Finite Group Databasebeta

← #120×D211 #27213630

Subgroups — #120×D211 #27213630

Order 13504.

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×D211C211yesmaximal, characteristic, complemented
2C2×D4×D211C213yesmaximal, complemented
3#20269947C213yesmaximal, complemented
4#20269946C211yesmaximal, characteristic, complemented
5#20269945C213yesmaximal, complemented
6C22⋊C4×D211C213yesmaximal, complemented
7#120×C211C211yesmaximal, characteristic, complemented
8C23×D211—212no—
9C23×D211C2213yes—
10C23×D211C2213yescomplemented
11C23×D211C2211yescharacteristic, complemented
12C2×D844C2213yescomplemented
13C2×C22⋊D211C2213yescomplemented
14C2×C22⋊D211C2213yescomplemented
15D4×D211—212no—
16C2×C4×D211C2213yescomplemented
17C22⋊Dic211C2213yescomplemented
18(C2×D211)⋊C4C2216yescomplemented
19C24×C211C2211yescharacteristic, complemented
20C2×D4×C211C2213yescomplemented
21C22⋊C4×C211C2213yescomplemented
22C22×D211—232no—
23C22×D211—26no—
24C22×D211—212no—
25C22×D211—212no—
26C22×D211—26no—
27C22×D211D4112yescomplemented
28C22×D211C2313yes—
29C22×D211C2311yescharacteristic
30C22×D211C2311yescharacteristic, complemented
31D844—26no—
32C22⋊D211—26no—
33C22⋊D211—26no—
34C2×Dic211C2313yescomplemented
35C4×D211—26no—
36C23×C211—26no—
37C23×C211C2313yes—
38C23×C211C2311yescharacteristic, complemented
39D4×C211—26no—
40C2×C4×C211C2313yescomplemented
41C22×C211—28no—
42C22×C211—26no—
43C22×C211—23no—
44C22×C211C2×D416yescomplemented
45C22×C211C2411yescharacteristic
46C2×D211—224no—
47C2×D211—26no—
48C2×D211—224no—
49C2×D211—23no—
50C2×D211—44no—
51C2×D211C2×D416yes—
52C2×D211C2×D416yescomplemented
53C4×C211—23no—
54Dic211—23no—
55C2×C211—26no—
56C2×C211—41no—
57C2×C211C22×D413yes—
58D211—26no—
59D211—41no—
60D211#12014yesdirect product factor
61C211C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—2111nomaximal, Sylow
63C25—2111no—
64C22×D4—2113no—
65#120—2111no—
66#120—2113no—
67#120—2113no—
68#120D21111yescharacteristic, direct product factor
69C2×C22⋊C4—2113no—
70C24—42212no—
71C24—2113no—
72C24—2111no—
73C24—2113no—
74C24C2×D21111yescharacteristic, complemented
75C2×D4—42212no—
76C2×D4—2113no—
77C2×D4—2113no—
78C2×D4—2113no—
79C2×D4C2×D21113yescomplemented
80C22×C4—2113no—
81C22⋊C4—2113no—
82C22⋊C4—2116no—
83C22⋊C4C2×D21113yescomplemented
84C23—26no—
85C23—4226no—
86C23—42232no—
87C23—42212no—
88C23—42212no—
89C23—4226no—
90C23—2111no—
91C23—21112no—
92C23—2113no—
93C23—2111no—
94C23C22×D21113yes—
95C23C22×D21111yescharacteristic, complemented
96D4—26no—
97D4—4226no—
98D4—4226no—
99D4—4226no—
100C2×C4—4226no—
101C2×C4—2113no—
102C2×C4C22×D21113yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—8444no—
107C22—42224no—
108C22—42224no—
109C22—4223no—
110C22—4226no—
111C22—2116no—
112C22—2116no—
113C22D4×D21116yescomplemented
114C22C23×D21111yescharacteristic
115C4—23no—
116C4—4223no—
117C2—26no—
118C2—41no—
119C2—8441no—
120C2—4226no—
121C2—2114no—
122C2C2×D4×D21113yes—
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
Fitting#120×C211C2
hypercenter#120D211
hypocenterC211C2×#120
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D211C2semidirect
2. C2×D4×D211C2semidirect
3. #20269947C2semidirect
4. #20269946C2semidirect
5. #20269945C2semidirect
6. C22⋊C4×D211C2semidirect
7. #120×C211C2semidirect
10. C23×D211C22semidirect
11. C23×D211C22semidirect
12. C2×D844C22semidirect
13. C2×C22⋊D211C22semidirect
14. C2×C22⋊D211C22semidirect
16. C2×C4×D211C22semidirect
17. C22⋊Dic211C22semidirect
18. (C2×D211)⋊C4C22semidirect
19. C24×C211C22semidirect
20. C2×D4×C211C22semidirect
21. C22⋊C4×C211C22semidirect
27. C22×D211D4semidirect
30. C22×D211C23semidirect
34. C2×Dic211C23semidirect
38. C23×C211C23semidirect
40. C2×C4×C211C23semidirect
44. C22×C211C2×D4semidirect
52. C2×D211C2×D4semidirect
60. D211#120direct (N × H)
61. C211C2×#120semidirect
68. #120D211direct (N × H)
74. C24C2×D211semidirect
79. C2×D4C2×D211semidirect
83. C22⋊C4C2×D211semidirect
95. C23C22×D211semidirect
102. C2×C4C22×D211semidirect
113. C22D4×D211semidirect

Maximal subgroups

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

subgroupindex
C24×D2112
C2×D4×D2112
#202699472
#202699462
#202699452
C22⋊C4×D2112
#120×C2112
C2×#120211