Finite Group Databasebeta

← #27213332

Subgroups — #27213332

Order 13504.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#20270022C211yesmaximal, characteristic, complemented
2#20269946C212yesmaximal, complemented
3#20269892C211yesmaximal, characteristic, complemented
4#20269891C212yesmaximal, complemented
5#120×C211C211yesmaximal, characteristic, complemented
6#16809962—21no—
7#16809961—22no—
8C2×C22⋊D211—22no—
9C2×C22⋊D211—21no—
10C2×C22⋊D211C2212yes—
11D4×D211—22no—
12C22⋊Dic211—22no—
13C22⋊Dic211C2212yescomplemented
14C24×C211C2211yescharacteristic, complemented
15C2×D4×C211—21no—
16C2×D4×C211C2211yescharacteristic
17C22⋊C4×C211—21no—
18C22⋊C4×C211C2211yescharacteristic, complemented
19C22×D211—21no—
20C22×D211D412yescomplemented
21Dic422—21no—
22D844—21no—
23C22⋊D211—22no—
24C22⋊D211—22no—
25C22⋊D211—22no—
26C22⋊D211—44no—
27C2×Dic211—21no—
28C2×Dic211—22no—
29C2×Dic211D412yescomplemented
30C4×D211—22no—
31C23×C211—21no—
32C23×C211—22no—
33C23×C211—42no—
34C23×C211D411yescharacteristic, complemented
35C23×C211C2311yescharacteristic
36D4×C211—22no—
37D4×C211—42no—
38C2×C4×C211—21no—
39C2×C4×C211D411yescharacteristic, complemented
40C22×C211—22no—
41C22×C211—24no—
42C22×C211—21no—
43C22×C211—41no—
44C22×C211—41no—
45C22×C211—42no—
46C22×C211—41no—
47C22×C211—41no—
48C22×C211C2×D412yes—
49C22×C211C2×D411yescharacteristic
50C2×D211—22no—
51C2×D211—42no—
52C2×D211—81no—
53C4×C211—21no—
54C4×C211—41no—
55Dic211—22no—
56Dic211—42no—
57C2×C211—22no—
58C2×C211—21no—
59C2×C211—41no—
60C2×C211—42no—
61C2×C211#12011yescharacteristic
62D211—42no—
63C211B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—2111nomaximal, Sylow
65H(5,𝔽2)—2111no—
66#120—2112no—
67#120D21111yescharacteristic, complemented
68#99—2111no—
69#99—2112no—
70C24C2×D21111yescharacteristic, complemented
71#41—4222no—
72#41—4221no—
73C2×D4—21no—
74C2×D4—4222no—
75C2×D4—4222no—
76C2×D4—4221no—
77C2×D4—2112no—
78C2×D4C2×D21111yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—4222no—
81C22⋊C4—2112no—
82C22⋊C4C2×D21111yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—4221no—
87C23—2112no—
88C23C22⋊D21111yescharacteristic, complemented
89C23C22×D21111yescharacteristic
90Q8—4221no—
91D4—22no—
92D4—42no—
93D4—8444no—
94D4—4221no—
95D4—4222no—
96D4—4222no—
97D4—4222no—
98C2×C4—21no—
99C2×C4—4222no—
100C2×C4—4222no—
101C2×C4—4221no—
102C2×C4—2112no—
103C2×C4C22⋊D21111yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—8442no—
113C22—16881no—
114C22—4222no—
115C22D4×D21112yes—
116C22C2×C22⋊D21111yescharacteristic
117C4—21no—
118C4—41no—
119C4—8442no—
120C4—4222no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—8442no—
126C2#2026994511yescharacteristic
subgroupG/H
centerC2#20269945
commutator, cosocleC23×C211C23
FrattiniC23C22×D211
Fitting#120×C211C2
socleC2×C211#120
hypercenter#120D211
hypocenterC211B(4,𝔽2)
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #20270022C2semidirect
2. #20269946C2semidirect
3. #20269892C2semidirect
4. #20269891C2semidirect
5. #120×C211C2semidirect
13. C22⋊Dic211C22semidirect
14. C24×C211C22semidirect
18. C22⋊C4×C211C22semidirect
20. C22×D211D4semidirect
29. C2×Dic211D4semidirect
34. C23×C211D4semidirect
39. C2×C4×C211D4semidirect
63. C211B(4,𝔽2)semidirect
67. #120D211semidirect
70. C24C2×D211semidirect
82. C22⋊C4C2×D211semidirect
88. C23C22⋊D211semidirect
103. C2×C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#202700222
#202699462
#202698922
#202698912
#120×C2112
B(4,𝔽2)211