Finite Group Databasebeta

← C22×#16861237 #30653465

Subgroups — C22×#16861237 #30653465

Order 16256.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C22×#14523248C211yesmaximal, characteristic, complemented
2C23×D508C211yesmaximal, characteristic, complemented
3C2×#16861237C2124yesmaximal, direct product factor
4C22×#14523237C212yesmaximal, complemented
5C22×D1016C212yesmaximal, complemented
6C22×M16×C127C211yesmaximal, characteristic, complemented
7C24×D127—21no—
8C2×#14523248—26no—
9C2×#14523248C22112yescomplemented
10C22×Dic254C2211yescharacteristic, complemented
11C22×D508—21no—
12C22×D508—26no—
13C22×D508C2212yes—
14C22×D508C22112yescomplemented
15C22×D508C2211yescharacteristic, complemented
16C22×C22⋊D127—21no—
17C22×C4×D127—21no—
18#16861237C22164yesdirect product factor
19C2×#14523237C22124yescomplemented
20C2×D1016C22124yescomplemented
21C23×C4×C127C2211yescharacteristic
22C2×M16×C127C22112yescomplemented
23C22×C8×C127C2212yescomplemented
24C23×D127—21no—
25C23×D127—22no—
26C23×D127—212no—
27C23×D127—48no—
28#14523248—224no—
29#14523248C23116yescomplemented
30C2×Dic254—23no—
31C2×Dic254C2316yescomplemented
32C2×D508—26no—
33C2×D508—212no—
34C2×D508—224no—
35C2×D508—23no—
36C2×D508C23112yes—
37C2×D508C23116yescomplemented
38C2×D508C2316yescomplemented
39C2×C22⋊D127—212no—
40C22×Dic127—21no—
41C2×C4×D127—212no—
42#14523237C23132yescomplemented
43D1016C23132yescomplemented
44C24×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D416yes—
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2316yes—
49M16×C127C23116yescomplemented
50C2×C8×C127C23112yescomplemented
51C22×D127—26no—
52C22×D127—212no—
53C22×D127—216no—
54C22×D127—448no—
55C22×D127—44no—
56C22×D127—48no—
57Dic254—26no—
58Dic254C2414yescomplemented
59D508—212no—
60D508—216no—
61D508—26no—
62D508C2418yes—
63D508C2414yescomplemented
64C22⋊D127—216no—
65C2×Dic127—26no—
66C4×D127—216no—
67C23×C127—24no—
68C23×C127C2×D416yes—
69C23×C127C2×D411yescharacteristic
70C2×C4×C127C2×D416yes—
71C2×C4×C127C2×D413yes—
72C2×C4×C127C2×D4112yes—
73C2×C4×C127C2413yes—
74C2×C4×C127C2414yes—
75C8×C127C2418yescomplemented
76C22×C127—212no—
77C22×C127AGL(1,ℤ/8)14yescomplemented
78C22×C127C22×D414yes—
79C22×C127C22×D413yes—
80C2×D127—24no—
81C2×D127—28no—
82C2×D127—432no—
83C2×D127—412no—
84C2×D127—424no—
85C4×C127C22×D414yes—
86C4×C127C22×D413yes—
87C4×C127C2511yescharacteristic
88Dic127—24no—
89C2×C127—24no—
90C2×C127C2×AGL(1,ℤ/8)16yescomplemented
91C2×C127C23×D411yescharacteristic
92D127—44no—
93D127—48no—
94C22×AGL(1,ℤ/8)—1271nomaximal, Sylow
95C127C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—1271no—
97C23×D4—1271no—
98C2×AGL(1,ℤ/8)—12724no—
99C22×SD16—1272no—
100C22×D8—1272no—
101C22×M16D12711yescharacteristic, complemented
102C25—2541no—
103C2×#41—12712no—
104C2×#41—2546no—
105C22×Q8—1271no—
106C22×D4—1271no—
107C22×D4—1272no—
108C22×D4—12712no—
109C22×D4—2541no—
110C22×D4—2546no—
111C22×D4—2541no—
112C23×C4—2541no—
113C23×C4C2×D12711yescharacteristic
114AGL(1,ℤ/8)—12764no—
115C2×SD16—12724no—
116C2×D8—12724no—
117C2×M16C2×D127112yescomplemented
118C22×C8C2×D12712yescomplemented
119C24—5088no—
120C24—25412no—
121C24—2541no—
122C24—2542no—
123C24D50811yescharacteristic
124#41—12716no—
125#41—25424no—
126C2×Q8—1276no—
127C2×Q8—2543no—
128C2×D4—12712no—
129C2×D4—1276no—
130C2×D4—12716no—
131C2×D4—25412no—
132C2×D4—25412no—
133C2×D4—2546no—
134C2×D4—25424no—
135C2×D4—2543no—
136C22×C4—2541no—
137C22×C4—25412no—
138C22×C4D50816yes—
139C22×C4D50811yescharacteristic
140C22×C4C22×D12716yes—
141C22×C4C22×D12711yescharacteristic
142SD16—12732no—
143D8—12732no—
144M16C22×D127116yescomplemented
145C2×C8C22×D127112yescomplemented
146C23—24no—
147C23—50848no—
148C23—5088no—
149C23—5084no—
150C23—2546no—
151C23—25416no—
152C23—25412no—
153C23C2×D50816yes—
154C23C2×D50811yescharacteristic
155Q8—1274no—
156Q8—2546no—
157D4—1274no—
158D4—1278no—
159D4—25416no—
160D4—25416no—
161D4—25412no—
162D4—2546no—
163C2×C4—2546no—
164C2×C4—25416no—
165C2×C4C2×D50816yes—
166C2×C4C2×D50813yes—
167C2×C4C2×D508112yes—
168C2×C4C23×D12713yes—
169C2×C4C23×D12714yes—
170C8C23×D12718yescomplemented
171C22—212no—
172C22—50832no—
173C22—50824no—
174C22—50812no—
175C22—2544no—
176C22—2548no—
177C22#1686123714yesdirect product factor
178C22C22×D50814yes—
179C22C22×D50813yes—
180C4—2544no—
181C4C22×D50814yes—
182C4C22×D50813yes—
183C4C24×D12711yescharacteristic
184C2—24no—
185C2—5088no—
186C2—5084no—
187C2C2×#1686123716yesdirect product factor
188C2C23×D50811yescharacteristic
subgroupG/H
centerC23C2×D508
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
FittingC22×M16×C127C2
socleC23×C127C2×D4
hypercenterC22×M16D127
hypocenterC127C22×AGL(1,ℤ/8)
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C23×D508C2semidirect
3. C2×#16861237C2direct (N × H)
4. C22×#14523237C2semidirect
5. C22×D1016C2semidirect
6. C22×M16×C127C2semidirect
9. C2×#14523248C22semidirect
10. C22×Dic254C22semidirect
14. C22×D508C22semidirect
15. C22×D508C22semidirect
18. #16861237C22direct (N × H)
19. C2×#14523237C22semidirect
20. C2×D1016C22semidirect
22. C2×M16×C127C22semidirect
23. C22×C8×C127C22semidirect
29. #14523248C23semidirect
31. C2×Dic254C23semidirect
37. C2×D508C23semidirect
38. C2×D508C23semidirect
42. #14523237C23semidirect
43. D1016C23semidirect
49. M16×C127C23semidirect
50. C2×C8×C127C23semidirect
58. Dic254C24semidirect
63. D508C24semidirect
75. C8×C127C24semidirect
77. C22×C127AGL(1,ℤ/8)semidirect
90. C2×C127C2×AGL(1,ℤ/8)semidirect
95. C127C22×AGL(1,ℤ/8)semidirect
101. C22×M16D127semidirect
117. C2×M16C2×D127semidirect
118. C22×C8C2×D127semidirect
144. M16C22×D127semidirect
145. C2×C8C22×D127semidirect
170. C8C23×D127semidirect
177. C22#16861237direct (N × H)
187. C2C2×#16861237direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232482
C23×D5082
C2×#168612372
C22×#145232372
C22×D10162
C22×M16×C1272
C22×AGL(1,ℤ/8)127