Finite Group Databasebeta

← C23×#13133186 #28290723

Subgroups — C23×#13133186 #28290723

Order 13568.

42862 conjugacy classes of subgroups; only the 5451 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 5451 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
1C24×C22⋊D53C213yesmaximal, complemented
2C22×#13133186C2156yesmaximal, direct product factor
3C23×C22⋊Dic53C213yesmaximal, complemented
4C27×C53C211yesmaximal, characteristic, complemented
5C25×D53C2211yescharacteristic, complemented
6C23×C22⋊D53—248no—
7C23×C22⋊D53C22184yescomplemented
8C24×Dic53C2213yescomplemented
9C2×#13133186C221448yesdirect product factor
10C22×C22⋊Dic53C22184yescomplemented
11C26×C53—248no—
12C26×C53C2213yes—
13C26×C53C22128yescomplemented
14C24×D53—224no—
15C24×D53C23114yescomplemented
16C22×C22⋊D53—2672no—
17C22×C22⋊D53C231336yescomplemented
18C23×Dic53—224no—
19C23×Dic53C23142yescomplemented
20#13133186C231512yesdirect product factor
21C2×C22⋊Dic53C231336yescomplemented
22C25×C53—2512no—
23C25×C53—248no—
24C25×C53—2672no—
25C25×C53D4148yescomplemented
26C25×C53C23142yes—
27C25×C53C2311yescharacteristic
28C25×C53C231112yescomplemented
29C24×C53—23584no—
30C24×C53—2384no—
31C24×C53—2336no—
32C24×C53—21344no—
33C24×C53C2×D4124yes—
34C24×C53C2×D41336yescomplemented
35C24×C53C24184yes—
36C24×C53C2417yes—
37C24×C53C24164yescomplemented
38C23×D53—2168no—
39C23×D53—464no—
40C23×D53C24128yescomplemented
41C2×C22⋊D53—21344no—
42C2×C22⋊D53C241192yescomplemented
43C22×Dic53—2168no—
44C22×Dic53C24184yescomplemented
45C22⋊Dic53C241192yescomplemented
46C23×C53—23584no—
47C23×C53—21344no—
48C23×C53—2336no—
49C23×C53—2384no—
50C23×C53#120164yescomplemented
51C23×C53C22×D4184yes—
52C23×C53C22×D41336yescomplemented
53C23×C53C25124yes—
54C23×C53C2517yes—
55C22×D53—2168no—
56C22×D53—4224no—
57C22×D53C2518yescomplemented
58C22⋊D53—2384no—
59C2×Dic53—2168no—
60C2×Dic53C25124yescomplemented
61C23×#120—531nomaximal, Sylow
62C22×C53—2512no—
63C22×C53—248no—
64C22×C53—2672no—
65C22×C53C2×#1201112yescomplemented
66C22×C53C23×D4142yes—
67C22×C53C23×D4148yescomplemented
68C22×C53C2611yescharacteristic
69C2×D53—224no—
70C2×D53—4112no—
71Dic53—224no—
72C27D5311yescharacteristic, complemented
73C24×D4—533no—
74C22×#120—5356no—
75C23×C22⋊C4—533no—
76C2×C53—248no—
77C2×C53C22×#120128yescomplemented
78C2×C53C24×D413yes—
79D53—48no—
80C26—248no—
81C26—531no—
82C26C2×D5313yes—
83C26C2×D53128yescomplemented
84C23×D4—5384no—
85C23×D4—10648no—
86C24×C4—533no—
87C2×#120—53448no—
88C22×C22⋊C4—5384no—
89C53C23×#12011yescharacteristic, complemented, Sylow
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—5314no—
94C25—10624no—
95C25C22⋊D53148yescomplemented
96C25C22×D53142yes—
97C25C22×D5311yescharacteristic
98C25C22×D531112yescomplemented
99C22×D4—53336no—
100C22×D4—106672no—
101C23×C4—5342no—
102C23×C4—10624no—
103#120—53512no—
104C2×C22⋊C4—53336no—
105C24—23584no—
106C24—2384no—
107C24—2336no—
108C24—21344no—
109C24—5328no—
110C24—106168no—
111C24—21264no—
112C24C2×C22⋊D53124yes—
113C24C2×C22⋊D531336yescomplemented
114C24C23×D53184yes—
115C24C23×D5317yes—
116C24C23×D53164yescomplemented
117C2×D4—53192no—
118C2×D4—1061344no—
119C22×C4—5384no—
120C22×C4—106168no—
121C22⋊C4—53192no—
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—538no—
127C23—106168no—
128C23—212224no—
129C23#13133186164yesdirect product factor
130C23C22×C22⋊D53184yes—
131C23C22×C22⋊D531336yescomplemented
132C23C24×D53124yes—
133C23C24×D5317yes—
134D4—106384no—
135C2×C4—5324no—
136C2×C4—106168no—
137C22—2512no—
138C22—2672no—
139C22—248no—
140C22—10624no—
141C22—212112no—
142C22C2×#131331861112yesdirect product factor
143C22C23×C22⋊D53142yes—
144C22C23×C22⋊D53148yescomplemented
145C22C25×D5311yescharacteristic
146C4—10624no—
147C2—248no—
148C2—2128no—
149C2C22×#13133186128yesdirect product factor
150C2C24×C22⋊D5313yes—
subgroupG/H
centerC25C22×D53
commutator, cosocleC22×C53C26
FrattiniC22C25×D53
FittingC27×C53C2
socleC25×C53C23
hypercenterC27D53
hypocenterC53C23×#120
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C25 ◁ C27

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊D53C2semidirect
2. C22×#13133186C2direct (N × H)
3. C23×C22⋊Dic53C2semidirect
4. C27×C53C2semidirect
5. C25×D53C22semidirect
7. C23×C22⋊D53C22semidirect
8. C24×Dic53C22semidirect
9. C2×#13133186C22direct (N × H)
10. C22×C22⋊Dic53C22semidirect
13. C26×C53C22semidirect
15. C24×D53C23semidirect
17. C22×C22⋊D53C23semidirect
19. C23×Dic53C23semidirect
20. #13133186C23direct (N × H)
21. C2×C22⋊Dic53C23semidirect
25. C25×C53D4semidirect
28. C25×C53C23semidirect
34. C24×C53C2×D4semidirect
37. C24×C53C24semidirect
40. C23×D53C24semidirect
42. C2×C22⋊D53C24semidirect
44. C22×Dic53C24semidirect
45. C22⋊Dic53C24semidirect
50. C23×C53#120semidirect
52. C23×C53C22×D4semidirect
57. C22×D53C25semidirect
60. C2×Dic53C25semidirect
65. C22×C53C2×#120semidirect
67. C22×C53C23×D4semidirect
72. C27D53semidirect
77. C2×C53C22×#120semidirect
83. C26C2×D53semidirect
89. C53C23×#120semidirect
95. C25C22⋊D53semidirect
98. C25C22×D53semidirect
113. C24C2×C22⋊D53semidirect
116. C24C23×D53semidirect
129. C23#13133186direct (N × H)
131. C23C22×C22⋊D53semidirect
142. C22C2×#13133186direct (N × H)
144. C22C23×C22⋊D53semidirect
149. C2C22×#13133186direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊D532
C22×#131331862
C23×C22⋊Dic532
C27×C532
C23×#12053