Finite Group Databasebeta

← C22×C32×#157 #13049875

Subgroups — C22×C32×#157 #13049875

Order 1296.

736 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 192 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×C32×C3⋊D3C211yesmaximal, characteristic
2C2×C32×#157C216yesmaximal, direct product factor
3C22×C3×#157C314yesmaximal, direct product factor
4C22×C34C411yescharacteristic, complemented
5C2×C32×C3⋊D3C413yescomplemented
6C2×C32×C3⋊D3C2213yes—
7C32×#157C2214yesdirect product factor
8C22×C3×C3⋊D3C2×C314yes—
9C22×C32×D3—62no—
10C2×C3×#157C2×C3124yesdirect product factor
11C2×C34C2×C413yescomplemented
12C32×C3⋊D3C2×C413yescomplemented
13C32×C3⋊D3C2311yescharacteristic
14C22×#157C3211yescharacteristic, direct product factor
15C22×C32×C4—91nomaximal
16C22×C33—22no—
17C22×C33—48no—
18C22×C33C3×C414yescomplemented
19C2×C3×C3⋊D3C3×C4112yescomplemented
20C2×C3×C3⋊D3C22×C3112yes—
21C2×C32×D3—66no—
22C2×C32×D3—66no—
23C3×#157C22×C3116yesdirect product factor
24C34C22×C411yescharacteristic, complemented, Sylow
25C23×C32—91no—
26C22×C3⋊D3C2×C3211yescharacteristic
27C22×C3×D3—68no—
28C2×#157C2×C3216yesdirect product factor
29C2×C32×C4—96no—
30C2×C33—26no—
31C2×C33—424no—
32C2×C33C2×C3×C4112yescomplemented
33C3×C3⋊D3C2×C3×C4112yescomplemented
34C3×C3⋊D3C23×C314yes—
35C32×D3—66no—
36C32×D3—62no—
37C22×C3×C4—94no—
38C22×C32—28no—
39C22×C32—48no—
40C22×C32—412no—
41C22×C32—48no—
42C22×C32—93no—
43C22×C32—93no—
44C22×C32C32×C411yescharacteristic, complemented
45C22×C32#15711yescharacteristic, direct product factor
46C2×C3⋊D3C32×C413yescomplemented
47C2×C3⋊D3C22×C3213yes—
48C2×C3×D3—624no—
49C2×C3×D3—624no—
50#157C22×C3214yesdirect product factor
51C32×C4—94no—
52C33—22no—
53C33—48no—
54C33C22×C3×C414yescomplemented
55C23×C3—94no—
56C22×D3—62no—
57C2×C3×C4—924no—
58C2×C32—224no—
59C2×C32—424no—
60C2×C32—436no—
61C2×C32—424no—
62C2×C32—91no—
63C2×C32—93no—
64C2×C32C2×C32×C413yescomplemented
65C2×C32C2×#15713yesdirect product factor
66C3⋊D3C2×C32×C413yescomplemented
67C3⋊D3C23×C3211yescharacteristic
68C3×D3—624no—
69C3×D3—68no—
70C22×C4—91noSylow
71C22×C3—22no—
72C22×C3—48no—
73C22×C3—912no—
74C22×C3—912no—
75C22×C3C3×#15714yesdirect product factor
76C2×D3—66no—
77C2×D3—66no—
78C3×C4—916no—
79C32—28no—
80C32—48no—
81C32—412no—
82C32—48no—
83C32C22×C32×C411yescharacteristic, complemented
84C32C22×#15711yescharacteristic, direct product factor
85C23—91no—
86C2×C4—96no—
87C2×C3—26no—
88C2×C3—424no—
89C2×C3—94no—
90C2×C3—912no—
91C2×C3C2×C3×#157112yesdirect product factor
92D3—66no—
93D3—62no—
94C22—93no—
95C22—93no—
96C22C32×#15711yescharacteristic, direct product factor
97C4—94no—
98C3—22no—
99C3—48no—
100C3C22×C3×#15714yesdirect product factor
101C2—91no—
102C2—93no—
103C2C2×C32×#15713yesdirect product factor
subgroupG/H
center, hypercenterC22×C32#157
commutator, hypocenterC32C22×C32×C4
FrattiniC1G
Fitting, socleC22×C34C4
prime radicalC22×C32×C3⋊D3C2
cosocleC3⋊D3C23×C32
LCSG ▷ C32
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
2. C2×C32×#157C2direct (N × H)
3. C22×C3×#157C3direct (N × H)
4. C22×C34C4semidirect
5. C2×C32×C3⋊D3C4semidirect
7. C32×#157C22direct (N × H)
10. C2×C3×#157C2×C3direct (N × H)
11. C2×C34C2×C4semidirect
12. C32×C3⋊D3C2×C4semidirect
14. C22×#157C32direct (N × H)
18. C22×C33C3×C4semidirect
19. C2×C3×C3⋊D3C3×C4semidirect
23. C3×#157C22×C3direct (N × H)
24. C34C22×C4semidirect
28. C2×#157C2×C32direct (N × H)
32. C2×C33C2×C3×C4semidirect
33. C3×C3⋊D3C2×C3×C4semidirect
44. C22×C32C32×C4semidirect
45. C22×C32#157direct (N × H)
46. C2×C3⋊D3C32×C4semidirect
50. #157C22×C32direct (N × H)
54. C33C22×C3×C4semidirect
64. C2×C32C2×C32×C4semidirect
65. C2×C32C2×#157direct (N × H)
66. C3⋊D3C2×C32×C4semidirect
75. C22×C3C3×#157direct (N × H)
83. C32C22×C32×C4semidirect
84. C32C22×#157direct (N × H)
91. C2×C3C2×C3×#157direct (N × H)
96. C22C32×#157direct (N × H)
100. C3C22×C3×#157direct (N × H)
103. C2C2×C32×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C3⋊D32
C2×C32×#1572
C22×C3×#1573
C22×C32×C49