
210C8—1278no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—101612no—
215C22—5086no—
216C22—2546no—
217C22C22×C22⋊D12716yes—
218C22C22×C22⋊D12711yescharacteristic
219C4—24no—
220C4—2542no—
221C4C22×C22⋊D12711yescharacteristic
222C4C22×C22⋊D12716yes—
223C4C24×D12711yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—5086no—
228C2C23×C22⋊D12711yescharacteristic
subgroupG/H
centerC2C23×C22⋊D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
Fitting#585×C127C2
socleC2×C127C23×D4
hypercenter#585D127
hypocenterC127#3585
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. #22550684C2semidirect
3. #22550679C2semidirect
4. #22550674C2semidirect
5. C2×#16861234C2semidirect
6. #22550643C2semidirect
7. #585×C127C2semidirect
9. #16861255C22semidirect
11. C2×#14523248C22semidirect
12. C22×D508C22semidirect
14. #16861240C22semidirect
15. #16861239C22semidirect
16. #16861235C22semidirect
17. #16861234C22semidirect
18. #16861234C22semidirect
19. #16861229C22semidirect
20. C2×#14523235C22semidirect
21. C2×#14523233C22semidirect
22. #16861216C22semidirect
23. C2×#14523232C22semidirect
25. 2−1+4×C127C22semidirect
27. H(5,𝔽2)×C127C22semidirect
30. C2×#41×C127C22semidirect
43. #14523238C23semidirect
44. #14523236C23semidirect
45. #14523235C23semidirect
46. #14523233C23semidirect
48. #14523232C23semidirect
49. C2×C127⋊C8C23semidirect
57. #41×C127C23semidirect
60. C2×Q8×C127C23semidirect
64. C2×D4×C127C23semidirect
115. C127#3585semidirect
116. #585D127semidirect
124. 2−1+4C2×D127semidirect
128. H(5,𝔽2)C2×D127semidirect
133. C2×#41C2×D127semidirect
157. #41C22×D127semidirect
161. C2×Q8C22×D127semidirect
171. C2×D4C22×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168612552
#225506842
#225506792
#225506742
C2×#168612342
#225506432
#585×C1272
#3585127