Finite Group Databasebeta

← C22×#22548760 #44565166

Subgroups — C22×#22548760 #44565166

Order 32128.

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×#22548760C2148yesmaximal, direct product factor
2C22×#16860810C212yesmaximal, complemented
3C22×#16860809C214yesmaximal, complemented
4C23×C22⋊D251C214yesmaximal, complemented
5C22×D4×D251C214yesmaximal, complemented
6C23×D4×C251C211yesmaximal, characteristic, complemented
7C24×D251C2214yescomplemented
8#22548760C221256yesdirect product factor
9C2×#16860810C22148yescomplemented
10C2×#16860809C22196yescomplemented
11C22×Dic502C2212yescomplemented
12C22×D1004C2212yescomplemented
13C22×C22⋊D251C2218yescomplemented
14C22×C22⋊D251C2214yescomplemented
15C22×C22⋊D251C22196yescomplemented
16C2×D4×D251C22196yescomplemented
17C23×Dic251C2214yescomplemented
18C22×C4×D251C2214yescomplemented
19C25×C251C2212yescomplemented
20C22×D4×C251C2214yescomplemented
21C22×D4×C251C22124yescomplemented
22C23×C4×C251C2211yescharacteristic, complemented
23C23×D251—416no—
24C23×D251C2314yes—
25C23×D251C23148yescomplemented
26#16860810C231128yescomplemented
27#16860809C231256yescomplemented
28C2×Dic502C23124yescomplemented
29C2×D1004C23124yescomplemented
30C2×C22⋊D251C23196yescomplemented
31C2×C22⋊D251C23148yescomplemented
32C2×C22⋊D251C231256yescomplemented
33D4×D251C231256yescomplemented
34C22×Dic251C2314yes—
35C22×Dic251C23148yescomplemented
36C2×C4×D251C23148yescomplemented
37C24×C251—48no—
38C24×C251C2314yes—
39C24×C251C2311yescharacteristic
40C24×C251C23124yescomplemented
41C2×D4×C251C23148yescomplemented
42C2×D4×C251C23164yescomplemented
43C22×C4×C251C2312yes—
44C22×C4×C251C23112yescomplemented
45C22×D251—216no—
46C22×D251—496no—
47C22×D251C24124yes—
48C22×D251C24164yescomplemented
49Dic502C24132yescomplemented
50D1004C24132yescomplemented
51C22⋊D251C241128yescomplemented
52C22⋊D251C24164yescomplemented
53C2×Dic251C24124yes—
54C2×Dic251C24164yescomplemented
55C4×D251C24164yescomplemented
56C23×C251—24no—
57C23×C251—216no—
58C23×C251—448no—
59C23×C251C2416yes—
60C23×C251C24124yes—
61C23×C251C2411yescharacteristic
62C23×C251C24132yescomplemented
63D4×C251C24164yescomplemented
64C2×C4×C251C24112yes—
65C2×C4×C251C24116yescomplemented
66C22×C251—212no—
67C22×C251—248no—
68C22×C251—432no—
69C22×C251H(5,𝔽2)14yescomplemented
70C22×C251C2514yes—
71C22×C251C25116yes—
72C22×C251C2513yes—
73C2×D251—248no—
74C2×D251—464no—
75C2×D251C25116yes—
76C4×C251C2518yes—
77Dic251C25116yes—
78C2×C251—216no—
79C2×C251—24no—
80C2×C251C2×H(5,𝔽2)16yescomplemented
81C2×C251C2611yescharacteristic
82D251—216no—
83C251C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
84C22×H(5,𝔽2)—2511nomaximal, Sylow
85C2×H(5,𝔽2)—25148no—
86C22×#41—2514no—
87C22×#41—2512no—
88C23×D4—2514no—
89C23×D4—2514no—
90C23×D4D25111yescharacteristic, complemented
91C25—2514no—
92C25C2×D25112yescomplemented
93H(5,𝔽2)—251256no—
94C2×#41—25196no—
95C2×#41—25148no—
96C22×Q8—2512no—
97C22×D4—2518no—
98C22×D4—2514no—
99C22×D4—25196no—
100C22×D4—25196no—
101C22×D4—2512no—
102C22×D4C2×D25114yescomplemented
103C22×D4C2×D251124yescomplemented
104C23×C4—2514no—
105C23×C4—2514no—
106C23×C4C2×D25111yescharacteristic, complemented
107C24—48no—
108C24—100416no—
109C24—25148no—
110C24—2514no—
111C24C22×D25114yes—
112C24C22×D25111yescharacteristic
113C24C22×D251124yescomplemented
114#41—251256no—
115#41—251128no—
116C2×Q8—25124no—
117C2×D4—25196no—
118C2×D4—25148no—
119C2×D4—251256no—
120C2×D4—251256no—
121C2×D4—25124no—
122C2×D4C22×D251148yescomplemented
123C2×D4C22×D251164yescomplemented
124C22×C4—25148no—
125C22×C4—2514no—
126C22×C4—25148no—
127C22×C4C22×D25112yes—
128C22×C4C22×D251112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—100496no—
133C23—50216no—
134C23—25164no—
135C23—25124no—
136C23C23×D251124yes—
137C23C23×D25116yes—
138C23C23×D25111yescharacteristic
139C23C23×D251132yescomplemented
140Q8—25132no—
141D4—251128no—
142D4—25164no—
143D4—25132no—
144D4C23×D251164yescomplemented
145C2×C4—25124no—
146C2×C4—25164no—
147C2×C4—25164no—
148C2×C4C23×D251112yes—
149C2×C4C23×D251116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—100464no—
154C22—50248no—
155C22—25116no—
156C22#2254876014yesdirect product factor
157C22C24×D251116yes—
158C22C24×D25114yes—
159C22C24×D25113yes—
160C4—25116no—
161C4C24×D25118yes—
162C2—24no—
163C2—216no—
164C2—50216no—
165C2C2×#2254876016yesdirect product factor
166C2C25×D25111yescharacteristic
subgroupG/H
centerC23C23×D251
commutator, cosocleC2×C251C26
FrattiniC2C25×D251
FittingC23×D4×C251C2
socleC23×C251C24
hypercenterC23×D4D251
hypocenterC251C22×H(5,𝔽2)
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22548760C2direct (N × H)
2. C22×#16860810C2semidirect
3. C22×#16860809C2semidirect
4. C23×C22⋊D251C2semidirect
5. C22×D4×D251C2semidirect
6. C23×D4×C251C2semidirect
7. C24×D251C22semidirect
8. #22548760C22direct (N × H)
9. C2×#16860810C22semidirect
10. C2×#16860809C22semidirect
11. C22×Dic502C22semidirect
12. C22×D1004C22semidirect
13. C22×C22⋊D251C22semidirect
14. C22×C22⋊D251C22semidirect
15. C22×C22⋊D251C22semidirect
16. C2×D4×D251C22semidirect
17. C23×Dic251C22semidirect
18. C22×C4×D251C22semidirect
19. C25×C251C22semidirect
20. C22×D4×C251C22semidirect
21. C22×D4×C251C22semidirect
22. C23×C4×C251C22semidirect
25. C23×D251C23semidirect
26. #16860810C23semidirect
27. #16860809C23semidirect
28. C2×Dic502C23semidirect
29. C2×D1004C23semidirect
30. C2×C22⋊D251C23semidirect
31. C2×C22⋊D251C23semidirect
32. C2×C22⋊D251C23semidirect
33. D4×D251C23semidirect
35. C22×Dic251C23semidirect
36. C2×C4×D251C23semidirect
40. C24×C251C23semidirect
41. C2×D4×C251C23semidirect
42. C2×D4×C251C23semidirect
44. C22×C4×C251C23semidirect
48. C22×D251C24semidirect
49. Dic502C24semidirect
50. D1004C24semidirect
51. C22⋊D251C24semidirect
52. C22⋊D251C24semidirect
54. C2×Dic251C24semidirect
55. C4×D251C24semidirect
62. C23×C251C24semidirect
63. D4×C251C24semidirect
65. C2×C4×C251C24semidirect
69. C22×C251H(5,𝔽2)semidirect
80. C2×C251C2×H(5,𝔽2)semidirect
83. C251C22×H(5,𝔽2)semidirect
90. C23×D4D251semidirect
92. C25C2×D251semidirect
102. C22×D4C2×D251semidirect
103. C22×D4C2×D251semidirect
106. C23×C4C2×D251semidirect
113. C24C22×D251semidirect
122. C2×D4C22×D251semidirect
123. C2×D4C22×D251semidirect
128. C22×C4C22×D251semidirect
139. C23C23×D251semidirect
144. D4C23×D251semidirect
149. C2×C4C23×D251semidirect
156. C22#22548760direct (N × H)
165. C2C2×#22548760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487602
C22×#168608102
C22×#168608092
C23×C22⋊D2512
C22×D4×D2512
C23×D4×C2512
C22×H(5,𝔽2)251