Finite Group Databasebeta

← C22×#17984133 #31795590

Subgroups — C22×#17984133 #31795590

Order 17536.

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×#17984133C2148yesmaximal, direct product factor
2C22×#14555510C212yesmaximal, complemented
3C22×#14555509C214yesmaximal, complemented
4C23×C22⋊D137C214yesmaximal, complemented
5C22×D4×D137C214yesmaximal, complemented
6C23×D4×C137C211yesmaximal, characteristic, complemented
7C24×D137C2214yescomplemented
8#17984133C221256yesdirect product factor
9C2×#14555510C22148yescomplemented
10C2×#14555509C22196yescomplemented
11C22×Dic274C2212yescomplemented
12C22×D548C2212yescomplemented
13C22×C22⋊D137C2218yescomplemented
14C22×C22⋊D137C2214yescomplemented
15C22×C22⋊D137C22196yescomplemented
16C2×D4×D137C22196yescomplemented
17C23×Dic137C2214yescomplemented
18C22×C4×D137C2214yescomplemented
19C25×C137C2212yescomplemented
20C22×D4×C137C2214yescomplemented
21C22×D4×C137C22124yescomplemented
22C23×C4×C137C2211yescharacteristic, complemented
23C23×D137—416no—
24C23×D137C2314yes—
25C23×D137C23148yescomplemented
26#14555510C231128yescomplemented
27#14555509C231256yescomplemented
28C2×Dic274C23124yescomplemented
29C2×D548C23124yescomplemented
30C2×C22⋊D137C23196yescomplemented
31C2×C22⋊D137C23148yescomplemented
32C2×C22⋊D137C231256yescomplemented
33D4×D137C231256yescomplemented
34C22×Dic137C2314yes—
35C22×Dic137C23148yescomplemented
36C2×C4×D137C23148yescomplemented
37C24×C137—48no—
38C24×C137C2314yes—
39C24×C137C2311yescharacteristic
40C24×C137C23124yescomplemented
41C2×D4×C137C23148yescomplemented
42C2×D4×C137C23164yescomplemented
43C22×C4×C137C2312yes—
44C22×C4×C137C23112yescomplemented
45C22×D137—216no—
46C22×D137—496no—
47C22×D137C24124yes—
48C22×D137C24164yescomplemented
49Dic274C24132yescomplemented
50D548C24132yescomplemented
51C22⋊D137C241128yescomplemented
52C22⋊D137C24164yescomplemented
53C2×Dic137C24124yes—
54C2×Dic137C24164yescomplemented
55C4×D137C24164yescomplemented
56C23×C137—24no—
57C23×C137—216no—
58C23×C137—448no—
59C23×C137C2416yes—
60C23×C137C24124yes—
61C23×C137C2411yescharacteristic
62C23×C137C24132yescomplemented
63D4×C137C24164yescomplemented
64C2×C4×C137C24112yes—
65C2×C4×C137C24116yescomplemented
66C22×C137—212no—
67C22×C137—248no—
68C22×C137—432no—
69C22×C137H(5,𝔽2)14yescomplemented
70C22×C137C2514yes—
71C22×C137C25116yes—
72C22×C137C2513yes—
73C2×D137—248no—
74C2×D137—464no—
75C2×D137C25116yes—
76C4×C137C2518yes—
77Dic137C25116yes—
78C2×C137—216no—
79C2×C137—24no—
80C2×C137C2×H(5,𝔽2)16yescomplemented
81C2×C137C2611yescharacteristic
82D137—216no—
83C137C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—1371nomaximal, Sylow
85C2×H(5,𝔽2)—13748no—
86C22×#41—1374no—
87C22×#41—1372no—
88C23×D4—1374no—
89C23×D4—1374no—
90C23×D4D13711yescharacteristic, complemented
91C25—1374no—
92C25C2×D13712yescomplemented
93H(5,𝔽2)—137256no—
94C2×#41—13796no—
95C2×#41—13748no—
96C22×Q8—1372no—
97C22×D4—1378no—
98C22×D4—1374no—
99C22×D4—13796no—
100C22×D4—13796no—
101C22×D4—1372no—
102C22×D4C2×D13714yescomplemented
103C22×D4C2×D137124yescomplemented
104C23×C4—1374no—
105C23×C4—1374no—
106C23×C4C2×D13711yescharacteristic, complemented
107C24—48no—
108C24—54816no—
109C24—1374no—
110C24—13748no—
111C24C22×D13714yes—
112C24C22×D13711yescharacteristic
113C24C22×D137124yescomplemented
114#41—137256no—
115#41—137128no—
116C2×Q8—13724no—
117C2×D4—13796no—
118C2×D4—13748no—
119C2×D4—137256no—
120C2×D4—137256no—
121C2×D4—13724no—
122C2×D4C22×D137148yescomplemented
123C2×D4C22×D137164yescomplemented
124C22×C4—13748no—
125C22×C4—1374no—
126C22×C4—13748no—
127C22×C4C22×D13712yes—
128C22×C4C22×D137112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—27416no—
133C23—54896no—
134C23—13724no—
135C23—13764no—
136C23C23×D137124yes—
137C23C23×D13716yes—
138C23C23×D13711yescharacteristic
139C23C23×D137132yescomplemented
140Q8—13732no—
141D4—137128no—
142D4—13764no—
143D4—13732no—
144D4C23×D137164yescomplemented
145C2×C4—13724no—
146C2×C4—13764no—
147C2×C4—13764no—
148C2×C4C23×D137112yes—
149C2×C4C23×D137116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—27448no—
154C22—54864no—
155C22—13716no—
156C22#1798413314yesdirect product factor
157C22C24×D137116yes—
158C22C24×D13714yes—
159C22C24×D13713yes—
160C4—13716no—
161C4C24×D13718yes—
162C2—24no—
163C2—216no—
164C2—27416no—
165C2C2×#1798413316yesdirect product factor
166C2C25×D13711yescharacteristic
subgroupG/H
centerC23C23×D137
commutator, cosocleC2×C137C26
FrattiniC2C25×D137
FittingC23×D4×C137C2
socleC23×C137C24
hypercenterC23×D4D137
hypocenterC137C22×H(5,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#17984133C2direct (N × H)
2. C22×#14555510C2semidirect
3. C22×#14555509C2semidirect
4. C23×C22⋊D137C2semidirect
5. C22×D4×D137C2semidirect
6. C23×D4×C137C2semidirect
7. C24×D137C22semidirect
8. #17984133C22direct (N × H)
9. C2×#14555510C22semidirect
10. C2×#14555509C22semidirect
11. C22×Dic274C22semidirect
12. C22×D548C22semidirect
13. C22×C22⋊D137C22semidirect
14. C22×C22⋊D137C22semidirect
15. C22×C22⋊D137C22semidirect
16. C2×D4×D137C22semidirect
17. C23×Dic137C22semidirect
18. C22×C4×D137C22semidirect
19. C25×C137C22semidirect
20. C22×D4×C137C22semidirect
21. C22×D4×C137C22semidirect
22. C23×C4×C137C22semidirect
25. C23×D137C23semidirect
26. #14555510C23semidirect
27. #14555509C23semidirect
28. C2×Dic274C23semidirect
29. C2×D548C23semidirect
30. C2×C22⋊D137C23semidirect
31. C2×C22⋊D137C23semidirect
32. C2×C22⋊D137C23semidirect
33. D4×D137C23semidirect
35. C22×Dic137C23semidirect
36. C2×C4×D137C23semidirect
40. C24×C137C23semidirect
41. C2×D4×C137C23semidirect
42. C2×D4×C137C23semidirect
44. C22×C4×C137C23semidirect
48. C22×D137C24semidirect
49. Dic274C24semidirect
50. D548C24semidirect
51. C22⋊D137C24semidirect
52. C22⋊D137C24semidirect
54. C2×Dic137C24semidirect
55. C4×D137C24semidirect
62. C23×C137C24semidirect
63. D4×C137C24semidirect
65. C2×C4×C137C24semidirect
69. C22×C137H(5,𝔽2)semidirect
80. C2×C137C2×H(5,𝔽2)semidirect
83. C137C22×H(5,𝔽2)semidirect
90. C23×D4D137semidirect
92. C25C2×D137semidirect
102. C22×D4C2×D137semidirect
103. C22×D4C2×D137semidirect
106. C23×C4C2×D137semidirect
113. C24C22×D137semidirect
122. C2×D4C22×D137semidirect
123. C2×D4C22×D137semidirect
128. C22×C4C22×D137semidirect
139. C23C23×D137semidirect
144. D4C23×D137semidirect
149. C2×C4C23×D137semidirect
156. C22#17984133direct (N × H)
165. C2C2×#17984133direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841332
C22×#145555102
C22×#145555092
C23×C22⋊D1372
C22×D4×D1372
C23×D4×C1372
C22×H(5,𝔽2)137