Finite Group Databasebeta

← C2×#121×C227 #28408686

Subgroups — C2×#121×C227 #28408686

Order 14528.

426 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C22×D4×C227C212yesmaximal, complemented
2C22×D4×C227C211yesmaximal, characteristic, complemented
3C23×C4×C227C211yesmaximal, characteristic, complemented
4#121×C227C218yesmaximal, direct product factor
5C2×C4⋊C4×C227C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×C227C212yesmaximal, complemented
7C24×C227C2212yes—
8C24×C227C2211yescharacteristic
9C2×D4×C227—28no—
10C2×D4×C227—24no—
11C2×D4×C227C2218yescomplemented
12C2×D4×C227C2214yescomplemented
13C22×C4×C227—24no—
14C22×C4×C227C2211yescharacteristic
15C22×C4×C227C2212yes—
16C22×C4×C227C2211yescharacteristic
17C22×C4×C227C2214yescomplemented
18C4⋊C4×C227C2214yescomplemented
19C22⋊C4×C227C2218yescomplemented
20C23×C227—22no—
21C23×C227—22no—
22C23×C227—24no—
23C23×C227—24no—
24C23×C227—24no—
25C23×C227D414yescomplemented
26C23×C227C2314yes—
27C23×C227C2312yes—
28C23×C227C2311yescharacteristic
29D4×C227—216no—
30D4×C227—28no—
31C2×C4×C227—22no—
32C2×C4×C227—24no—
33C2×C4×C227—28no—
34C2×C4×C227D414yescomplemented
35C2×C4×C227C2312yes—
36C2×C4×C227C2314yes—
37C2×C4×C227C2312yes—
38C22×C227—28no—
39C22×C227—22no—
40C22×C227—22no—
41C22×C227—24no—
42C22×C227—24no—
43C22×C227—24no—
44C22×C227—48no—
45C22×C227C2×D412yes—
46C22×C227C2×D412yes—
47C22×C227C2×D414yescomplemented
48C22×C227#4112yes—
49C22×C227C2411yescharacteristic
50C4×C227—22no—
51C4×C227—24no—
52C4×C227C2×D414yescomplemented
53C2×C227—24no—
54C2×C227—44no—
55C2×C227#12114yesdirect product factor
56C2×C227C22×D411yescharacteristic
57C2×C227C22×D411yescharacteristic
58C2×C227C2×#4111yescharacteristic
59C227C2×#12111yescharacteristic, direct product factor, Sylow
60C2×#121C22711yesmaximal, characteristic, direct product factor, Sylow
61C22×D4C2×C22712yescomplemented
62C22×D4C2×C22711yescharacteristic, complemented
63C23×C4C2×C22711yescharacteristic, complemented
64#121C2×C22718yesdirect product factor
65C2×C4⋊C4C2×C22711yescharacteristic, complemented
66C2×C22⋊C4C2×C22712yescomplemented
67C24C22×C22712yes—
68C24C22×C22711yescharacteristic
69C2×D4—28no—
70C2×D4—24no—
71C2×D4C22×C22718yescomplemented
72C2×D4C22×C22714yescomplemented
73C22×C4—24no—
74C22×C4C22×C22711yescharacteristic
75C22×C4C22×C22712yes—
76C22×C4C22×C22711yescharacteristic
77C22×C4C22×C22714yescomplemented
78C4⋊C4C22×C22714yescomplemented
79C22⋊C4C22×C22718yescomplemented
80C23—22no—
81C23—22no—
82C23—24no—
83C23—24no—
84C23—24no—
85C23D4×C22714yescomplemented
86C23C23×C22714yes—
87C23C23×C22712yes—
88C23C23×C22711yescharacteristic
89D4—216no—
90D4—28no—
91C2×C4—22no—
92C2×C4—24no—
93C2×C4—28no—
94C2×C4D4×C22714yescomplemented
95C2×C4C23×C22712yes—
96C2×C4C23×C22714yes—
97C2×C4C23×C22712yes—
98C22—28no—
99C22—22no—
100C22—22no—
101C22—24no—
102C22—24no—
103C22—24no—
104C22—48no—
105C22C2×D4×C22712yes—
106C22C2×D4×C22712yes—
107C22C2×D4×C22714yescomplemented
108C22#41×C22712yes—
109C22C24×C22711yescharacteristic
110C4—22no—
111C4—24no—
112C4C2×D4×C22714yescomplemented
113C2—24no—
114C2—44no—
115C2#121×C22714yesdirect product factor
116C2C22×D4×C22711yescharacteristic
117C2C22×D4×C22711yescharacteristic
118C2C2×#41×C22711yescharacteristic
subgroupG/H
center, socleC23×C227C23
commutator, Frattini, cosocleC22C24×C227
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D4×C227C2semidirect
2. C22×D4×C227C2semidirect
3. C23×C4×C227C2semidirect
4. #121×C227C2direct (N × H)
5. C2×C4⋊C4×C227C2semidirect
6. C2×C22⋊C4×C227C2semidirect
11. C2×D4×C227C22semidirect
12. C2×D4×C227C22semidirect
17. C22×C4×C227C22semidirect
18. C4⋊C4×C227C22semidirect
19. C22⋊C4×C227C22semidirect
25. C23×C227D4semidirect
34. C2×C4×C227D4semidirect
47. C22×C227C2×D4semidirect
52. C4×C227C2×D4semidirect
55. C2×C227#121direct (N × H)
59. C227C2×#121direct (N × H)
60. C2×#121C227direct (N × H)
61. C22×D4C2×C227semidirect
62. C22×D4C2×C227semidirect
63. C23×C4C2×C227semidirect
64. #121C2×C227direct (N × H)
65. C2×C4⋊C4C2×C227semidirect
66. C2×C22⋊C4C2×C227semidirect
71. C2×D4C22×C227semidirect
72. C2×D4C22×C227semidirect
77. C22×C4C22×C227semidirect
78. C4⋊C4C22×C227semidirect
79. C22⋊C4C22×C227semidirect
85. C23D4×C227semidirect
94. C2×C4D4×C227semidirect
107. C22C2×D4×C227semidirect
112. C4C2×D4×C227semidirect
115. C2#121×C227direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×C2272
C23×C4×C2272
#121×C2272
C2×C4⋊C4×C2272
C2×C22⋊C4×C2272
C2×#121227