Finite Group Databasebeta

← C22×#120×C227 #44397387

Subgroups — C22×#120×C227 #44397387

Order 29056.

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×C227C211yesmaximal, characteristic, complemented
2C23×D4×C227C213yesmaximal, complemented
3C2×#120×C227C2124yesmaximal, direct product factor
4C22×C22⋊C4×C227C213yesmaximal, complemented
5C25×C227—224no—
6C25×C227C2213yes—
7C25×C227C22112yescomplemented
8C25×C227C2211yescharacteristic, complemented
9C22×D4×C227—224no—
10C22×D4×C227C22136yescomplemented
11C23×C4×C227C2213yescomplemented
12#120×C227C22164yesdirect product factor
13C2×C22⋊C4×C227C22136yescomplemented
14C24×C227—2128no—
15C24×C227—224no—
16C24×C227—2144no—
17C24×C227—212no—
18C24×C227D4124yescomplemented
19C24×C227C23118yes—
20C24×C227C2311yescharacteristic
21C24×C227C23116yescomplemented
22C24×C227C2316yescomplemented
23C2×D4×C227—2144no—
24C2×D4×C227C23148yescomplemented
25C22×C4×C227—212no—
26C22×C4×C227C23118yescomplemented
27C22⋊C4×C227C23148yescomplemented
28C23×C227—2384no—
29C23×C227—296no—
30C23×C227—272no—
31C23×C227—296no—
32C23×C227—236no—
33C23×C227—416no—
34C23×C227C2×D4112yes—
35C23×C227C2×D4172yescomplemented
36C23×C227C24112yes—
37C23×C227C2413yes—
38C23×C227C2414yescomplemented
39D4×C227—296no—
40C2×C4×C227—236no—
41C2×C4×C227C24112yescomplemented
42C22×C227—2128no—
43C22×C227—2144no—
44C22×C227—224no—
45C22×C227—212no—
46C22×C227—424no—
47C22×C227#120116yesdirect product factor
48C22×C227C22×D4118yes—
49C22×C227C22×D4124yescomplemented
50C22×C227C2511yescharacteristic
51C4×C227—212no—
52C2×C227—224no—
53C2×C227—44no—
54C2×C227C2×#120112yesdirect product factor
55C2×C227C23×D413yes—
56C227C22×#12011yescharacteristic, direct product factor, Sylow
57C22×#120C22711yesmaximal, characteristic, direct product factor, Sylow
58C26C2×C22711yescharacteristic, complemented
59C23×D4C2×C22713yescomplemented
60C2×#120C2×C227124yesdirect product factor
61C22×C22⋊C4C2×C22713yescomplemented
62C25—224no—
63C25C22×C22713yes—
64C25C22×C227112yescomplemented
65C25C22×C22711yescharacteristic, complemented
66C22×D4—224no—
67C22×D4C22×C227136yescomplemented
68C23×C4C22×C22713yescomplemented
69#120C22×C227164yesdirect product factor
70C2×C22⋊C4C22×C227136yescomplemented
71C24—2128no—
72C24—224no—
73C24—2144no—
74C24—212no—
75C24D4×C227124yescomplemented
76C24C23×C227118yes—
77C24C23×C22711yescharacteristic
78C24C23×C227116yescomplemented
79C24C23×C22716yescomplemented
80C2×D4—2144no—
81C2×D4C23×C227148yescomplemented
82C22×C4—212no—
83C22×C4C23×C227118yescomplemented
84C22⋊C4C23×C227148yescomplemented
85C23—2384no—
86C23—296no—
87C23—272no—
88C23—296no—
89C23—236no—
90C23—416no—
91C23C2×D4×C227112yes—
92C23C2×D4×C227172yescomplemented
93C23C24×C227112yes—
94C23C24×C22713yes—
95C23C24×C22714yescomplemented
96D4—296no—
97C2×C4—236no—
98C2×C4C24×C227112yescomplemented
99C22—2128no—
100C22—2144no—
101C22—224no—
102C22—212no—
103C22—424no—
104C22#120×C227116yesdirect product factor
105C22C22×D4×C227118yes—
106C22C22×D4×C227124yescomplemented
107C22C25×C22711yescharacteristic
108C4—212no—
109C2—224no—
110C2—44no—
111C2C2×#120×C227112yesdirect product factor
112C2C23×D4×C22713yes—
subgroupG/H
center, socleC24×C227C23
commutator, Frattini, cosocleC22C25×C227
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. C26×C227C2semidirect
2. C23×D4×C227C2semidirect
3. C2×#120×C227C2direct (N × H)
4. C22×C22⋊C4×C227C2semidirect
7. C25×C227C22semidirect
8. C25×C227C22semidirect
10. C22×D4×C227C22semidirect
11. C23×C4×C227C22semidirect
12. #120×C227C22direct (N × H)
13. C2×C22⋊C4×C227C22semidirect
18. C24×C227D4semidirect
21. C24×C227C23semidirect
22. C24×C227C23semidirect
24. C2×D4×C227C23semidirect
26. C22×C4×C227C23semidirect
27. C22⋊C4×C227C23semidirect
35. C23×C227C2×D4semidirect
38. C23×C227C24semidirect
41. C2×C4×C227C24semidirect
47. C22×C227#120direct (N × H)
49. C22×C227C22×D4semidirect
54. C2×C227C2×#120direct (N × H)
56. C227C22×#120direct (N × H)
57. C22×#120C227direct (N × H)
58. C26C2×C227semidirect
59. C23×D4C2×C227semidirect
60. C2×#120C2×C227direct (N × H)
61. C22×C22⋊C4C2×C227semidirect
64. C25C22×C227semidirect
65. C25C22×C227semidirect
67. C22×D4C22×C227semidirect
68. C23×C4C22×C227semidirect
69. #120C22×C227direct (N × H)
70. C2×C22⋊C4C22×C227semidirect
75. C24D4×C227semidirect
78. C24C23×C227semidirect
79. C24C23×C227semidirect
81. C2×D4C23×C227semidirect
83. C22×C4C23×C227semidirect
84. C22⋊C4C23×C227semidirect
92. C23C2×D4×C227semidirect
95. C23C24×C227semidirect
98. C2×C4C24×C227semidirect
104. C22#120×C227direct (N × H)
106. C22C22×D4×C227semidirect
111. C2C2×#120×C227direct (N × H)

Maximal subgroups

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

subgroupindex
C26×C2272
C23×D4×C2272
C2×#120×C2272
C22×C22⋊C4×C2272
C22×#120227