Finite Group Databasebeta

← #120×D257 #30657221

Subgroups — #120×D257 #30657221

Order 16448.

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×D257C211yesmaximal, characteristic, complemented
2C2×D4×D257C213yesmaximal, complemented
3#22551562C213yesmaximal, complemented
4#22551561C211yesmaximal, characteristic, complemented
5#22551560C213yesmaximal, complemented
6C22⋊C4×D257C213yesmaximal, complemented
7#120×C257C211yesmaximal, characteristic, complemented
8C23×D257—212no—
9C23×D257C2213yes—
10C23×D257C2213yescomplemented
11C23×D257C2211yescharacteristic, complemented
12C2×D1028C2213yescomplemented
13C2×C22⋊D257C2213yescomplemented
14C2×C22⋊D257C2213yescomplemented
15D4×D257—212no—
16C2×C4×D257C2213yescomplemented
17C22⋊Dic257C2213yescomplemented
18#16861595C2216yescomplemented
19C24×C257C2211yescharacteristic, complemented
20C2×D4×C257C2213yescomplemented
21C22⋊C4×C257C2213yescomplemented
22C22×D257—232no—
23C22×D257—26no—
24C22×D257—212no—
25C22×D257—212no—
26C22×D257—26no—
27C22×D257D4112yescomplemented
28C22×D257C2313yes—
29C22×D257C2311yescharacteristic
30C22×D257C2311yescharacteristic, complemented
31D1028—26no—
32C22⋊D257—26no—
33C22⋊D257—26no—
34C2×Dic257C2313yescomplemented
35C4×D257—26no—
36C23×C257—26no—
37C23×C257C2313yes—
38C23×C257C2311yescharacteristic, complemented
39D4×C257—26no—
40C2×C4×C257C2313yescomplemented
41C22×C257—28no—
42C22×C257—26no—
43C22×C257—23no—
44C22×C257C2×D416yescomplemented
45C22×C257C2411yescharacteristic
46C2×D257—224no—
47C2×D257—26no—
48C2×D257—224no—
49C2×D257—23no—
50C2×D257—44no—
51C2×D257C2×D416yes—
52C2×D257C2×D416yescomplemented
53C4×C257—23no—
54Dic257—23no—
55C2×C257—26no—
56C2×C257—41no—
57C2×C257C22×D413yes—
58D257—26no—
59D257—41no—
60D257#12014yesdirect product factor
61C257C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—2571nomaximal, Sylow
63C25—2571no—
64C22×D4—2573no—
65#120—2571no—
66#120—2573no—
67#120—2573no—
68#120D25711yescharacteristic, direct product factor
69C2×C22⋊C4—2573no—
70C24—2573no—
71C24—2573no—
72C24—2571no—
73C24—51412no—
74C24C2×D25711yescharacteristic, complemented
75C2×D4—2573no—
76C2×D4—2573no—
77C2×D4—2573no—
78C2×D4—51412no—
79C2×D4C2×D25713yescomplemented
80C22×C4—2573no—
81C22⋊C4—2573no—
82C22⋊C4—2576no—
83C22⋊C4C2×D25713yescomplemented
84C23—2571no—
85C23—25712no—
86C23—2573no—
87C23—2571no—
88C23—26no—
89C23—51412no—
90C23—51412no—
91C23—5146no—
92C23—51432no—
93C23—5146no—
94C23C22×D25713yes—
95C23C22×D25711yescharacteristic, complemented
96D4—26no—
97D4—5146no—
98D4—5146no—
99D4—5146no—
100C2×C4—2573no—
101C2×C4—5146no—
102C2×C4C22×D25713yescomplemented
103C22—2576no—
104C22—2576no—
105C22—28no—
106C22—26no—
107C22—23no—
108C22—51424no—
109C22—51424no—
110C22—5146no—
111C22—5143no—
112C22—10284no—
113C22D4×D25716yescomplemented
114C22C23×D25711yescharacteristic
115C4—23no—
116C4—5143no—
117C2—2574no—
118C2—26no—
119C2—5146no—
120C2—41no—
121C2—10281no—
122C2C2×D4×D25713yes—
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#120×C257C2
hypercenter#120D257
hypocenterC257C2×#120
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D257C2semidirect
2. C2×D4×D257C2semidirect
3. #22551562C2semidirect
4. #22551561C2semidirect
5. #22551560C2semidirect
6. C22⋊C4×D257C2semidirect
7. #120×C257C2semidirect
10. C23×D257C22semidirect
11. C23×D257C22semidirect
12. C2×D1028C22semidirect
13. C2×C22⋊D257C22semidirect
14. C2×C22⋊D257C22semidirect
16. C2×C4×D257C22semidirect
17. C22⋊Dic257C22semidirect
18. #16861595C22semidirect
19. C24×C257C22semidirect
20. C2×D4×C257C22semidirect
21. C22⋊C4×C257C22semidirect
27. C22×D257D4semidirect
30. C22×D257C23semidirect
34. C2×Dic257C23semidirect
38. C23×C257C23semidirect
40. C2×C4×C257C23semidirect
44. C22×C257C2×D4semidirect
52. C2×D257C2×D4semidirect
60. D257#120direct (N × H)
61. C257C2×#120semidirect
68. #120D257direct (N × H)
74. C24C2×D257semidirect
79. C2×D4C2×D257semidirect
83. C22⋊C4C2×D257semidirect
95. C23C22×D257semidirect
102. C2×C4C22×D257semidirect
113. C22D4×D257semidirect

Maximal subgroups

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

subgroupindex
C24×D2572
C2×D4×D2572
#225515622
#225515612
#225515602
C22⋊C4×D2572
#120×C2572
C2×#120257