Finite Group Databasebeta

← C22×#16861235 #30653463

Subgroups — C22×#16861235 #30653463

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
2C2×#16861235C2124yesmaximal, direct product factor
3C22×#14523236C212yesmaximal, complemented
4C22×#14523233C212yesmaximal, complemented
5C22×#14523232C211yesmaximal, characteristic, complemented
6C23×D4×C127C211yesmaximal, characteristic, complemented
7C2×#14523248—26no—
8C2×#14523248C22112yescomplemented
9C22×Dic254C2211yescharacteristic, complemented
10C22×D508C2211yescharacteristic, complemented
11C22×C22⋊D127—21no—
12C22×C4×D127—21no—
13#16861235C22164yesdirect product factor
14C2×#14523236C22124yescomplemented
15C2×#14523233C22124yescomplemented
16C2×#14523232C22112yescomplemented
17C22×C127⋊C8C2212yescomplemented
18C25×C127—21no—
19C22×D4×C127—21no—
20C22×D4×C127—26no—
21C22×D4×C127C2212yes—
22C22×D4×C127C22112yescomplemented
23C23×C4×C127C2211yescharacteristic
24C23×D127—21no—
25#14523248—224no—
26#14523248C23116yescomplemented
27C2×Dic254—23no—
28C2×Dic254C2316yescomplemented
29C2×D508—23no—
30C2×D508C2316yescomplemented
31C2×C22⋊D127—212no—
32C22×Dic127—21no—
33C2×C4×D127—212no—
34#14523236C23132yescomplemented
35#14523233C23132yescomplemented
36#14523232C23116yescomplemented
37C2×C127⋊C8C23112yescomplemented
38C24×C127—22no—
39C24×C127—212no—
40C24×C127—48no—
41C24×C127D411yescharacteristic
42C2×D4×C127—26no—
43C2×D4×C127—212no—
44C2×D4×C127—224no—
45C2×D4×C127C23112yes—
46C2×D4×C127C23116yescomplemented
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D416yes—
49C22×C4×C127C2311yescharacteristic
50C22×C4×C127C2316yes—
51C22×D127—26no—
52C22×D127—44no—
53Dic254—26no—
54Dic254C2414yescomplemented
55D508—26no—
56D508C2414yescomplemented
57C22⋊D127—216no—
58C2×Dic127—26no—
59C4×D127—216no—
60C127⋊C8C2418yescomplemented
61C23×C127—24no—
62C23×C127—212no—
63C23×C127—216no—
64C23×C127—448no—
65C23×C127—48no—
66C23×C127C2×D416yes—
67C23×C127C2×D411yescharacteristic
68D4×C127—212no—
69D4×C127—216no—
70D4×C127C2418yes—
71C2×C4×C127C2×D416yes—
72C2×C4×C127C2×D413yes—
73C2×C4×C127C2×D4112yes—
74C2×C4×C127C2413yes—
75C2×C4×C127C2414yes—
76C22×C127—212no—
77C22×C127—28no—
78C22×C127—432no—
79C22×C127—424no—
80C22×C127AGL(1,ℤ/8)14yescomplemented
81C22×C127C22×D414yes—
82C22×C127C22×D413yes—
83C2×D127—24no—
84C2×D127—412no—
85C4×C127C22×D414yes—
86C4×C127C22×D413yes—
87C4×C127C2511yescharacteristic
88Dic127—24no—
89C2×C127—24no—
90C2×C127—48no—
91C2×C127C2×AGL(1,ℤ/8)16yescomplemented
92C2×C127C23×D411yescharacteristic
93D127—44no—
94C22×AGL(1,ℤ/8)—1271nomaximal, Sylow
95C127C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—1271no—
97C23×D4D12711yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—12724no—
99C22×SD16—1272no—
100C22×D8—1272no—
101C22×M16—1271no—
102C25—21no—
103C2×#41—12712no—
104C2×#41—2546no—
105C22×Q8—1271no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—1271no—
109C22×D4—2541no—
110C22×D4C2×D12712yes—
111C22×D4C2×D127112yescomplemented
112C23×C4—2541no—
113C23×C4C2×D12711yescharacteristic
114AGL(1,ℤ/8)—12764no—
115C2×SD16—12724no—
116C2×D8—12724no—
117C2×M16—12712no—
118C22×C8—1272no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—2541no—
123C24C22⋊D12711yescharacteristic
124#41—12716no—
125#41—25424no—
126C2×Q8—1276no—
127C2×Q8—2543no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—1276no—
132C2×D4—25412no—
133C2×D4—2543no—
134C2×D4C22×D127112yes—
135C2×D4C22×D127116yescomplemented
136C22×C4—2541no—
137C22×C4—25412no—
138C22×C4C22⋊D12716yes—
139C22×C4C22⋊D12711yescharacteristic
140C22×C4C22×D12716yes—
141C22×C4C22×D12711yescharacteristic
142SD16—12732no—
143D8—12732no—
144M16—12716no—
145C2×C8—12712no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—5084no—
152C23—2546no—
153C23C2×C22⋊D12716yes—
154C23C2×C22⋊D12711yescharacteristic
155Q8—1274no—
156Q8—2546no—
157D4—212no—
158D4—216no—
159D4—1274no—
160D4—25416no—
161D4—2546no—
162D4C23×D12718yes—
163C2×C4—2546no—
164C2×C4—25416no—
165C2×C4C2×C22⋊D12716yes—
166C2×C4C2×C22⋊D12713yes—
167C2×C4C2×C22⋊D127112yes—
168C2×C4C23×D12713yes—
169C2×C4C23×D12714yes—
170C8—1278no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—50812no—
176C22—2544no—
177C22#1686123514yesdirect product factor
178C22C22×C22⋊D12714yes—
179C22C22×C22⋊D12713yes—
180C4—2544no—
181C4C22×C22⋊D12714yes—
182C4C22×C22⋊D12713yes—
183C4C24×D12711yescharacteristic
184C2—24no—
185C2—48no—
186C2—5084no—
187C2C2×#1686123516yesdirect product factor
188C2C23×C22⋊D12711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
FittingC23×D4×C127C2
socleC23×C127C2×D4
hypercenterC23×D4D127
hypocenterC127C22×AGL(1,ℤ/8)
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C2×#16861235C2direct (N × H)
3. C22×#14523236C2semidirect
4. C22×#14523233C2semidirect
5. C22×#14523232C2semidirect
6. C23×D4×C127C2semidirect
8. C2×#14523248C22semidirect
9. C22×Dic254C22semidirect
10. C22×D508C22semidirect
13. #16861235C22direct (N × H)
14. C2×#14523236C22semidirect
15. C2×#14523233C22semidirect
16. C2×#14523232C22semidirect
17. C22×C127⋊C8C22semidirect
22. C22×D4×C127C22semidirect
26. #14523248C23semidirect
28. C2×Dic254C23semidirect
30. C2×D508C23semidirect
34. #14523236C23semidirect
35. #14523233C23semidirect
36. #14523232C23semidirect
37. C2×C127⋊C8C23semidirect
46. C2×D4×C127C23semidirect
54. Dic254C24semidirect
56. D508C24semidirect
60. C127⋊C8C24semidirect
80. C22×C127AGL(1,ℤ/8)semidirect
91. C2×C127C2×AGL(1,ℤ/8)semidirect
95. C127C22×AGL(1,ℤ/8)semidirect
97. C23×D4D127semidirect
111. C22×D4C2×D127semidirect
135. C2×D4C22×D127semidirect
177. C22#16861235direct (N × H)
187. C2C2×#16861235direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168612352
C22×#145232362
C22×#145232332
C22×#145232322
C23×D4×C1272
C22×AGL(1,ℤ/8)127