Finite Group Databasebeta

← C22×#16813668 #28405376

Subgroups — C22×#16813668 #28405376

Order 14464.

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×#16813668C2148yesmaximal, direct product factor
2C22×#14268648C212yesmaximal, complemented
3C22×#14268647C213yesmaximal, complemented
4C23×D452C213yesmaximal, complemented
5C22×D4×D113C216yesmaximal, complemented
6C22×#41×C113C211yesmaximal, characteristic, complemented
7C24×D113C2216yescomplemented
8#16813668C221256yesdirect product factor
9C2×#14268648C22148yescomplemented
10C2×#14268647C22172yescomplemented
11C22×Dic226C2211yescharacteristic, complemented
12C22×D452C2216yescomplemented
13C22×D452C2213yescomplemented
14C22×D452C22172yescomplemented
15C22×C22⋊D113C2216yescomplemented
16C2×D4×D113C221144yescomplemented
17C22×C4×D113C2216yescomplemented
18C2×#41×C113C22124yescomplemented
19C22×Q8×C113C2211yescharacteristic, complemented
20C22×D4×C113C2213yescomplemented
21C23×C4×C113C2213yescomplemented
22C23×D113—424no—
23C23×D113C2316yes—
24C23×D113C23172yescomplemented
25#14268648C231128yescomplemented
26#14268647C231192yescomplemented
27C2×Dic226C23112yescomplemented
28C2×D452C23172yescomplemented
29C2×D452C23136yescomplemented
30C2×D452C231192yescomplemented
31C2×C22⋊D113C23172yescomplemented
32D4×D113C231384yescomplemented
33C22×Dic113C2312yes—
34C2×C4×D113C23172yescomplemented
35C24×C113C2313yes—
36#41×C113C23164yescomplemented
37C2×Q8×C113C23112yescomplemented
38C2×D4×C113C23136yescomplemented
39C22×C4×C113C2313yes—
40C22×C4×C113C2311yescharacteristic
41C22×C4×C113C23136yescomplemented
42C23×C113—212no—
43C23×C113C24118yes—
44C23×C113C2411yescharacteristic
45C22×D113—224no—
46C22×D113—4144no—
47C22×D113C24136yes—
48C22×D113C24196yescomplemented
49Q8×C113C24116yescomplemented
50D4×C113C24148yescomplemented
51C2×C4×C113C2416yes—
52C2×C4×C113C24118yes—
53C2×C4×C113C24148yescomplemented
54C22⋊D113C24196yescomplemented
55C2×Dic113C24112yes—
56D452C24196yescomplemented
57D452C24148yescomplemented
58C4×D113C24196yescomplemented
59Dic226C24116yescomplemented
60C22×C113—236no—
61C22×C113H(5,𝔽2)14yescomplemented
62C22×C113C25112yes—
63C22×C113C2513yes—
64C2×D113—272no—
65C2×D113—496no—
66C2×D113C25124yes—
67C4×C113C2514yes—
68C4×C113C25112yes—
69Dic113C2518yes—
70C2×C113—212no—
71C2×C113C2×H(5,𝔽2)16yescomplemented
72C2×C113C2611yescharacteristic
73D113—224no—
74C22×H(5,𝔽2)—1131nomaximal, Sylow
75C113C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—11348no—
77C22×#41—1132no—
78C22×#41—1133no—
79C22×#41D11311yescharacteristic, complemented
80C23×D4—1136no—
81C23×D4—1133no—
82C25—1136no—
83H(5,𝔽2)—113256no—
84C2×#41—11348no—
85C2×#41—11372no—
86C2×#41C2×D113124yescomplemented
87C22×Q8—1131no—
88C22×Q8C2×D11311yescharacteristic, complemented
89C22×D4—1136no—
90C22×D4—113144no—
91C22×D4—1136no—
92C22×D4—1133no—
93C22×D4—11372no—
94C22×D4C2×D11313yescomplemented
95C23×C4—1136no—
96C23×C4C2×D11313yescomplemented
97C24—11372no—
98C24—1136no—
99C24—45224no—
100C24C22×D11313yes—
101#41—113128no—
102#41—113192no—
103#41C22×D113164yescomplemented
104C2×Q8—11312no—
105C2×Q8C22×D113112yescomplemented
106C2×D4—11372no—
107C2×D4—113384no—
108C2×D4—11372no—
109C2×D4—11336no—
110C2×D4—113192no—
111C2×D4C22×D113136yescomplemented
112C22×C4—1132no—
113C22×C4—11372no—
114C22×C4C22×D11313yes—
115C22×C4C22×D11311yescharacteristic
116C22×C4C22×D113136yescomplemented
117C23—212no—
118C23—11396no—
119C23—11336no—
120C23—452144no—
121C23—22624no—
122C23C23×D113118yes—
123C23C23×D11311yescharacteristic
124Q8—11316no—
125Q8C23×D113116yescomplemented
126D4—11396no—
127D4—11396no—
128D4—11348no—
129D4C23×D113148yescomplemented
130C2×C4—11312no—
131C2×C4—11396no—
132C2×C4C23×D113118yes—
133C2×C4C23×D11316yes—
134C2×C4C23×D113148yescomplemented
135C22—236no—
136C22—11324no—
137C22—45296no—
138C22—22672no—
139C22#1681366814yesdirect product factor
140C22C24×D113112yes—
141C22C24×D11313yes—
142C4—1138no—
143C4C24×D113112yes—
144C4C24×D11314yes—
145C2—212no—
146C2—22624no—
147C2C2×#1681366816yesdirect product factor
148C2C25×D11311yescharacteristic
subgroupG/H
centerC23C23×D113
commutator, cosocleC2×C113C26
FrattiniC2C25×D113
FittingC22×#41×C113C2
socleC23×C113C24
hypercenterC22×#41D113
hypocenterC113C22×H(5,𝔽2)
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2direct (N × H)
2. C22×#14268648C2semidirect
3. C22×#14268647C2semidirect
4. C23×D452C2semidirect
5. C22×D4×D113C2semidirect
6. C22×#41×C113C2semidirect
7. C24×D113C22semidirect
8. #16813668C22direct (N × H)
9. C2×#14268648C22semidirect
10. C2×#14268647C22semidirect
11. C22×Dic226C22semidirect
12. C22×D452C22semidirect
13. C22×D452C22semidirect
14. C22×D452C22semidirect
15. C22×C22⋊D113C22semidirect
16. C2×D4×D113C22semidirect
17. C22×C4×D113C22semidirect
18. C2×#41×C113C22semidirect
19. C22×Q8×C113C22semidirect
20. C22×D4×C113C22semidirect
21. C23×C4×C113C22semidirect
24. C23×D113C23semidirect
25. #14268648C23semidirect
26. #14268647C23semidirect
27. C2×Dic226C23semidirect
28. C2×D452C23semidirect
29. C2×D452C23semidirect
30. C2×D452C23semidirect
31. C2×C22⋊D113C23semidirect
32. D4×D113C23semidirect
34. C2×C4×D113C23semidirect
36. #41×C113C23semidirect
37. C2×Q8×C113C23semidirect
38. C2×D4×C113C23semidirect
41. C22×C4×C113C23semidirect
48. C22×D113C24semidirect
49. Q8×C113C24semidirect
50. D4×C113C24semidirect
53. C2×C4×C113C24semidirect
54. C22⋊D113C24semidirect
56. D452C24semidirect
57. D452C24semidirect
58. C4×D113C24semidirect
59. Dic226C24semidirect
61. C22×C113H(5,𝔽2)semidirect
71. C2×C113C2×H(5,𝔽2)semidirect
75. C113C22×H(5,𝔽2)semidirect
79. C22×#41D113semidirect
86. C2×#41C2×D113semidirect
88. C22×Q8C2×D113semidirect
94. C22×D4C2×D113semidirect
96. C23×C4C2×D113semidirect
103. #41C22×D113semidirect
105. C2×Q8C22×D113semidirect
111. C2×D4C22×D113semidirect
116. C22×C4C22×D113semidirect
125. Q8C23×D113semidirect
129. D4C23×D113semidirect
134. C2×C4C23×D113semidirect
139. C22#16813668direct (N × H)
147. C2C2×#16813668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136682
C22×#142686482
C22×#142686472
C23×D4522
C22×D4×D1132
C22×#41×C1132
C22×H(5,𝔽2)113