Finite Group Databasebeta

← C2×#126×D127 #30651984

Subgroups — C2×#126×D127 #30651984

Order 16256.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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×#16861204C213yesmaximal, complemented
2C2×#16861203C211yesmaximal, characteristic, complemented
3C2×#16861202C213yesmaximal, complemented
4#126×D127C2116yesmaximal, direct product factor
5C2×C4⋊C4×D127C213yesmaximal, complemented
6C2×C22⋊C4×D127C213yesmaximal
7C2×C42×D127C211yesmaximal, characteristic, complemented
8C2×#126×C127C211yesmaximal, characteristic, complemented
9C24×D127C2211yescharacteristic
10C22×C4×D127C2213yes—
11C22×C4×D127C2213yes—
12#16861204C22124yescomplemented
13#16861203C2218yescomplemented
14#16861202C22124yescomplemented
15C2×C4⋊Dic127C2213yescomplemented
16C2×Dic127⋊C4C2213yescomplemented
17C2×Dic127⋊C4C2213yescomplemented
18C4⋊C4×D127C22124yescomplemented
19C2×C22⋊Dic127C2213yes—
20C2×(C2×D127)⋊C4C2213yes—
21C2×(C2×D127)⋊C4C2213yes—
22C22⋊C4×D127C22124yes—
23C2×C4×Dic127C2213yescomplemented
24C42×D127C2218yescomplemented
25#126×C127C2218yescomplemented
26C2×C4⋊C4×C127C2213yescomplemented
27C2×C22⋊C4×C127C2213yes—
28C2×C42×C127C2211yescharacteristic, complemented
29C23×D127—212no—
30C23×D127C2314yes—
31C23×D127C2311yescharacteristic
32C23×D127C2311yescharacteristic
33C22×Dic127C2313yes—
34C22×Dic127C2313yes—
35C2×C4×D127—212no—
36C2×C4×D127—212no—
37C2×C4×D127C23112yes—
38C2×C4×D127C23112yes—
39C4⋊Dic127C23112yescomplemented
40Dic127⋊C4C23112yescomplemented
41Dic127⋊C4C23112yescomplemented
42C22⋊Dic127C23112yes—
43(C2×D127)⋊C4C23112yes—
44(C2×D127)⋊C4C23112yes—
45C4×Dic127C23112yescomplemented
46C24×C127C2311yescharacteristic
47C22×C4×C127C2313yes—
48C22×C4×C127C2313yes—
49C4⋊C4×C127C23112yescomplemented
50C22⋊C4×C127C23112yes—
51C42×C127C2314yescomplemented
52C22×D127—26no—
53C22×D127—224no—
54C22×D127—416no—
55C22×D127#41112yes—
56C22×D127C2412yes—
57C22×D127C2412yes—
58C2×Dic127—26no—
59C2×Dic127—26no—
60C2×Dic127C2416yes—
61C2×Dic127C2416yes—
62C4×D127—224no—
63C4×D127—224no—
64C23×C127—26no—
65C23×C127C2412yes—
66C23×C127C2411yescharacteristic
67C2×C4×C127—26no—
68C2×C4×C127—26no—
69C2×C4×C127C2416yes—
70C2×C4×C127C2416yes—
71C22×C127—26no—
72C22×C127—44no—
73C22×C127C2×#4116yes—
74C22×C127C2511yescharacteristic
75C2×D127—26no—
76C2×D127—44no—
77C2×D127—416no—
78C2×D127#126116yesdirect product factor
79C2×D127C2×#41112yes—
80C4×C127—26no—
81C4×C127—26no—
82Dic127—26no—
83Dic127—26no—
84C2×C127—42no—
85C2×C127C2×#12614yescomplemented
86C2×C127C22×#4113yes—
87D127—42no—
88D127C2×#12618yesdirect product factor
89C22×#126—1271nomaximal, Sylow
90C127C22×#12611yescharacteristic, complemented, Sylow
91C2×#126—12716no—
92C2×#126—1273no—
93C2×#126—1273no—
94C2×#126—1271no—
95C2×#126D12711yescharacteristic, direct product factor
96C22×C4⋊C4—1273no—
97C22×C22⋊C4—1273no—
98C22×C42—1271no—
99C25—1271no—
100C23×C4—1273no—
101C23×C4—1273no—
102#126—12724no—
103#126—12724no—
104#126—1278no—
105#126C2×D12718yesdirect product factor
106C2×C4⋊C4—1273no—
107C2×C4⋊C4—1273no—
108C2×C4⋊C4—1273no—
109C2×C4⋊C4—12724no—
110C2×C4⋊C4C2×D12713yescomplemented
111C2×C22⋊C4—1273no—
112C2×C22⋊C4—1273no—
113C2×C22⋊C4—1273no—
114C2×C22⋊C4—12724no—
115C2×C22⋊C4C2×D12713yes—
116C2×C42—1273no—
117C2×C42—1278no—
118C2×C42C2×D12711yescharacteristic, complemented
119C24—1271no—
120C24—1274no—
121C24—1271no—
122C24—25412no—
123C24C22×D12711yescharacteristic
124C22×C4—1273no—
125C22×C4—1273no—
126C22×C4—12712no—
127C22×C4—12712no—
128C22×C4—25412no—
129C22×C4—25412no—
130C22×C4C22×D12713yes—
131C22×C4C22×D12713yes—
132C4⋊C4—12712no—
133C4⋊C4—12712no—
134C4⋊C4—12712no—
135C4⋊C4C22×D127112yescomplemented
136C22⋊C4—12712no—
137C22⋊C4—12712no—
138C22⋊C4—12712no—
139C22⋊C4C22×D127112yes—
140C42—12712no—
141C42C22×D12714yescomplemented
142C23—26no—
143C23—12712no—
144C23—1272no—
145C23—1272no—
146C23—50816no—
147C23—25424no—
148C23—2546no—
149C23C23×D12712yes—
150C23C23×D12711yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—1276no—
154C2×C4—1276no—
155C2×C4—25424no—
156C2×C4—25424no—
157C2×C4—2546no—
158C2×C4—2546no—
159C2×C4C23×D12716yes—
160C2×C4C23×D12716yes—
161C22—26no—
162C22—44no—
163C22—12716no—
164C22—12712no—
165C22—50816no—
166C22—5084no—
167C22—2546no—
168C22#41×D12716yes—
169C22C24×D12711yescharacteristic
170C4—26no—
171C4—26no—
172C4—2546no—
173C4—2546no—
174C2—42no—
175C2—1278no—
176C2—5082no—
177C2#126×D12714yesdirect product factor
178C2C2×#41×D12713yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#126×C127C2
socleC23×C127C24
prime radicalC24×D127C22
hypercenterC2×#126D127
hypocenterC127C22×#126
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16861204C2semidirect
2. C2×#16861203C2semidirect
3. C2×#16861202C2semidirect
4. #126×D127C2direct (N × H)
5. C2×C4⋊C4×D127C2semidirect
7. C2×C42×D127C2semidirect
8. C2×#126×C127C2semidirect
12. #16861204C22semidirect
13. #16861203C22semidirect
14. #16861202C22semidirect
15. C2×C4⋊Dic127C22semidirect
16. C2×Dic127⋊C4C22semidirect
17. C2×Dic127⋊C4C22semidirect
18. C4⋊C4×D127C22semidirect
23. C2×C4×Dic127C22semidirect
24. C42×D127C22semidirect
25. #126×C127C22semidirect
26. C2×C4⋊C4×C127C22semidirect
28. C2×C42×C127C22semidirect
39. C4⋊Dic127C23semidirect
40. Dic127⋊C4C23semidirect
41. Dic127⋊C4C23semidirect
45. C4×Dic127C23semidirect
49. C4⋊C4×C127C23semidirect
51. C42×C127C23semidirect
78. C2×D127#126direct (N × H)
85. C2×C127C2×#126semidirect
88. D127C2×#126direct (N × H)
90. C127C22×#126semidirect
95. C2×#126D127direct (N × H)
105. #126C2×D127direct (N × H)
110. C2×C4⋊C4C2×D127semidirect
118. C2×C42C2×D127semidirect
135. C4⋊C4C22×D127semidirect
141. C42C22×D127semidirect
177. C2#126×D127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612042
C2×#168612032
C2×#168612022
#126×D1272
C2×C4⋊C4×D1272
C2×C22⋊C4×D1272
C2×C42×D1272
C2×#126×C1272
C22×#126127