Finite Group Databasebeta

← C22×#15692301 #27185934

Subgroups — C22×#15692301 #27185934

Order 12928.

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×#15692301C2148yesmaximal, direct product factor
2C22×#13108075C212yesmaximal, complemented
3C22×#13108074C213yesmaximal, complemented
4C23×D404C213yesmaximal, complemented
5C22×D4×D101C216yesmaximal, complemented
6C22×#41×C101C211yesmaximal, characteristic, complemented
7C24×D101C2216yescomplemented
8#15692301C221256yesdirect product factor
9C2×#13108075C22148yescomplemented
10C2×#13108074C22172yescomplemented
11C22×Dic202C2211yescharacteristic, complemented
12C22×D404C2216yescomplemented
13C22×D404C2213yescomplemented
14C22×D404C22172yescomplemented
15C22×C22⋊D101C2216yescomplemented
16C2×D4×D101C221144yescomplemented
17C22×C4×D101C2216yescomplemented
18C2×#41×C101C22124yescomplemented
19C22×Q8×C101C2211yescharacteristic, complemented
20C22×D4×C101C2213yescomplemented
21C23×C4×C101C2213yescomplemented
22C23×D101—424no—
23C23×D101C2316yes—
24C23×D101C23172yescomplemented
25#13108075C231128yescomplemented
26#13108074C231192yescomplemented
27C2×Dic202C23112yescomplemented
28C2×D404C23172yescomplemented
29C2×D404C23136yescomplemented
30C2×D404C231192yescomplemented
31C2×C22⋊D101C23172yescomplemented
32D4×D101C231384yescomplemented
33C22×Dic101C2312yes—
34C2×C4×D101C23172yescomplemented
35C24×C101C2313yes—
36#41×C101C23164yescomplemented
37C2×Q8×C101C23112yescomplemented
38C2×D4×C101C23136yescomplemented
39C22×C4×C101C2313yes—
40C22×C4×C101C2311yescharacteristic
41C22×C4×C101C23136yescomplemented
42C23×C101—212no—
43C23×C101C24118yes—
44C23×C101C2411yescharacteristic
45C22×D101—224no—
46C22×D101—4144no—
47C22×D101C24136yes—
48C22×D101C24196yescomplemented
49Q8×C101C24116yescomplemented
50D4×C101C24148yescomplemented
51C2×C4×C101C2416yes—
52C2×C4×C101C24118yes—
53C2×C4×C101C24148yescomplemented
54C22⋊D101C24196yescomplemented
55C2×Dic101C24112yes—
56D404C24196yescomplemented
57D404C24148yescomplemented
58C4×D101C24196yescomplemented
59Dic202C24116yescomplemented
60C22×C101—236no—
61C22×C101H(5,𝔽2)14yescomplemented
62C22×C101C25112yes—
63C22×C101C2513yes—
64C2×D101—272no—
65C2×D101—496no—
66C2×D101C25124yes—
67C4×C101C2514yes—
68C4×C101C25112yes—
69Dic101C2518yes—
70C2×C101—212no—
71C2×C101C2×H(5,𝔽2)16yescomplemented
72C2×C101C2611yescharacteristic
73D101—224no—
74C22×H(5,𝔽2)—1011nomaximal, Sylow
75C101C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—10148no—
77C22×#41—1012no—
78C22×#41—1013no—
79C22×#41D10111yescharacteristic, complemented
80C23×D4—1016no—
81C23×D4—1013no—
82C25—1016no—
83H(5,𝔽2)—101256no—
84C2×#41—10148no—
85C2×#41—10172no—
86C2×#41C2×D101124yescomplemented
87C22×Q8—1011no—
88C22×Q8C2×D10111yescharacteristic, complemented
89C22×D4—1016no—
90C22×D4—101144no—
91C22×D4—1016no—
92C22×D4—1013no—
93C22×D4—10172no—
94C22×D4C2×D10113yescomplemented
95C23×C4—1016no—
96C23×C4C2×D10113yescomplemented
97C24—10172no—
98C24—1016no—
99C24—40424no—
100C24C22×D10113yes—
101#41—101128no—
102#41—101192no—
103#41C22×D101164yescomplemented
104C2×Q8—10112no—
105C2×Q8C22×D101112yescomplemented
106C2×D4—10172no—
107C2×D4—101384no—
108C2×D4—10172no—
109C2×D4—10136no—
110C2×D4—101192no—
111C2×D4C22×D101136yescomplemented
112C22×C4—1012no—
113C22×C4—10172no—
114C22×C4C22×D10113yes—
115C22×C4C22×D10111yescharacteristic
116C22×C4C22×D101136yescomplemented
117C23—212no—
118C23—10196no—
119C23—10136no—
120C23—404144no—
121C23—20224no—
122C23C23×D101118yes—
123C23C23×D10111yescharacteristic
124Q8—10116no—
125Q8C23×D101116yescomplemented
126D4—10196no—
127D4—10196no—
128D4—10148no—
129D4C23×D101148yescomplemented
130C2×C4—10112no—
131C2×C4—10196no—
132C2×C4C23×D101118yes—
133C2×C4C23×D10116yes—
134C2×C4C23×D101148yescomplemented
135C22—236no—
136C22—10124no—
137C22—40496no—
138C22—20272no—
139C22#1569230114yesdirect product factor
140C22C24×D101112yes—
141C22C24×D10113yes—
142C4—1018no—
143C4C24×D101112yes—
144C4C24×D10114yes—
145C2—212no—
146C2—20224no—
147C2C2×#1569230116yesdirect product factor
148C2C25×D10111yescharacteristic
subgroupG/H
centerC23C23×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC22×#41×C101C2
socleC23×C101C24
hypercenterC22×#41D101
hypocenterC101C22×H(5,𝔽2)
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15692301C2direct (N × H)
2. C22×#13108075C2semidirect
3. C22×#13108074C2semidirect
4. C23×D404C2semidirect
5. C22×D4×D101C2semidirect
6. C22×#41×C101C2semidirect
7. C24×D101C22semidirect
8. #15692301C22direct (N × H)
9. C2×#13108075C22semidirect
10. C2×#13108074C22semidirect
11. C22×Dic202C22semidirect
12. C22×D404C22semidirect
13. C22×D404C22semidirect
14. C22×D404C22semidirect
15. C22×C22⋊D101C22semidirect
16. C2×D4×D101C22semidirect
17. C22×C4×D101C22semidirect
18. C2×#41×C101C22semidirect
19. C22×Q8×C101C22semidirect
20. C22×D4×C101C22semidirect
21. C23×C4×C101C22semidirect
24. C23×D101C23semidirect
25. #13108075C23semidirect
26. #13108074C23semidirect
27. C2×Dic202C23semidirect
28. C2×D404C23semidirect
29. C2×D404C23semidirect
30. C2×D404C23semidirect
31. C2×C22⋊D101C23semidirect
32. D4×D101C23semidirect
34. C2×C4×D101C23semidirect
36. #41×C101C23semidirect
37. C2×Q8×C101C23semidirect
38. C2×D4×C101C23semidirect
41. C22×C4×C101C23semidirect
48. C22×D101C24semidirect
49. Q8×C101C24semidirect
50. D4×C101C24semidirect
53. C2×C4×C101C24semidirect
54. C22⋊D101C24semidirect
56. D404C24semidirect
57. D404C24semidirect
58. C4×D101C24semidirect
59. Dic202C24semidirect
61. C22×C101H(5,𝔽2)semidirect
71. C2×C101C2×H(5,𝔽2)semidirect
75. C101C22×H(5,𝔽2)semidirect
79. C22×#41D101semidirect
86. C2×#41C2×D101semidirect
88. C22×Q8C2×D101semidirect
94. C22×D4C2×D101semidirect
96. C23×C4C2×D101semidirect
103. #41C22×D101semidirect
105. C2×Q8C22×D101semidirect
111. C2×D4C22×D101semidirect
116. C22×C4C22×D101semidirect
125. Q8C23×D101semidirect
129. D4C23×D101semidirect
134. C2×C4C23×D101semidirect
139. C22#15692301direct (N × H)
147. C2C2×#15692301direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923012
C22×#131080752
C22×#131080742
C23×D4042
C22×D4×D1012
C22×#41×C1012
C22×H(5,𝔽2)101