Finite Group Databasebeta

← #1820×C211 #41009440

Subgroups — #1820×C211 #41009440

Order 27008.

300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×#117×C211C211yesmaximal, characteristic
2#381×C211C212yesmaximal
3#342×C211C214yesmaximal
4C23×C4×C211C2211yescharacteristic
5#117×C211—24no—
6C2×C4⋊C4×C211C412yescomplemented
7C2×C22⋊C4×C211C2214yes—
8C2×C22⋊C4×C211C2212yes—
9C2×C42×C211C412yescomplemented
10#95×C211—24no—
11C24×C211C2311yescharacteristic
12C22×C4×C211—24no—
13C22×C4×C211—24no—
14C22×C4×C211C2×C411yescharacteristic
15C22×C4×C211C2×C411yescharacteristic
16C22×C4×C211C2×C414yes—
17C22×C4×C211D414yes—
18C4⋊C4×C211—24no—
19C22⋊C4×C211—24no—
20C22⋊C4×C211—28no—
21C42×C211—24no—
22C23×C211—42no—
23C23×C211C22×C411yescharacteristic
24C23×C211C2×D412yes—
25C23×C211#4114yes—
26C2×C4×C211—22no—
27C2×C4×C211—22no—
28C2×C4×C211—28no—
29C2×C4×C211—24no—
30C2×C4×C211—44no—
31C2×C4×C211—48no—
32C2×C4×C211—44no—
33C2×C4×C211C22⋊C412yes—
34C2×C4×C211C22⋊C412yes—
35C22×C211—22no—
36C22×C211—24no—
37C22×C211—44no—
38C22×C211C2×C22⋊C411yescharacteristic
39C22×C211#11712yes—
40C22×C211#12314yes—
41C4×C211—22no—
42C4×C211—22no—
43C4×C211—44no—
44C4×C211—84no—
45C2×C211—24no—
46C2×C211—22no—
47C2×C211#38111yescharacteristic
48C2×C211#41012yes—
49C211#182011yescharacteristic, direct product factor, Sylow
50#1820C21111yesmaximal, characteristic, direct product factor, Sylow
51C2×#117C2×C21111yescharacteristic
52#381C2×C21112yes—
53#342C2×C21114yes—
54C23×C4C22×C21111yescharacteristic
55#117—24no—
56C2×C4⋊C4C4×C21112yescomplemented
57C2×C22⋊C4C22×C21114yes—
58C2×C22⋊C4C22×C21112yes—
59C2×C42C4×C21112yescomplemented
60#95—24no—
61C24C23×C21111yescharacteristic
62C22×C4—24no—
63C22×C4—24no—
64C22×C4C2×C4×C21111yescharacteristic
65C22×C4C2×C4×C21111yescharacteristic
66C22×C4C2×C4×C21114yes—
67C22×C4D4×C21114yes—
68C4⋊C4—24no—
69C22⋊C4—24no—
70C22⋊C4—28no—
71C42—24no—
72C23—42no—
73C23C22×C4×C21111yescharacteristic
74C23C2×D4×C21112yes—
75C23#41×C21114yes—
76C2×C4—22no—
77C2×C4—22no—
78C2×C4—28no—
79C2×C4—24no—
80C2×C4—44no—
81C2×C4—48no—
82C2×C4—44no—
83C2×C4C22⋊C4×C21112yes—
84C2×C4C22⋊C4×C21112yes—
85C22—22no—
86C22—24no—
87C22—44no—
88C22C2×C22⋊C4×C21111yescharacteristic
89C22#117×C21112yes—
90C22#123×C21114yes—
91C4—22no—
92C4—22no—
93C4—44no—
94C4—84no—
95C2—24no—
96C2—22no—
97C2#381×C21111yescharacteristic
98C2#410×C21112yes—
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC24C23×C211
prime radicalC24×C211C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4⋊C4×C211C4semidirect
9. C2×C42×C211C4semidirect
49. C211#1820direct (N × H)
50. #1820C211direct (N × H)
56. C2×C4⋊C4C4×C211semidirect
59. C2×C42C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#117×C2112
#381×C2112
#342×C2112
#1820211