Finite Group Databasebeta

← #120×D421 #41003324

Subgroups — #120×D421 #41003324

Order 26944.

662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C24×D421C211yesmaximal, characteristic, complemented
2C2×D4×D421C213yesmaximal, complemented
3#27212336C213yesmaximal, complemented
4#27212335C211yesmaximal, characteristic, complemented
5#27212334C213yesmaximal, complemented
6C22⋊C4×D421C213yesmaximal, complemented
7#120×C421C211yesmaximal, characteristic, complemented
8C23×D421—212no—
9C23×D421C2213yes—
10C23×D421C2213yescomplemented
11C23×D421C2211yescharacteristic, complemented
12C2×D1684C2213yescomplemented
13C2×C22⋊D421C2213yescomplemented
14C2×C22⋊D421C2213yescomplemented
15D4×D421—212no—
16C2×C4×D421C2213yescomplemented
17C22⋊Dic421C2213yescomplemented
18#20269748C2216yescomplemented
19C24×C421C2211yescharacteristic, complemented
20C2×D4×C421C2213yescomplemented
21C22⋊C4×C421C2213yescomplemented
22C22×D421—232no—
23C22×D421—26no—
24C22×D421—212no—
25C22×D421—212no—
26C22×D421—26no—
27C22×D421D4112yescomplemented
28C22×D421C2313yes—
29C22×D421C2311yescharacteristic
30C22×D421C2311yescharacteristic, complemented
31D1684—26no—
32C22⋊D421—26no—
33C22⋊D421—26no—
34C2×Dic421C2313yescomplemented
35C4×D421—26no—
36C23×C421—26no—
37C23×C421C2313yes—
38C23×C421C2311yescharacteristic, complemented
39D4×C421—26no—
40C2×C4×C421C2313yescomplemented
41C22×C421—28no—
42C22×C421—26no—
43C22×C421—23no—
44C22×C421C2×D416yescomplemented
45C22×C421C2411yescharacteristic
46C2×D421—224no—
47C2×D421—26no—
48C2×D421—224no—
49C2×D421—23no—
50C2×D421—44no—
51C2×D421C2×D416yes—
52C2×D421C2×D416yescomplemented
53C4×C421—23no—
54Dic421—23no—
55C2×C421—26no—
56C2×C421—41no—
57C2×C421C22×D413yes—
58D421—26no—
59D421—41no—
60D421#12014yesdirect product factor
61C421C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—4211nomaximal, Sylow
63C25—4211no—
64C22×D4—4213no—
65#120—4211no—
66#120—4213no—
67#120—4213no—
68#120D42111yescharacteristic, direct product factor
69C2×C22⋊C4—4213no—
70C24—84212no—
71C24—4213no—
72C24—4211no—
73C24—4213no—
74C24C2×D42111yescharacteristic, complemented
75C2×D4—84212no—
76C2×D4—4213no—
77C2×D4—4213no—
78C2×D4—4213no—
79C2×D4C2×D42113yescomplemented
80C22×C4—4213no—
81C22⋊C4—4213no—
82C22⋊C4—4216no—
83C22⋊C4C2×D42113yescomplemented
84C23—26no—
85C23—84232no—
86C23—84212no—
87C23—84212no—
88C23—8426no—
89C23—8426no—
90C23—4211no—
91C23—42112no—
92C23—4213no—
93C23—4211no—
94C23C22×D42113yes—
95C23C22×D42111yescharacteristic, complemented
96D4—26no—
97D4—8426no—
98D4—8426no—
99D4—8426no—
100C2×C4—8426no—
101C2×C4—4213no—
102C2×C4C22×D42113yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—84224no—
107C22—84224no—
108C22—8426no—
109C22—8423no—
110C22—16844no—
111C22—4216no—
112C22—4216no—
113C22D4×D42116yescomplemented
114C22C23×D42111yescharacteristic
115C4—23no—
116C4—8423no—
117C2—26no—
118C2—41no—
119C2—8426no—
120C2—16841no—
121C2—4214no—
122C2C2×D4×D42113yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#120×C421C2
hypercenter#120D421
hypocenterC421C2×#120
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D421C2semidirect
2. C2×D4×D421C2semidirect
3. #27212336C2semidirect
4. #27212335C2semidirect
5. #27212334C2semidirect
6. C22⋊C4×D421C2semidirect
7. #120×C421C2semidirect
10. C23×D421C22semidirect
11. C23×D421C22semidirect
12. C2×D1684C22semidirect
13. C2×C22⋊D421C22semidirect
14. C2×C22⋊D421C22semidirect
16. C2×C4×D421C22semidirect
17. C22⋊Dic421C22semidirect
18. #20269748C22semidirect
19. C24×C421C22semidirect
20. C2×D4×C421C22semidirect
21. C22⋊C4×C421C22semidirect
27. C22×D421D4semidirect
30. C22×D421C23semidirect
34. C2×Dic421C23semidirect
38. C23×C421C23semidirect
40. C2×C4×C421C23semidirect
44. C22×C421C2×D4semidirect
52. C2×D421C2×D4semidirect
60. D421#120direct (N × H)
61. C421C2×#120semidirect
68. #120D421direct (N × H)
74. C24C2×D421semidirect
79. C2×D4C2×D421semidirect
83. C22⋊C4C2×D421semidirect
95. C23C22×D421semidirect
102. C2×C4C22×D421semidirect
113. C22D4×D421semidirect

Maximal subgroups

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

subgroupindex
C24×D4212
C2×D4×D4212
#272123362
#272123352
#272123342
C22⋊C4×D4212
#120×C4212
C2×#120421