Finite Group Databasebeta

← C22×C23×#120 #15671050

Subgroups — C22×C23×#120 #15671050

Order 2944.

4544 conjugacy classes of subgroups; only the 1112 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 1112 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×C23C211yesmaximal, characteristic, complemented
2C23×D4×C23C213yesmaximal, complemented
3C2×C23×#120C2124yesmaximal, direct product factor
4C22×C22⋊C4×C23C213yesmaximal, complemented
5C25×C23—224no—
6C25×C23C2213yes—
7C25×C23C22112yescomplemented
8C25×C23C2211yescharacteristic, complemented
9C22×D4×C23—224no—
10C22×D4×C23C22136yescomplemented
11C23×C4×C23C2213yescomplemented
12C23×#120C22164yesdirect product factor
13C2×C22⋊C4×C23C22136yescomplemented
14C24×C23—2128no—
15C24×C23—224no—
16C24×C23—2144no—
17C24×C23—212no—
18C24×C23D4124yescomplemented
19C24×C23C23118yes—
20C24×C23C2311yescharacteristic
21C24×C23C23116yescomplemented
22C24×C23C2316yescomplemented
23C2×D4×C23—2144no—
24C2×D4×C23C23148yescomplemented
25C22×C4×C23—212no—
26C22×C4×C23C23118yescomplemented
27C22⋊C4×C23C23148yescomplemented
28C23×C23—2384no—
29C23×C23—296no—
30C23×C23—272no—
31C23×C23—296no—
32C23×C23—236no—
33C23×C23—416no—
34C23×C23C2×D4112yes—
35C23×C23C2×D4172yescomplemented
36C23×C23C24112yes—
37C23×C23C2413yes—
38C23×C23C2414yescomplemented
39D4×C23—296no—
40C2×C4×C23—236no—
41C2×C4×C23C24112yescomplemented
42C22×#120C2311yesmaximal, characteristic, direct product factor, Sylow
43C22×C23—2128no—
44C22×C23—2144no—
45C22×C23—224no—
46C22×C23—212no—
47C22×C23—424no—
48C22×C23#120116yesdirect product factor
49C22×C23C22×D4118yes—
50C22×C23C22×D4124yescomplemented
51C22×C23C2511yescharacteristic
52C4×C23—212no—
53C26C2×C2311yescharacteristic, complemented
54C23×D4C2×C2313yescomplemented
55C2×#120C2×C23124yesdirect product factor
56C22×C22⋊C4C2×C2313yescomplemented
57C2×C23—224no—
58C2×C23—44no—
59C2×C23C2×#120112yesdirect product factor
60C2×C23C23×D413yes—
61C25—224no—
62C25C22×C2313yes—
63C25C22×C23112yescomplemented
64C25C22×C2311yescharacteristic, complemented
65C22×D4—224no—
66C22×D4C22×C23136yescomplemented
67C23×C4C22×C2313yescomplemented
68#120C22×C23164yesdirect product factor
69C2×C22⋊C4C22×C23136yescomplemented
70C23C22×#12011yescharacteristic, direct product factor, Sylow
71C24—2128no—
72C24—224no—
73C24—2144no—
74C24—212no—
75C24D4×C23124yescomplemented
76C24C23×C23118yes—
77C24C23×C2311yescharacteristic
78C24C23×C23116yescomplemented
79C24C23×C2316yescomplemented
80C2×D4—2144no—
81C2×D4C23×C23148yescomplemented
82C22×C4—212no—
83C22×C4C23×C23118yescomplemented
84C22⋊C4C23×C23148yescomplemented
85C23—2384no—
86C23—296no—
87C23—272no—
88C23—296no—
89C23—236no—
90C23—416no—
91C23C2×D4×C23112yes—
92C23C2×D4×C23172yescomplemented
93C23C24×C23112yes—
94C23C24×C2313yes—
95C23C24×C2314yescomplemented
96D4—296no—
97C2×C4—236no—
98C2×C4C24×C23112yescomplemented
99C22—2128no—
100C22—224no—
101C22—2144no—
102C22—212no—
103C22—424no—
104C22C23×#120116yesdirect product factor
105C22C22×D4×C23118yes—
106C22C22×D4×C23124yescomplemented
107C22C25×C2311yescharacteristic
108C4—212no—
109C2—224no—
110C2—44no—
111C2C2×C23×#120112yesdirect product factor
112C2C23×D4×C2313yes—
subgroupG/H
center, socleC24×C23C23
commutator, Frattini, cosocleC22C25×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C26×C23C2semidirect
2. C23×D4×C23C2semidirect
3. C2×C23×#120C2direct (N × H)
4. C22×C22⋊C4×C23C2semidirect
7. C25×C23C22semidirect
8. C25×C23C22semidirect
10. C22×D4×C23C22semidirect
11. C23×C4×C23C22semidirect
12. C23×#120C22direct (N × H)
13. C2×C22⋊C4×C23C22semidirect
18. C24×C23D4semidirect
21. C24×C23C23semidirect
22. C24×C23C23semidirect
24. C2×D4×C23C23semidirect
26. C22×C4×C23C23semidirect
27. C22⋊C4×C23C23semidirect
35. C23×C23C2×D4semidirect
38. C23×C23C24semidirect
41. C2×C4×C23C24semidirect
42. C22×#120C23direct (N × H)
48. C22×C23#120direct (N × H)
50. C22×C23C22×D4semidirect
53. C26C2×C23semidirect
54. C23×D4C2×C23semidirect
55. C2×#120C2×C23direct (N × H)
56. C22×C22⋊C4C2×C23semidirect
59. C2×C23C2×#120direct (N × H)
63. C25C22×C23semidirect
64. C25C22×C23semidirect
66. C22×D4C22×C23semidirect
67. C23×C4C22×C23semidirect
68. #120C22×C23direct (N × H)
69. C2×C22⋊C4C22×C23semidirect
70. C23C22×#120direct (N × H)
75. C24D4×C23semidirect
78. C24C23×C23semidirect
79. C24C23×C23semidirect
81. C2×D4C23×C23semidirect
83. C22×C4C23×C23semidirect
84. C22⋊C4C23×C23semidirect
92. C23C2×D4×C23semidirect
95. C23C24×C23semidirect
98. C2×C4C24×C23semidirect
104. C22C23×#120direct (N × H)
106. C22C22×D4×C23semidirect
111. C2C2×C23×#120direct (N × H)

Maximal subgroups

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

subgroupindex
C26×C232
C23×D4×C232
C2×C23×#1202
C22×C22⋊C4×C232
C22×#12023