Finite Group Databasebeta

← C22×#16861255 #30653532

Subgroups — C22×#16861255 #30653532

Order 16256.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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×#16861255C2148yesmaximal, direct product factor
2C22×#14523249C212yesmaximal, complemented
3C22×#14523248C213yesmaximal, complemented
4C23×D508C213yesmaximal, complemented
5C22×D4×D127C216yesmaximal, complemented
6C22×#41×C127C211yesmaximal, characteristic, complemented
7C24×D127C2216yescomplemented
8#16861255C221256yesdirect product factor
9C2×#14523249C22148yescomplemented
10C2×#14523248C22172yescomplemented
11C22×Dic254C2211yescharacteristic, complemented
12C22×D508C2216yescomplemented
13C22×D508C2213yescomplemented
14C22×D508C22172yescomplemented
15C22×C22⋊D127C2216yescomplemented
16C2×D4×D127C221144yescomplemented
17C22×C4×D127C2216yescomplemented
18C2×#41×C127C22124yescomplemented
19C22×Q8×C127C2211yescharacteristic, complemented
20C22×D4×C127C2213yescomplemented
21C23×C4×C127C2213yescomplemented
22C23×D127—424no—
23C23×D127C2316yes—
24C23×D127C23172yescomplemented
25#14523249C231128yescomplemented
26#14523248C231192yescomplemented
27C2×Dic254C23112yescomplemented
28C2×D508C23172yescomplemented
29C2×D508C23136yescomplemented
30C2×D508C231192yescomplemented
31C2×C22⋊D127C23172yescomplemented
32D4×D127C231384yescomplemented
33C22×Dic127C2312yes—
34C2×C4×D127C23172yescomplemented
35C24×C127C2313yes—
36#41×C127C23164yescomplemented
37C2×Q8×C127C23112yescomplemented
38C2×D4×C127C23136yescomplemented
39C22×C4×C127C2313yes—
40C22×C4×C127C2311yescharacteristic
41C22×C4×C127C23136yescomplemented
42C22×D127—224no—
43C22×D127—4144no—
44C22×D127C24136yes—
45C22×D127C24196yescomplemented
46Dic254C24116yescomplemented
47D508C24196yescomplemented
48D508C24148yescomplemented
49C22⋊D127C24196yescomplemented
50C2×Dic127C24112yes—
51C4×D127C24196yescomplemented
52C23×C127—212no—
53C23×C127C24118yes—
54C23×C127C2411yescharacteristic
55Q8×C127C24116yescomplemented
56D4×C127C24148yescomplemented
57C2×C4×C127C2416yes—
58C2×C4×C127C24118yes—
59C2×C4×C127C24148yescomplemented
60C22×C127—236no—
61C22×C127H(5,𝔽2)14yescomplemented
62C22×C127C25112yes—
63C22×C127C2513yes—
64C2×D127—272no—
65C2×D127—496no—
66C2×D127C25124yes—
67C4×C127C2514yes—
68C4×C127C25112yes—
69Dic127C2518yes—
70C2×C127—212no—
71C2×C127C2×H(5,𝔽2)16yescomplemented
72C2×C127C2611yescharacteristic
73D127—224no—
74C22×H(5,𝔽2)—1271nomaximal, Sylow
75C127C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—12748no—
77C22×#41—1272no—
78C22×#41—1273no—
79C22×#41D12711yescharacteristic, complemented
80C23×D4—1276no—
81C23×D4—1273no—
82C25—1276no—
83H(5,𝔽2)—127256no—
84C2×#41—12748no—
85C2×#41—12772no—
86C2×#41C2×D127124yescomplemented
87C22×Q8—1271no—
88C22×Q8C2×D12711yescharacteristic, complemented
89C22×D4—1276no—
90C22×D4—127144no—
91C22×D4—1276no—
92C22×D4—1273no—
93C22×D4—12772no—
94C22×D4C2×D12713yescomplemented
95C23×C4—1276no—
96C23×C4C2×D12713yescomplemented
97C24—12772no—
98C24—1276no—
99C24—50824no—
100C24C22×D12713yes—
101#41—127128no—
102#41—127192no—
103#41C22×D127164yescomplemented
104C2×Q8—12712no—
105C2×Q8C22×D127112yescomplemented
106C2×D4—12772no—
107C2×D4—127384no—
108C2×D4—12772no—
109C2×D4—12736no—
110C2×D4—127192no—
111C2×D4C22×D127136yescomplemented
112C22×C4—1272no—
113C22×C4—12772no—
114C22×C4C22×D12713yes—
115C22×C4C22×D12711yescharacteristic
116C22×C4C22×D127136yescomplemented
117C23—212no—
118C23—12796no—
119C23—12736no—
120C23—508144no—
121C23—25424no—
122C23C23×D127118yes—
123C23C23×D12711yescharacteristic
124Q8—12716no—
125Q8C23×D127116yescomplemented
126D4—12796no—
127D4—12796no—
128D4—12748no—
129D4C23×D127148yescomplemented
130C2×C4—12712no—
131C2×C4—12796no—
132C2×C4C23×D127118yes—
133C2×C4C23×D12716yes—
134C2×C4C23×D127148yescomplemented
135C22—236no—
136C22—12724no—
137C22—50896no—
138C22—25472no—
139C22#1686125514yesdirect product factor
140C22C24×D127112yes—
141C22C24×D12713yes—
142C4—1278no—
143C4C24×D127112yes—
144C4C24×D12714yes—
145C2—212no—
146C2—25424no—
147C2C2×#1686125516yesdirect product factor
148C2C25×D12711yescharacteristic
subgroupG/H
centerC23C23×D127
commutator, cosocleC2×C127C26
FrattiniC2C25×D127
FittingC22×#41×C127C2
socleC23×C127C24
hypercenterC22×#41D127
hypocenterC127C22×H(5,𝔽2)
LCSG ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2direct (N × H)
2. C22×#14523249C2semidirect
3. C22×#14523248C2semidirect
4. C23×D508C2semidirect
5. C22×D4×D127C2semidirect
6. C22×#41×C127C2semidirect
7. C24×D127C22semidirect
8. #16861255C22direct (N × H)
9. C2×#14523249C22semidirect
10. C2×#14523248C22semidirect
11. C22×Dic254C22semidirect
12. C22×D508C22semidirect
13. C22×D508C22semidirect
14. C22×D508C22semidirect
15. C22×C22⋊D127C22semidirect
16. C2×D4×D127C22semidirect
17. C22×C4×D127C22semidirect
18. C2×#41×C127C22semidirect
19. C22×Q8×C127C22semidirect
20. C22×D4×C127C22semidirect
21. C23×C4×C127C22semidirect
24. C23×D127C23semidirect
25. #14523249C23semidirect
26. #14523248C23semidirect
27. C2×Dic254C23semidirect
28. C2×D508C23semidirect
29. C2×D508C23semidirect
30. C2×D508C23semidirect
31. C2×C22⋊D127C23semidirect
32. D4×D127C23semidirect
34. C2×C4×D127C23semidirect
36. #41×C127C23semidirect
37. C2×Q8×C127C23semidirect
38. C2×D4×C127C23semidirect
41. C22×C4×C127C23semidirect
45. C22×D127C24semidirect
46. Dic254C24semidirect
47. D508C24semidirect
48. D508C24semidirect
49. C22⋊D127C24semidirect
51. C4×D127C24semidirect
55. Q8×C127C24semidirect
56. D4×C127C24semidirect
59. C2×C4×C127C24semidirect
61. C22×C127H(5,𝔽2)semidirect
71. C2×C127C2×H(5,𝔽2)semidirect
75. C127C22×H(5,𝔽2)semidirect
79. C22×#41D127semidirect
86. C2×#41C2×D127semidirect
88. C22×Q8C2×D127semidirect
94. C22×D4C2×D127semidirect
96. C23×C4C2×D127semidirect
103. #41C22×D127semidirect
105. C2×Q8C22×D127semidirect
111. C2×D4C22×D127semidirect
116. C22×C4C22×D127semidirect
125. Q8C23×D127semidirect
129. D4C23×D127semidirect
134. C2×C4C23×D127semidirect
139. C22#16861255direct (N × H)
147. C2C2×#16861255direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612552
C22×#145232492
C22×#145232482
C23×D5082
C22×D4×D1272
C22×#41×C1272
C22×H(5,𝔽2)127