Finite Group Databasebeta

← #37559611

Subgroups — #37559611

Order 23872.

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

normal subgroups only

This group has 160 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#26045837C211yesmaximal, characteristic
2#26045775C212yesmaximal, complemented
3#26045707C211yesmaximal, characteristic, complemented
4#26045706C212yesmaximal, complemented
5#123×C373C211yesmaximal, characteristic, complemented
6#20238610—21no—
7#20238609—22no—
8C2×C22⋊D373—21no—
9C2×C22⋊D373C2212yes—
10D4×D373—22no—
11#20238589—22no—
12C22⋊Dic373C2212yescomplemented
13#20238587—22no—
14C2×D4×C373C2211yescharacteristic
15C22×C4×C373C2211yescharacteristic, complemented
16C4⋊C4×C373—21no—
17C22⋊C4×C373—21no—
18C22⋊C4×C373C2211yescharacteristic, complemented
19C22×D373—21no—
20C22×D373D412yes—
21Dic746—21no—
22D1492—21no—
23C22⋊D373—22no—
24C22⋊D373—22no—
25C22⋊D373—22no—
26C2×Dic373—21no—
27C2×Dic373—22no—
28C2×Dic373D412yes—
29C4×D373—22no—
30C23×C373D411yescharacteristic
31C23×C373C2311yescharacteristic
32D4×C373—22no—
33C2×C4×C373—21no—
34C2×C4×C373—21no—
35C2×C4×C373—22no—
36C2×C4×C373D411yescharacteristic
37C22×C373—21no—
38C22×C373—41no—
39C22×C373—41no—
40C22×C373C2×D412yes—
41C22×C373C2×D411yescharacteristic
42C2×D373—22no—
43C2×D373—42no—
44C2×D373—81no—
45C4×C373—21no—
46C4×C373—41no—
47C4×C373—41no—
48Dic373—22no—
49Dic373—42no—
50C2×C373—22no—
51C2×C373—21no—
52C2×C373—41no—
53C2×C373#12011yescharacteristic
54D373—42no—
55C373#45811yescharacteristic, complemented, Sylow
56#458—3731nomaximal, Sylow
57H(5,𝔽2)—3731no—
58#123—3732no—
59#123D37311yescharacteristic, complemented
60#99—3731no—
61#99—3732no—
62#41—7462no—
63#41—7461no—
64C2×D4—3732no—
65C2×D4—7462no—
66C2×D4—7461no—
67C2×D4C2×D37311yescharacteristic
68C22×C4C2×D37311yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—7462no—
71C22⋊C4—21no—
72C22⋊C4—3732no—
73C22⋊C4—7462no—
74C22⋊C4C2×D37311yescharacteristic, complemented
75C23—3732no—
76C23—7461no—
77C23C22⋊D37311yescharacteristic
78C23C22×D37311yescharacteristic
79Q8—7461no—
80D4—22no—
81D4—7461no—
82D4—7462no—
83D4—7462no—
84D4—7462no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3732no—
89C2×C4—7462no—
90C2×C4—7462no—
91C2×C4—7461no—
92C2×C4C22⋊D37311yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—29841no—
97C22—14922no—
98C22—7462no—
99C22D4×D37312yes—
100C22C2×C22⋊D37311yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—14922no—
105C4—7462no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—14922no—
110C2#2604576011yescharacteristic
subgroupG/H
centerC2#26045760
commutator, cosocleC23×C373C23
FrattiniC23C22×D373
Fitting#123×C373C2
socleC2×C373#120
prime radical#26045837C2
hypercenter#123D373
hypocenterC373#458
LCSG ▷ C23×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #26045775C2semidirect
3. #26045707C2semidirect
4. #26045706C2semidirect
5. #123×C373C2semidirect
12. C22⋊Dic373C22semidirect
15. C22×C4×C373C22semidirect
18. C22⋊C4×C373C22semidirect
55. C373#458semidirect
59. #123D373semidirect
68. C22×C4C2×D373semidirect
74. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
#260458372
#260457752
#260457072
#260457062
#123×C3732
#458373