Finite Group Databasebeta

← #120×D193 #27135743

Subgroups — #120×D193 #27135743

Order 12352.

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×D193C211yesmaximal, characteristic, complemented
2C2×D4×D193C213yesmaximal, complemented
3#20259869C213yesmaximal, complemented
4#20259868C211yesmaximal, characteristic, complemented
5#20259867C213yesmaximal, complemented
6C22⋊C4×D193C213yesmaximal, complemented
7#120×C193C211yesmaximal, characteristic, complemented
8C23×D193—212no—
9C23×D193C2213yes—
10C23×D193C2213yescomplemented
11C23×D193C2211yescharacteristic, complemented
12C2×D772C2213yescomplemented
13C2×C22⋊D193C2213yescomplemented
14C2×C22⋊D193C2213yescomplemented
15D4×D193—212no—
16C2×C4×D193C2213yescomplemented
17C22⋊Dic193C2213yescomplemented
18#15690532C2216yescomplemented
19C24×C193C2211yescharacteristic, complemented
20C2×D4×C193C2213yescomplemented
21C22⋊C4×C193C2213yescomplemented
22C22×D193—232no—
23C22×D193—26no—
24C22×D193—212no—
25C22×D193—212no—
26C22×D193—26no—
27C22×D193D4112yescomplemented
28C22×D193C2313yes—
29C22×D193C2311yescharacteristic
30C22×D193C2311yescharacteristic, complemented
31D772—26no—
32C22⋊D193—26no—
33C22⋊D193—26no—
34C2×Dic193C2313yescomplemented
35C4×D193—26no—
36C23×C193—26no—
37C23×C193C2313yes—
38C23×C193C2311yescharacteristic, complemented
39D4×C193—26no—
40C2×C4×C193C2313yescomplemented
41C22×C193—28no—
42C22×C193—26no—
43C22×C193—23no—
44C22×C193C2×D416yescomplemented
45C22×C193C2411yescharacteristic
46C2×D193—224no—
47C2×D193—26no—
48C2×D193—224no—
49C2×D193—23no—
50C2×D193—44no—
51C2×D193C2×D416yes—
52C2×D193C2×D416yescomplemented
53C4×C193—23no—
54Dic193—23no—
55C2×C193—26no—
56C2×C193—41no—
57C2×C193C22×D413yes—
58D193—26no—
59D193—41no—
60D193#12014yesdirect product factor
61C193C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—1931nomaximal, Sylow
63C25—1931no—
64C22×D4—1933no—
65#120—1931no—
66#120—1933no—
67#120—1933no—
68#120D19311yescharacteristic, direct product factor
69C2×C22⋊C4—1933no—
70C24—38612no—
71C24—1933no—
72C24—1931no—
73C24—1933no—
74C24C2×D19311yescharacteristic, complemented
75C2×D4—38612no—
76C2×D4—1933no—
77C2×D4—1933no—
78C2×D4—1933no—
79C2×D4C2×D19313yescomplemented
80C22×C4—1933no—
81C22⋊C4—1933no—
82C22⋊C4—1936no—
83C22⋊C4C2×D19313yescomplemented
84C23—26no—
85C23—3866no—
86C23—38632no—
87C23—38612no—
88C23—38612no—
89C23—3866no—
90C23—1931no—
91C23—19312no—
92C23—1933no—
93C23—1931no—
94C23C22×D19313yes—
95C23C22×D19311yescharacteristic, complemented
96D4—26no—
97D4—3866no—
98D4—3866no—
99D4—3866no—
100C2×C4—3866no—
101C2×C4—1933no—
102C2×C4C22×D19313yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—7724no—
107C22—38624no—
108C22—38624no—
109C22—3863no—
110C22—3866no—
111C22—1936no—
112C22—1936no—
113C22D4×D19316yescomplemented
114C22C23×D19311yescharacteristic
115C4—23no—
116C4—3863no—
117C2—26no—
118C2—41no—
119C2—7721no—
120C2—3866no—
121C2—1934no—
122C2C2×D4×D19313yes—
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#120×C193C2
hypercenter#120D193
hypocenterC193C2×#120
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D193C2semidirect
2. C2×D4×D193C2semidirect
3. #20259869C2semidirect
4. #20259868C2semidirect
5. #20259867C2semidirect
6. C22⋊C4×D193C2semidirect
7. #120×C193C2semidirect
10. C23×D193C22semidirect
11. C23×D193C22semidirect
12. C2×D772C22semidirect
13. C2×C22⋊D193C22semidirect
14. C2×C22⋊D193C22semidirect
16. C2×C4×D193C22semidirect
17. C22⋊Dic193C22semidirect
18. #15690532C22semidirect
19. C24×C193C22semidirect
20. C2×D4×C193C22semidirect
21. C22⋊C4×C193C22semidirect
27. C22×D193D4semidirect
30. C22×D193C23semidirect
34. C2×Dic193C23semidirect
38. C23×C193C23semidirect
40. C2×C4×C193C23semidirect
44. C22×C193C2×D4semidirect
52. C2×D193C2×D4semidirect
60. D193#120direct (N × H)
61. C193C2×#120semidirect
68. #120D193direct (N × H)
74. C24C2×D193semidirect
79. C2×D4C2×D193semidirect
83. C22⋊C4C2×D193semidirect
95. C23C22×D193semidirect
102. C2×C4C22×D193semidirect
113. C22D4×D193semidirect

Maximal subgroups

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

subgroupindex
C24×D1932
C2×D4×D1932
#202598692
#202598682
#202598672
C22⋊C4×D1932
#120×C1932
C2×#120193