Finite Group Databasebeta

← C22×#16862292 #30680763

Subgroups — C22×#16862292 #30680763

Order 16768.

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×#16862292C2148yesmaximal, direct product factor
2C22×#14523514C212yesmaximal, complemented
3C22×#14523513C213yesmaximal, complemented
4C23×D524C213yesmaximal, complemented
5C22×D4×D131C216yesmaximal, complemented
6C22×#41×C131C211yesmaximal, characteristic, complemented
7C24×D131C2216yescomplemented
8#16862292C221256yesdirect product factor
9C2×#14523514C22148yescomplemented
10C2×#14523513C22172yescomplemented
11C22×Dic262C2211yescharacteristic, complemented
12C22×D524C2216yescomplemented
13C22×D524C2213yescomplemented
14C22×D524C22172yescomplemented
15C22×C22⋊D131C2216yescomplemented
16C2×D4×D131C221144yescomplemented
17C22×C4×D131C2216yescomplemented
18C2×#41×C131C22124yescomplemented
19C22×Q8×C131C2211yescharacteristic, complemented
20C22×D4×C131C2213yescomplemented
21C23×C4×C131C2213yescomplemented
22C23×D131—424no—
23C23×D131C2316yes—
24C23×D131C23172yescomplemented
25#14523514C231128yescomplemented
26#14523513C231192yescomplemented
27C2×Dic262C23112yescomplemented
28C2×D524C23172yescomplemented
29C2×D524C23136yescomplemented
30C2×D524C231192yescomplemented
31C2×C22⋊D131C23172yescomplemented
32D4×D131C231384yescomplemented
33C22×Dic131C2312yes—
34C2×C4×D131C23172yescomplemented
35C24×C131C2313yes—
36#41×C131C23164yescomplemented
37C2×Q8×C131C23112yescomplemented
38C2×D4×C131C23136yescomplemented
39C22×C4×C131C2313yes—
40C22×C4×C131C2311yescharacteristic
41C22×C4×C131C23136yescomplemented
42C22×D131—224no—
43C22×D131—4144no—
44C22×D131C24136yes—
45C22×D131C24196yescomplemented
46Dic262C24116yescomplemented
47D524C24196yescomplemented
48D524C24148yescomplemented
49C22⋊D131C24196yescomplemented
50C2×Dic131C24112yes—
51C4×D131C24196yescomplemented
52C23×C131—212no—
53C23×C131C24118yes—
54C23×C131C2411yescharacteristic
55Q8×C131C24116yescomplemented
56D4×C131C24148yescomplemented
57C2×C4×C131C2416yes—
58C2×C4×C131C24118yes—
59C2×C4×C131C24148yescomplemented
60C22×C131—236no—
61C22×C131H(5,𝔽2)14yescomplemented
62C22×C131C25112yes—
63C22×C131C2513yes—
64C2×D131—272no—
65C2×D131—496no—
66C2×D131C25124yes—
67C4×C131C2514yes—
68C4×C131C25112yes—
69Dic131C2518yes—
70C2×C131—212no—
71C2×C131C2×H(5,𝔽2)16yescomplemented
72C2×C131C2611yescharacteristic
73D131—224no—
74C131C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1311nomaximal, Sylow
76C2×H(5,𝔽2)—13148no—
77C22×#41—1312no—
78C22×#41—1313no—
79C22×#41D13111yescharacteristic, complemented
80C23×D4—1316no—
81C23×D4—1313no—
82C25—1316no—
83H(5,𝔽2)—131256no—
84C2×#41—13148no—
85C2×#41—13172no—
86C2×#41C2×D131124yescomplemented
87C22×Q8—1311no—
88C22×Q8C2×D13111yescharacteristic, complemented
89C22×D4—1316no—
90C22×D4—131144no—
91C22×D4—1316no—
92C22×D4—1313no—
93C22×D4—13172no—
94C22×D4C2×D13113yescomplemented
95C23×C4—1316no—
96C23×C4C2×D13113yescomplemented
97C24—52424no—
98C24—1316no—
99C24—13172no—
100C24C22×D13113yes—
101#41—131128no—
102#41—131192no—
103#41C22×D131164yescomplemented
104C2×Q8—13112no—
105C2×Q8C22×D131112yescomplemented
106C2×D4—13172no—
107C2×D4—131384no—
108C2×D4—13172no—
109C2×D4—13136no—
110C2×D4—131192no—
111C2×D4C22×D131136yescomplemented
112C22×C4—1312no—
113C22×C4—13172no—
114C22×C4C22×D13113yes—
115C22×C4C22×D13111yescharacteristic
116C22×C4C22×D131136yescomplemented
117C23—212no—
118C23—26224no—
119C23—524144no—
120C23—13136no—
121C23—13196no—
122C23C23×D131118yes—
123C23C23×D13111yescharacteristic
124Q8—13116no—
125Q8C23×D131116yescomplemented
126D4—13196no—
127D4—13196no—
128D4—13148no—
129D4C23×D131148yescomplemented
130C2×C4—13112no—
131C2×C4—13196no—
132C2×C4C23×D131118yes—
133C2×C4C23×D13116yes—
134C2×C4C23×D131148yescomplemented
135C22—236no—
136C22—26272no—
137C22—52496no—
138C22—13124no—
139C22#1686229214yesdirect product factor
140C22C24×D131112yes—
141C22C24×D13113yes—
142C4—1318no—
143C4C24×D131112yes—
144C4C24×D13114yes—
145C2—212no—
146C2—26224no—
147C2C2×#1686229216yesdirect product factor
148C2C25×D13111yescharacteristic
subgroupG/H
centerC23C23×D131
commutator, cosocleC2×C131C26
FrattiniC2C25×D131
FittingC22×#41×C131C2
socleC23×C131C24
hypercenterC22×#41D131
hypocenterC131C22×H(5,𝔽2)
LCSG ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16862292C2direct (N × H)
2. C22×#14523514C2semidirect
3. C22×#14523513C2semidirect
4. C23×D524C2semidirect
5. C22×D4×D131C2semidirect
6. C22×#41×C131C2semidirect
7. C24×D131C22semidirect
8. #16862292C22direct (N × H)
9. C2×#14523514C22semidirect
10. C2×#14523513C22semidirect
11. C22×Dic262C22semidirect
12. C22×D524C22semidirect
13. C22×D524C22semidirect
14. C22×D524C22semidirect
15. C22×C22⋊D131C22semidirect
16. C2×D4×D131C22semidirect
17. C22×C4×D131C22semidirect
18. C2×#41×C131C22semidirect
19. C22×Q8×C131C22semidirect
20. C22×D4×C131C22semidirect
21. C23×C4×C131C22semidirect
24. C23×D131C23semidirect
25. #14523514C23semidirect
26. #14523513C23semidirect
27. C2×Dic262C23semidirect
28. C2×D524C23semidirect
29. C2×D524C23semidirect
30. C2×D524C23semidirect
31. C2×C22⋊D131C23semidirect
32. D4×D131C23semidirect
34. C2×C4×D131C23semidirect
36. #41×C131C23semidirect
37. C2×Q8×C131C23semidirect
38. C2×D4×C131C23semidirect
41. C22×C4×C131C23semidirect
45. C22×D131C24semidirect
46. Dic262C24semidirect
47. D524C24semidirect
48. D524C24semidirect
49. C22⋊D131C24semidirect
51. C4×D131C24semidirect
55. Q8×C131C24semidirect
56. D4×C131C24semidirect
59. C2×C4×C131C24semidirect
61. C22×C131H(5,𝔽2)semidirect
71. C2×C131C2×H(5,𝔽2)semidirect
74. C131C22×H(5,𝔽2)semidirect
79. C22×#41D131semidirect
86. C2×#41C2×D131semidirect
88. C22×Q8C2×D131semidirect
94. C22×D4C2×D131semidirect
96. C23×C4C2×D131semidirect
103. #41C22×D131semidirect
105. C2×Q8C22×D131semidirect
111. C2×D4C22×D131semidirect
116. C22×C4C22×D131semidirect
125. Q8C23×D131semidirect
129. D4C23×D131semidirect
134. C2×C4C23×D131semidirect
139. C22#16862292direct (N × H)
147. C2C2×#16862292direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622922
C22×#145235142
C22×#145235132
C23×D5242
C22×D4×D1312
C22×#41×C1312
C22×H(5,𝔽2)131