Finite Group Databasebeta

← #120×D199 #27164502

Subgroups — #120×D199 #27164502

Order 12736.

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×D199C211yesmaximal, characteristic, complemented
2C2×D4×D199C213yesmaximal, complemented
3#20263925C213yesmaximal, complemented
4#20263924C211yesmaximal, characteristic, complemented
5#20263923C213yesmaximal, complemented
6C22⋊C4×D199C213yesmaximal, complemented
7#120×C199C211yesmaximal, characteristic, complemented
8C23×D199—212no—
9C23×D199C2213yes—
10C23×D199C2213yescomplemented
11C23×D199C2211yescharacteristic, complemented
12C2×D796C2213yescomplemented
13C2×C22⋊D199C2213yescomplemented
14C2×C22⋊D199C2213yescomplemented
15D4×D199—212no—
16C2×C4×D199C2213yescomplemented
17C22⋊Dic199C2213yescomplemented
18(C2×D199)⋊C4C2216yescomplemented
19C24×C199C2211yescharacteristic, complemented
20C2×D4×C199C2213yescomplemented
21C22⋊C4×C199C2213yescomplemented
22C22×D199—232no—
23C22×D199—26no—
24C22×D199—212no—
25C22×D199—212no—
26C22×D199—26no—
27C22×D199D4112yescomplemented
28C22×D199C2313yes—
29C22×D199C2311yescharacteristic
30C22×D199C2311yescharacteristic, complemented
31D796—26no—
32C22⋊D199—26no—
33C22⋊D199—26no—
34C2×Dic199C2313yescomplemented
35C4×D199—26no—
36C23×C199—26no—
37C23×C199C2313yes—
38C23×C199C2311yescharacteristic, complemented
39D4×C199—26no—
40C2×C4×C199C2313yescomplemented
41C22×C199—28no—
42C22×C199—26no—
43C22×C199—23no—
44C22×C199C2×D416yescomplemented
45C22×C199C2411yescharacteristic
46C2×D199—224no—
47C2×D199—26no—
48C2×D199—224no—
49C2×D199—23no—
50C2×D199—44no—
51C2×D199C2×D416yes—
52C2×D199C2×D416yescomplemented
53C4×C199—23no—
54Dic199—23no—
55C2×C199—26no—
56C2×C199—41no—
57C2×C199C22×D413yes—
58D199—26no—
59D199—41no—
60D199#12014yesdirect product factor
61C199C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—1991nomaximal, Sylow
63C25—1991no—
64C22×D4—1993no—
65#120—1991no—
66#120—1993no—
67#120—1993no—
68#120D19911yescharacteristic, direct product factor
69C2×C22⋊C4—1993no—
70C24—39812no—
71C24—1993no—
72C24—1991no—
73C24—1993no—
74C24C2×D19911yescharacteristic, complemented
75C2×D4—39812no—
76C2×D4—1993no—
77C2×D4—1993no—
78C2×D4—1993no—
79C2×D4C2×D19913yescomplemented
80C22×C4—1993no—
81C22⋊C4—1993no—
82C22⋊C4—1996no—
83C22⋊C4C2×D19913yescomplemented
84C23—26no—
85C23—3986no—
86C23—39832no—
87C23—39812no—
88C23—39812no—
89C23—3986no—
90C23—1991no—
91C23—19912no—
92C23—1993no—
93C23—1991no—
94C23C22×D19913yes—
95C23C22×D19911yescharacteristic, complemented
96D4—26no—
97D4—3986no—
98D4—3986no—
99D4—3986no—
100C2×C4—3986no—
101C2×C4—1993no—
102C2×C4C22×D19913yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—7964no—
107C22—39824no—
108C22—39824no—
109C22—3983no—
110C22—3986no—
111C22—1996no—
112C22—1996no—
113C22D4×D19916yescomplemented
114C22C23×D19911yescharacteristic
115C4—23no—
116C4—3983no—
117C2—26no—
118C2—41no—
119C2—7961no—
120C2—3986no—
121C2—1994no—
122C2C2×D4×D19913yes—
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#120×C199C2
hypercenter#120D199
hypocenterC199C2×#120
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D199C2semidirect
2. C2×D4×D199C2semidirect
3. #20263925C2semidirect
4. #20263924C2semidirect
5. #20263923C2semidirect
6. C22⋊C4×D199C2semidirect
7. #120×C199C2semidirect
10. C23×D199C22semidirect
11. C23×D199C22semidirect
12. C2×D796C22semidirect
13. C2×C22⋊D199C22semidirect
14. C2×C22⋊D199C22semidirect
16. C2×C4×D199C22semidirect
17. C22⋊Dic199C22semidirect
18. (C2×D199)⋊C4C22semidirect
19. C24×C199C22semidirect
20. C2×D4×C199C22semidirect
21. C22⋊C4×C199C22semidirect
27. C22×D199D4semidirect
30. C22×D199C23semidirect
34. C2×Dic199C23semidirect
38. C23×C199C23semidirect
40. C2×C4×C199C23semidirect
44. C22×C199C2×D4semidirect
52. C2×D199C2×D4semidirect
60. D199#120direct (N × H)
61. C199C2×#120semidirect
68. #120D199direct (N × H)
74. C24C2×D199semidirect
79. C2×D4C2×D199semidirect
83. C22⋊C4C2×D199semidirect
95. C23C22×D199semidirect
102. C2×C4C22×D199semidirect
113. C22D4×D199semidirect

Maximal subgroups

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

subgroupindex
C24×D1992
C2×D4×D1992
#202639252
#202639242
#202639232
C22⋊C4×D1992
#120×C1992
C2×#120199