Finite Group Databasebeta

← #1297×C211 #41008917

Subgroups — #1297×C211 #41008917

Order 27008.

246 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#407×C211C211yesmaximal, characteristic
2C4×C22⋊C4×C211C212yesmaximal
3C23×C4×C211C2211yescharacteristic
4C2×M16×C211C412yescomplemented
5C2×C22⋊C4×C211—22no—
6C2×C42×C211—22no—
7C22⋊C8×C211—22no—
8#95×C211C414yescomplemented
9C24×C211D411yescharacteristic
10C22×C4×C211—24no—
11C22×C4×C211—42no—
12C22×C4×C211C2×C411yescharacteristic
13C22×C4×C211C2×C412yes—
14C22×C4×C211D412yes—
15C22×C4×C211Q811yescharacteristic
16M16×C211—42no—
17C2×C8×C211—22no—
18C22⋊C4×C211—28no—
19C42×C211—44no—
20C23×C211—24no—
21C23×C211—41no—
22C23×C211C22⋊C412yes—
23C23×C211C4⋊C411yescharacteristic
24C2×C4×C211—24no—
25C2×C4×C211—22no—
26C2×C4×C211—28no—
27C2×C4×C211—42no—
28C2×C4×C211—41no—
29C2×C4×C211—44no—
30C2×C4×C211—44no—
31C2×C4×C211C4211yescharacteristic
32C2×C4×C211C22⋊C411yescharacteristic
33C2×C4×C211C4⋊C412yes—
34C8×C211—42no—
35C22×C211—22no—
36C22×C211—22no—
37C22×C211—22no—
38C22×C211—21no—
39C22×C211—44no—
40C22×C211#9511yescharacteristic
41C22×C211#10414yes—
42C4×C211—21no—
43C4×C211—22no—
44C4×C211—48no—
45C4×C211#9912yes—
46C2×C211—24no—
47C2×C211—41no—
48C2×C211#33912yes—
49C2×C211#34211yescharacteristic
50C211#129711yescharacteristic, direct product factor, Sylow
51#1297C21111yesmaximal, characteristic, direct product factor, Sylow
52#407C2×C21111yescharacteristic
53C4×C22⋊C4C2×C21112yes—
54C23×C4C22×C21111yescharacteristic
55C2×M16C4×C21112yescomplemented
56C2×C22⋊C4—22no—
57C2×C42—22no—
58C22⋊C8—22no—
59#95C4×C21114yescomplemented
60C24D4×C21111yescharacteristic
61C22×C4—24no—
62C22×C4—42no—
63C22×C4C2×C4×C21111yescharacteristic
64C22×C4C2×C4×C21112yes—
65C22×C4D4×C21112yes—
66C22×C4Q8×C21111yescharacteristic
67M16—42no—
68C2×C8—22no—
69C22⋊C4—28no—
70C42—44no—
71C23—24no—
72C23—41no—
73C23C22⋊C4×C21112yes—
74C23C4⋊C4×C21111yescharacteristic
75C2×C4—24no—
76C2×C4—22no—
77C2×C4—28no—
78C2×C4—42no—
79C2×C4—41no—
80C2×C4—44no—
81C2×C4—44no—
82C2×C4C42×C21111yescharacteristic
83C2×C4C22⋊C4×C21111yescharacteristic
84C2×C4C4⋊C4×C21112yes—
85C8—42no—
86C22—22no—
87C22—22no—
88C22—22no—
89C22—21no—
90C22—44no—
91C22#95×C21111yescharacteristic
92C22#104×C21114yes—
93C4—21no—
94C4—22no—
95C4—48no—
96C4#99×C21112yes—
97C2—24no—
98C2—41no—
99C2#339×C21112yes—
100C2#342×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C22⋊C4
commutatorC2×C4C42×C211
Frattini, cosocleC23×C4C22×C211
socleC22×C211#95
prime radicalC24×C211D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×M16×C211C4semidirect
8. #95×C211C4semidirect
50. C211#1297direct (N × H)
51. #1297C211direct (N × H)
55. C2×M16C4×C211semidirect
59. #95C4×C211semidirect

Maximal subgroups

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

subgroupindex
#407×C2112
C4×C22⋊C4×C2112
#1297211