Finite Group Databasebeta

← #27213334

Subgroups — #27213334

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#20269947C212yesmaximal, complemented
3#20269945C211yesmaximal, characteristic, complemented
4#20269891C212yesmaximal, complemented
5#99×C211C211yesmaximal, characteristic, complemented
6C23×D211C2211yescharacteristic, complemented
7#16809962—21no—
8#16809961—22no—
9C2×D844—22no—
10C2×C22⋊D211—21no—
11C2×C22⋊D211—21no—
12C2×C22⋊D211C2212yes—
13D4×D211—22no—
14C22⋊Dic211C2211yescharacteristic, complemented
15(C2×D211)⋊C4—22no—
16(C2×D211)⋊C4—21no—
17C2×D4×C211C2211yescharacteristic
18C22⋊C4×C211C2212yescomplemented
19C22×D211—21no—
20C22×D211—22no—
21C22×D211—21no—
22C22×D211—42no—
23C22×D211D412yescomplemented
24Dic422—21no—
25D844—21no—
26D844—44no—
27C22⋊D211—22no—
28C22⋊D211—22no—
29C22⋊D211—22no—
30C22⋊D211—42no—
31C2×Dic211—21no—
32C2×Dic211—21no—
33C2×Dic211D412yescomplemented
34C4×D211—22no—
35C23×C211D411yescharacteristic, complemented
36C23×C211C2311yescharacteristic
37D4×C211—22no—
38C2×C4×C211—22no—
39C2×C4×C211D411yescharacteristic, complemented
40C22×C211—21no—
41C22×C211—41no—
42C22×C211—41no—
43C22×C211C2×D412yes—
44C22×C211C2×D411yescharacteristic
45C2×D211—22no—
46C2×D211—24no—
47C2×D211—22no—
48C2×D211—41no—
49C2×D211—42no—
50C2×D211—41no—
51C2×D211—42no—
52C2×D211—81no—
53C4×C211—21no—
54C4×C211—42no—
55Dic211—22no—
56Dic211—41no—
57C2×C211—21no—
58C2×C211—22no—
59C2×C211—41no—
60C2×C211#12011yescharacteristic
61D211—42no—
62D211—42no—
63C211B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—2111nomaximal, Sylow
65H(5,𝔽2)—2111no—
66#120—2111no—
67#120—2112no—
68#99—2112no—
69#99D21111yescharacteristic, complemented
70C24—2111no—
71#41—4222no—
72#41—4221no—
73C2×D4—4222no—
74C2×D4—4222no—
75C2×D4—4221no—
76C2×D4—4221no—
77C2×D4—2112no—
78C2×D4C2×D21111yescharacteristic
79C22⋊C4—4222no—
80C22⋊C4—4221no—
81C22⋊C4—2111no—
82C22⋊C4C2×D21112yescomplemented
83C23—8442no—
84C23—4221no—
85C23—4222no—
86C23—4221no—
87C23—2112no—
88C23D84411yescharacteristic, complemented
89C23C22×D21111yescharacteristic
90Q8—4221no—
91D4—22no—
92D4—8444no—
93D4—8442no—
94D4—4221no—
95D4—4222no—
96D4—4222no—
97D4—4222no—
98C2×C4—22no—
99C2×C4—4222no—
100C2×C4—4221no—
101C2×C4—4221no—
102C2×C4—2112no—
103C2×C4D84411yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—8442no—
108C22—8442no—
109C22—8441no—
110C22—8441no—
111C22—16881no—
112C22—4224no—
113C22—4222no—
114C22—4222no—
115C22D4×D21112yes—
116C22C2×D84411yescharacteristic
117C4—21no—
118C4—42no—
119C4—8441no—
120C4—4222no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—8442no—
125C2—8442no—
126C2#2026994711yescharacteristic
subgroupG/H
centerC2#20269947
commutator, cosocleC23×C211C23
FrattiniC23C22×D211
Fitting#99×C211C2
socleC2×C211#120
hypercenter#99D211
hypocenterC211B(4,𝔽2)
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #20270022C2semidirect
2. #20269947C2semidirect
3. #20269945C2semidirect
4. #20269891C2semidirect
5. #99×C211C2semidirect
6. C23×D211C22semidirect
14. C22⋊Dic211C22semidirect
18. C22⋊C4×C211C22semidirect
23. C22×D211D4semidirect
33. C2×Dic211D4semidirect
35. C23×C211D4semidirect
39. C2×C4×C211D4semidirect
63. C211B(4,𝔽2)semidirect
69. #99D211semidirect
82. C22⋊C4C2×D211semidirect
88. C23D844semidirect
103. C2×C4D844semidirect

Maximal subgroups

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

subgroupindex
#202700222
#202699472
#202699452
#202698912
#99×C2112
B(4,𝔽2)211