Finite Group Databasebeta

← C24×D4×D53 #28297968

Subgroups — C24×D4×D53 #28297968

Order 13568.

111198 conjugacy classes of subgroups; only the 34795 normal subgroups are shown.

normal subgroups only

This group has 111198 conjugacy classes of subgroups — too many to draw. Only its 34795 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
1C26×D53C212yesmaximal, complemented
2C24×D212C211yesmaximal, characteristic, complemented
3C24×C22⋊D53C212yesmaximal, complemented
4C23×D4×D53C21120yesmaximal, direct product factor
5C24×C4×D53C211yesmaximal, characteristic, complemented
6C24×D4×C53C211yesmaximal, characteristic, complemented
7C25×D53—264no—
8C25×D53C2211yescharacteristic
9C25×D53C2212yescomplemented
10C25×D53C221120yescomplemented
11C23×D212C22160yescomplemented
12C23×C22⋊D53C221120yescomplemented
13C22×D4×D53C2212240yesdirect product factor
14C24×Dic53C2211yescharacteristic, complemented
15C23×C4×D53C22160yescomplemented
16C26×C53C2212yescomplemented
17C23×D4×C53C22160yescomplemented
18C24×C4×C53C2211yescharacteristic, complemented
19C24×D53—232no—
20C24×D53—21920no—
21C24×D53D4132yesdirect product factor
22C24×D53C23130yes—
23C24×D53C2311120yescomplemented
24C24×D53C23160yescomplemented
25C22×D212C231560yescomplemented
26C22×C22⋊D53C2311120yescomplemented
27C2×D4×D53C2317680yesdirect product factor
28C23×Dic53C23130yescomplemented
29C22×C4×D53C231560yescomplemented
30C25×C53—232no—
31C25×C53C2311yescharacteristic
32C25×C53C23160yescomplemented
33C22×D4×C53C231560yescomplemented
34C23×C4×C53C23130yescomplemented
35C24×C53—2480no—
36C24×C53C2×D4116yescomplemented
37C24×C53C24115yes—
38C24×C53C241280yescomplemented
39C23×D53—2480no—
40C23×D53—28960no—
41C23×D53C2×D41480yesdirect product factor
42C23×D53C241140yes—
43C23×D53C2411920yescomplemented
44C23×D53C241280yescomplemented
45C2×D4×C53C241960yescomplemented
46C22×C4×C53C241140yescomplemented
47C2×C22⋊D53C2411920yescomplemented
48C22×Dic53C241140yescomplemented
49D4×D53C2414096yesdirect product factor
50C2×D212C241960yescomplemented
51C2×C4×D53C241960yescomplemented
52C23×C53—21120no—
53C23×C53C22×D41120yescomplemented
54C23×C53C25135yes—
55C23×C53C251240yescomplemented
56C22×D53—21120no—
57C22×D53—27680no—
58C22×D53C22×D411120yesdirect product factor
59C22×D53C251120yes—
60C22×D53C251512yescomplemented
61C22×D53C251240yescomplemented
62D4×C53C251256yescomplemented
63C2×C4×C53C251120yescomplemented
64C22⋊D53C251512yescomplemented
65C2×Dic53C251120yescomplemented
66D212C251256yescomplemented
67C4×D53C251256yescomplemented
68C25×D4—531nomaximal, Sylow
69C22×C53—2480no—
70C22×C53C23×D41140yescomplemented
71C22×C53C26115yes—
72C22×C53C26132yescomplemented
73C2×D53—2480no—
74C2×D53—21024no—
75C2×D53C23×D41480yesdirect product factor
76C2×D53C26116yes—
77C2×D53C26132yescomplemented
78C4×C53C26116yescomplemented
79Dic53C26116yescomplemented
80C27—532no—
81C24×D4—532no—
82C24×D4—53120no—
83C24×D4—531no—
84C24×D4D5311yescharacteristic, direct product factor
85C25×C4—531no—
86C2×C53—232no—
87C2×C53C24×D4130yescomplemented
88C2×C53C2711yescharacteristic
89D53—232no—
90D53C24×D4132yesdirect product factor
91C26—531no—
92C26—532no—
93C26—53120no—
94C26—10664no—
95C26C2×D5312yescomplemented
96C23×D4—53120no—
97C23×D4—532240no—
98C23×D4—5360no—
99C23×D4C2×D53160yesdirect product factor
100C24×C4—531no—
101C24×C4—5360no—
102C24×C4C2×D5311yescharacteristic, complemented
103C53C25×D411yescharacteristic, complemented, Sylow
104C25—232no—
105C25—5332no—
106C25—5330no—
107C25—5360no—
108C25—531120no—
109C25—1061920no—
110C25—10632no—
111C25C22×D5311yescharacteristic
112C25C22×D53160yescomplemented
113C22×D4—531120no—
114C22×D4—537680no—
115C22×D4—53560no—
116C22×D4C22×D531560yesdirect product factor
117C23×C4—5330no—
118C23×C4—53560no—
119C23×C4C22×D53130yescomplemented
120C24—2480no—
121C24—53480no—
122C24—53140no—
123C24—53280no—
124C24—531920no—
125C24—1068960no—
126C24—106480no—
127C24D4×D53116yesdirect product factor
128C24C23×D53115yes—
129C24C23×D531280yescomplemented
130C2×D4—531920no—
131C2×D4—534096no—
132C2×D4—53960no—
133C2×D4C23×D531960yesdirect product factor
134C22×C4—53140no—
135C22×C4—53960no—
136C22×C4C23×D531140yescomplemented
137C23—21120no—
138C23—531120no—
139C23—53120no—
140C23—53240no—
141C23—53512no—
142C23—1067680no—
143C23—1061120no—
144C23C2×D4×D531120yesdirect product factor
145C23C24×D53135yes—
146C23C24×D531240yescomplemented
147D4—53512no—
148D4—53256no—
149D4C24×D531256yesdirect product factor
150C2×C4—53120no—
151C2×C4—53256no—
152C2×C4C24×D531120yescomplemented
153C22—2480no—
154C22—53480no—
155C22—5316no—
156C22—5332no—
157C22—1061024no—
158C22—106480no—
159C22C22×D4×D531140yesdirect product factor
160C22C25×D53115yes—
161C22C25×D53132yescomplemented
162C4—5316no—
163C4C25×D53116yescomplemented
164C2—232no—
165C2—5332no—
166C2—10632no—
167C2C23×D4×D53130yesdirect product factor
168C2C26×D5311yescharacteristic
subgroupG/H
centerC25C22×D53
commutator, cosocleC2×C53C27
FrattiniC2C26×D53
FittingC24×D4×C53C2
socleC25×C53C23
hypercenterC24×D4D53
hypocenterC53C25×D4
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C25 ◁ C24×D4

Complements

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

NH ≅ G/Ntype
1. C26×D53C2semidirect
2. C24×D212C2semidirect
3. C24×C22⋊D53C2semidirect
4. C23×D4×D53C2direct (N × H)
5. C24×C4×D53C2semidirect
6. C24×D4×C53C2semidirect
9. C25×D53C22semidirect
10. C25×D53C22semidirect
11. C23×D212C22semidirect
12. C23×C22⋊D53C22semidirect
13. C22×D4×D53C22direct (N × H)
14. C24×Dic53C22semidirect
15. C23×C4×D53C22semidirect
16. C26×C53C22semidirect
17. C23×D4×C53C22semidirect
18. C24×C4×C53C22semidirect
21. C24×D53D4direct (N × H)
23. C24×D53C23semidirect
24. C24×D53C23semidirect
25. C22×D212C23semidirect
26. C22×C22⋊D53C23semidirect
27. C2×D4×D53C23direct (N × H)
28. C23×Dic53C23semidirect
29. C22×C4×D53C23semidirect
32. C25×C53C23semidirect
33. C22×D4×C53C23semidirect
34. C23×C4×C53C23semidirect
36. C24×C53C2×D4semidirect
38. C24×C53C24semidirect
41. C23×D53C2×D4direct (N × H)
43. C23×D53C24semidirect
44. C23×D53C24semidirect
45. C2×D4×C53C24semidirect
46. C22×C4×C53C24semidirect
47. C2×C22⋊D53C24semidirect
48. C22×Dic53C24semidirect
49. D4×D53C24direct (N × H)
50. C2×D212C24semidirect
51. C2×C4×D53C24semidirect
53. C23×C53C22×D4semidirect
55. C23×C53C25semidirect
58. C22×D53C22×D4direct (N × H)
60. C22×D53C25semidirect
61. C22×D53C25semidirect
62. D4×C53C25semidirect
63. C2×C4×C53C25semidirect
64. C22⋊D53C25semidirect
65. C2×Dic53C25semidirect
66. D212C25semidirect
67. C4×D53C25semidirect
70. C22×C53C23×D4semidirect
72. C22×C53C26semidirect
75. C2×D53C23×D4direct (N × H)
77. C2×D53C26semidirect
78. C4×C53C26semidirect
79. Dic53C26semidirect
84. C24×D4D53direct (N × H)
87. C2×C53C24×D4semidirect
90. D53C24×D4direct (N × H)
95. C26C2×D53semidirect
99. C23×D4C2×D53direct (N × H)
102. C24×C4C2×D53semidirect
103. C53C25×D4semidirect
112. C25C22×D53semidirect
116. C22×D4C22×D53direct (N × H)
119. C23×C4C22×D53semidirect
127. C24D4×D53direct (N × H)
129. C24C23×D53semidirect
133. C2×D4C23×D53direct (N × H)
136. C22×C4C23×D53semidirect
144. C23C2×D4×D53direct (N × H)
146. C23C24×D53semidirect
149. D4C24×D53direct (N × H)
152. C2×C4C24×D53semidirect
159. C22C22×D4×D53direct (N × H)
161. C22C25×D53semidirect
163. C4C25×D53semidirect
167. C2C23×D4×D53direct (N × H)

Maximal subgroups

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

subgroupindex
C26×D532
C24×D2122
C24×C22⋊D532
C23×D4×D532
C24×C4×D532
C24×D4×C532
C25×D453