Finite Group Databasebeta

← C2×#22550681 #30653483

Subgroups — C2×#22550681 #30653483

Order 16256.

1512 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 549 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×#16861255C212yesmaximal, complemented
2#22550681C2116yesmaximal, direct product factor
3C2×#16861237C216yesmaximal, complemented
4C2×#16861231C213yesmaximal, complemented
5C22×D1016C213yesmaximal, complemented
6C2×#131×C127C211yesmaximal, characteristic, complemented
7#16861255—28no—
8#16861255C22116yescomplemented
9C2×#14523249—22no—
10C2×#14523248C2216yescomplemented
11C22×D508C2216yescomplemented
12C2×D4×D127—26no—
13#16861237C22148yescomplemented
14#16861231C22124yescomplemented
15C2×Dic508C2211yescharacteristic, complemented
16C2×#14523237C2216yescomplemented
17C2×D1016C2213yescomplemented
18C2×D1016C2216yescomplemented
19C2×D1016C22124yescomplemented
20C2×#41×C127C2211yescharacteristic
21#131×C127C2218yescomplemented
22C2×M16×C127C2213yescomplemented
23C22×C8×C127C2213yescomplemented
24C23×D127—26no—
25#14523249—216no—
26#14523248—212no—
27#14523248C23124yescomplemented
28C2×Dic254C2312yes—
29C2×D508—26no—
30C2×D508—212no—
31C2×D508C2316yes—
32C2×D508C23124yescomplemented
33C2×C22⋊D127—26no—
34D4×D127—248no—
35C2×C4×D127—26no—
36Dic508C2314yescomplemented
37#14523237C23124yescomplemented
38D1016C23112yescomplemented
39D1016C23124yescomplemented
40#41×C127D414yes—
41#41×C127C2314yes—
42C2×Q8×C127D411yescharacteristic
43C2×D4×C127D413yes—
44C22×C4×C127C2313yes—
45M16×C127C23112yescomplemented
46C2×C8×C127C2311yescharacteristic
47C2×C8×C127C2313yes—
48C2×C8×C127C23112yescomplemented
49C22×D127—26no—
50C22×D127—224no—
51C22×D127—812no—
52Dic254—22no—
53Dic254C2414yes—
54D508—224no—
55D508—26no—
56D508C24112yes—
57C22⋊D127—224no—
58C2×Dic127—22no—
59C4×D127—224no—
60C23×C127C2×D413yes—
61Q8×C127C2×D414yes—
62D4×C127C2×D4112yes—
63C2×C4×C127C2×D413yes—
64C2×C4×C127C2×D416yes—
65C2×C4×C127C2411yescharacteristic
66C2×C4×C127C2416yes—
67C8×C127C2412yes—
68C8×C127C2416yes—
69C22×C127—26no—
70C22×C127C22×D416yes—
71C22×C127C22×D411yescharacteristic
72C2×D127—212no—
73C2×D127—412no—
74C2×D127—824no—
75C4×C127C22×D411yescharacteristic
76C4×C127C22×D416yes—
77C4×C127C2511yescharacteristic
78Dic127—24no—
79C2×C127—26no—
80C2×C127#57612yescomplemented
81C2×C127C23×D411yescharacteristic
82D127—412no—
83C2×#576—1271nomaximal, Sylow
84C127C2×#57611yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—1272no—
86#576—12716no—
87C2×AGL(1,ℤ/8)—1276no—
88C2×#135—1273no—
89C22×D8—1273no—
90C2×#131D12711yescharacteristic, complemented
91H(5,𝔽2)—12716no—
92H(5,𝔽2)—2548no—
93C2×#41—1276no—
94C2×#41—2542no—
95C2×#41C2×D12711yescharacteristic
96C22×D4—1276no—
97C22×D4—2546no—
98AGL(1,ℤ/8)—12748no—
99#135—12724no—
100C2×Q16—1271no—
101C2×SD16—1276no—
102C2×D8—1273no—
103C2×D8—1276no—
104C2×D8—12724no—
105#131C2×D12718yescomplemented
106C2×M16C2×D12713yescomplemented
107C22×C8C2×D12713yescomplemented
108C24—2546no—
109#41—12724no—
110#41—25416no—
111#41—25412no—
112#41D50814yes—
113#41C22×D12714yes—
114C2×Q8—1272no—
115C2×Q8D50811yescharacteristic
116C2×D4—1276no—
117C2×D4—12724no—
118C2×D4—2546no—
119C2×D4—25448no—
120C2×D4—2546no—
121C2×D4—25412no—
122C2×D4D50813yes—
123C22×C4—2546no—
124C22×C4C22×D12713yes—
125Q16—1274no—
126SD16—12724no—
127D8—12712no—
128D8—12724no—
129M16C22×D127112yescomplemented
130C2×C8C22×D12711yescharacteristic
131C2×C8C22×D12713yes—
132C2×C8C22×D127112yescomplemented
133C23—101612no—
134C23—25424no—
135C23—2546no—
136C23C2×D50813yes—
137Q8—1274no—
138Q8—2542no—
139Q8C2×D50814yes—
140D4—12712no—
141D4—25424no—
142D4—25424no—
143D4—2546no—
144D4C2×D508112yes—
145C2×C4—2542no—
146C2×C4—25424no—
147C2×C4C2×D50816yes—
148C2×C4C2×D50813yes—
149C2×C4C23×D12716yes—
150C2×C4C23×D12711yescharacteristic
151C8C23×D12712yes—
152C8C23×D12716yes—
153C22—26no—
154C22—101624no—
155C22—50812no—
156C22—25412no—
157C22C22×D50816yes—
158C22C22×D50811yescharacteristic
159C4—2544no—
160C4C22×D50816yes—
161C4C22×D50811yescharacteristic
162C4C24×D12711yescharacteristic
163C2—26no—
164C2—50812no—
165C2#2255068112yesdirect product factor
166C2C23×D50811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
FittingC2×#131×C127C2
socleC22×C127C22×D4
hypercenterC2×#131D127
hypocenterC127C2×#576
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. #22550681C2direct (N × H)
3. C2×#16861237C2semidirect
4. C2×#16861231C2semidirect
5. C22×D1016C2semidirect
6. C2×#131×C127C2semidirect
8. #16861255C22semidirect
10. C2×#14523248C22semidirect
11. C22×D508C22semidirect
13. #16861237C22semidirect
14. #16861231C22semidirect
15. C2×Dic508C22semidirect
16. C2×#14523237C22semidirect
17. C2×D1016C22semidirect
18. C2×D1016C22semidirect
19. C2×D1016C22semidirect
21. #131×C127C22semidirect
22. C2×M16×C127C22semidirect
23. C22×C8×C127C22semidirect
27. #14523248C23semidirect
32. C2×D508C23semidirect
36. Dic508C23semidirect
37. #14523237C23semidirect
38. D1016C23semidirect
39. D1016C23semidirect
45. M16×C127C23semidirect
48. C2×C8×C127C23semidirect
80. C2×C127#576semidirect
84. C127C2×#576semidirect
90. C2×#131D127semidirect
105. #131C2×D127semidirect
106. C2×M16C2×D127semidirect
107. C22×C8C2×D127semidirect
129. M16C22×D127semidirect
132. C2×C8C22×D127semidirect
165. C2#22550681direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612552
#225506812
C2×#168612372
C2×#168612312
C22×D10162
C2×#131×C1272
C2×#576127