Finite Group Databasebeta

← #120×D173 #26011975

Subgroups — #120×D173 #26011975

Order 11072.

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×D173C211yesmaximal, characteristic, complemented
2C2×D4×D173C213yesmaximal, complemented
3#19149649C213yesmaximal, complemented
4#19149648C211yesmaximal, characteristic, complemented
5#19149647C213yesmaximal, complemented
6C22⋊C4×D173C213yesmaximal, complemented
7#120×C173C211yesmaximal, characteristic, complemented
8C23×D173—212no—
9C23×D173C2213yes—
10C23×D173C2213yescomplemented
11C23×D173C2211yescharacteristic, complemented
12C2×D692C2213yescomplemented
13C2×C22⋊D173C2213yescomplemented
14C2×C22⋊D173C2213yescomplemented
15D4×D173—212no—
16C2×C4×D173C2213yescomplemented
17C22⋊Dic173C2213yescomplemented
18#14583803C2216yescomplemented
19C24×C173C2211yescharacteristic, complemented
20C2×D4×C173C2213yescomplemented
21C22⋊C4×C173C2213yescomplemented
22C22×D173—232no—
23C22×D173—26no—
24C22×D173—212no—
25C22×D173—212no—
26C22×D173—26no—
27C22×D173D4112yescomplemented
28C22×D173C2313yes—
29C22×D173C2311yescharacteristic
30C22×D173C2311yescharacteristic, complemented
31D692—26no—
32C22⋊D173—26no—
33C22⋊D173—26no—
34C2×Dic173C2313yescomplemented
35C4×D173—26no—
36C23×C173—26no—
37C23×C173C2313yes—
38C23×C173C2311yescharacteristic, complemented
39D4×C173—26no—
40C2×C4×C173C2313yescomplemented
41C22×C173—28no—
42C22×C173—26no—
43C22×C173—23no—
44C22×C173C2×D416yescomplemented
45C22×C173C2411yescharacteristic
46C2×D173—224no—
47C2×D173—26no—
48C2×D173—224no—
49C2×D173—23no—
50C2×D173—44no—
51C2×D173C2×D416yes—
52C2×D173C2×D416yescomplemented
53C4×C173—23no—
54Dic173—23no—
55C2×C173—26no—
56C2×C173—41no—
57C2×C173C22×D413yes—
58D173—26no—
59D173—41no—
60D173#12014yesdirect product factor
61C173C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—1731nomaximal, Sylow
63C25—1731no—
64C22×D4—1733no—
65#120—1731no—
66#120—1733no—
67#120—1733no—
68#120D17311yescharacteristic, direct product factor
69C2×C22⋊C4—1733no—
70C24—34612no—
71C24—1733no—
72C24—1731no—
73C24—1733no—
74C24C2×D17311yescharacteristic, complemented
75C2×D4—34612no—
76C2×D4—1733no—
77C2×D4—1733no—
78C2×D4—1733no—
79C2×D4C2×D17313yescomplemented
80C22×C4—1733no—
81C22⋊C4—1733no—
82C22⋊C4—1736no—
83C22⋊C4C2×D17313yescomplemented
84C23—26no—
85C23—34632no—
86C23—34612no—
87C23—34612no—
88C23—3466no—
89C23—3466no—
90C23—1731no—
91C23—17312no—
92C23—1733no—
93C23—1731no—
94C23C22×D17313yes—
95C23C22×D17311yescharacteristic, complemented
96D4—26no—
97D4—3466no—
98D4—3466no—
99D4—3466no—
100C2×C4—3466no—
101C2×C4—1733no—
102C2×C4C22×D17313yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—34624no—
107C22—34624no—
108C22—3466no—
109C22—3463no—
110C22—1736no—
111C22—1736no—
112C22—6924no—
113C22D4×D17316yescomplemented
114C22C23×D17311yescharacteristic
115C4—23no—
116C4—3463no—
117C2—26no—
118C2—41no—
119C2—3466no—
120C2—1734no—
121C2—6921no—
122C2C2×D4×D17313yes—
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#120×C173C2
hypercenter#120D173
hypocenterC173C2×#120
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D173C2semidirect
2. C2×D4×D173C2semidirect
3. #19149649C2semidirect
4. #19149648C2semidirect
5. #19149647C2semidirect
6. C22⋊C4×D173C2semidirect
7. #120×C173C2semidirect
10. C23×D173C22semidirect
11. C23×D173C22semidirect
12. C2×D692C22semidirect
13. C2×C22⋊D173C22semidirect
14. C2×C22⋊D173C22semidirect
16. C2×C4×D173C22semidirect
17. C22⋊Dic173C22semidirect
18. #14583803C22semidirect
19. C24×C173C22semidirect
20. C2×D4×C173C22semidirect
21. C22⋊C4×C173C22semidirect
27. C22×D173D4semidirect
30. C22×D173C23semidirect
34. C2×Dic173C23semidirect
38. C23×C173C23semidirect
40. C2×C4×C173C23semidirect
44. C22×C173C2×D4semidirect
52. C2×D173C2×D4semidirect
60. D173#120direct (N × H)
61. C173C2×#120semidirect
68. #120D173direct (N × H)
74. C24C2×D173semidirect
79. C2×D4C2×D173semidirect
83. C22⋊C4C2×D173semidirect
95. C23C22×D173semidirect
102. C2×C4C22×D173semidirect
113. C22D4×D173semidirect

Maximal subgroups

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

subgroupindex
C24×D1732
C2×D4×D1732
#191496492
#191496482
#191496472
C22⋊C4×D1732
#120×C1732
C2×#120173