Finite Group Databasebeta

← #120×D353 #36378271

Subgroups — #120×D353 #36378271

Order 22592.

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×D353C211yesmaximal, characteristic, complemented
2C2×D4×D353C213yesmaximal, complemented
3#26014962C213yesmaximal, complemented
4#26014961C211yesmaximal, characteristic, complemented
5#26014960C213yesmaximal, complemented
6C22⋊C4×D353C213yesmaximal, complemented
7#120×C353C211yesmaximal, characteristic, complemented
8C23×D353—212no—
9C23×D353C2213yes—
10C23×D353C2213yescomplemented
11C23×D353C2211yescharacteristic, complemented
12C2×D1412C2213yescomplemented
13C2×C22⋊D353C2213yescomplemented
14C2×C22⋊D353C2213yescomplemented
15D4×D353—212no—
16C2×C4×D353C2213yescomplemented
17C22⋊Dic353C2213yescomplemented
18#19150714C2216yescomplemented
19C24×C353C2211yescharacteristic, complemented
20C2×D4×C353C2213yescomplemented
21C22⋊C4×C353C2213yescomplemented
22C22×D353—232no—
23C22×D353—26no—
24C22×D353—212no—
25C22×D353—212no—
26C22×D353—26no—
27C22×D353D4112yescomplemented
28C22×D353C2313yes—
29C22×D353C2311yescharacteristic
30C22×D353C2311yescharacteristic, complemented
31D1412—26no—
32C22⋊D353—26no—
33C22⋊D353—26no—
34C2×Dic353C2313yescomplemented
35C4×D353—26no—
36C23×C353—26no—
37C23×C353C2313yes—
38C23×C353C2311yescharacteristic, complemented
39D4×C353—26no—
40C2×C4×C353C2313yescomplemented
41C22×C353—28no—
42C22×C353—26no—
43C22×C353—23no—
44C22×C353C2×D416yescomplemented
45C22×C353C2411yescharacteristic
46C2×D353—224no—
47C2×D353—26no—
48C2×D353—224no—
49C2×D353—23no—
50C2×D353—44no—
51C2×D353C2×D416yes—
52C2×D353C2×D416yescomplemented
53C4×C353—23no—
54Dic353—23no—
55C2×C353—26no—
56C2×C353—41no—
57C2×C353C22×D413yes—
58D353—26no—
59D353—41no—
60D353#12014yesdirect product factor
61C353C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—3531nomaximal, Sylow
63C25—3531no—
64C22×D4—3533no—
65#120—3531no—
66#120—3533no—
67#120—3533no—
68#120D35311yescharacteristic, direct product factor
69C2×C22⋊C4—3533no—
70C24—3533no—
71C24—3533no—
72C24—3531no—
73C24—70612no—
74C24C2×D35311yescharacteristic, complemented
75C2×D4—3533no—
76C2×D4—3533no—
77C2×D4—3533no—
78C2×D4—70612no—
79C2×D4C2×D35313yescomplemented
80C22×C4—3533no—
81C22⋊C4—3533no—
82C22⋊C4—3536no—
83C22⋊C4C2×D35313yescomplemented
84C23—26no—
85C23—3531no—
86C23—35312no—
87C23—3533no—
88C23—3531no—
89C23—7066no—
90C23—70612no—
91C23—70612no—
92C23—70632no—
93C23—7066no—
94C23C22×D35313yes—
95C23C22×D35311yescharacteristic, complemented
96D4—26no—
97D4—7066no—
98D4—7066no—
99D4—7066no—
100C2×C4—3533no—
101C2×C4—7066no—
102C2×C4C22×D35313yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—3536no—
107C22—3536no—
108C22—14124no—
109C22—70624no—
110C22—70624no—
111C22—7063no—
112C22—7066no—
113C22D4×D35316yescomplemented
114C22C23×D35311yescharacteristic
115C4—23no—
116C4—7063no—
117C2—26no—
118C2—41no—
119C2—3534no—
120C2—14121no—
121C2—7066no—
122C2C2×D4×D35313yes—
subgroupG/H
center, FrattiniC22C23×D353
commutator, socle, cosocleC22×C353C24
Fitting#120×C353C2
hypercenter#120D353
hypocenterC353C2×#120
LCSG ▷ C22×C353 ▷ C353
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D353C2semidirect
2. C2×D4×D353C2semidirect
3. #26014962C2semidirect
4. #26014961C2semidirect
5. #26014960C2semidirect
6. C22⋊C4×D353C2semidirect
7. #120×C353C2semidirect
10. C23×D353C22semidirect
11. C23×D353C22semidirect
12. C2×D1412C22semidirect
13. C2×C22⋊D353C22semidirect
14. C2×C22⋊D353C22semidirect
16. C2×C4×D353C22semidirect
17. C22⋊Dic353C22semidirect
18. #19150714C22semidirect
19. C24×C353C22semidirect
20. C2×D4×C353C22semidirect
21. C22⋊C4×C353C22semidirect
27. C22×D353D4semidirect
30. C22×D353C23semidirect
34. C2×Dic353C23semidirect
38. C23×C353C23semidirect
40. C2×C4×C353C23semidirect
44. C22×C353C2×D4semidirect
52. C2×D353C2×D4semidirect
60. D353#120direct (N × H)
61. C353C2×#120semidirect
68. #120D353direct (N × H)
74. C24C2×D353semidirect
79. C2×D4C2×D353semidirect
83. C22⋊C4C2×D353semidirect
95. C23C22×D353semidirect
102. C2×C4C22×D353semidirect
113. C22D4×D353semidirect

Maximal subgroups

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

subgroupindex
C24×D3532
C2×D4×D3532
#260149622
#260149612
#260149602
C22⋊C4×D3532
#120×C3532
C2×#120353