Finite Group Databasebeta

← #37559572

Subgroups — #37559572

Order 23872.

188 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C22×D1492C211yesmaximal, characteristic
2C2×#20238597C212yesmaximal, complemented
3#26045712C212yesmaximal, complemented
4C22⋊#15688412C211yesmaximal, characteristic, complemented
5C22⋊Q8×C373C211yesmaximal, characteristic, complemented
6C23×D373—21no—
7C2×D1492—24no—
8C2×D1492—21no—
9C2×D1492C2212yes—
10#20238597—24no—
11C2×#15688412C2212yescomplemented
12C2×Q8×C373C2211yescharacteristic, complemented
13C22×C4×C373C2211yescharacteristic
14C4⋊C4×C373—21no—
15C4⋊C4×C373C2211yescharacteristic, complemented
16C22⋊C4×C373—21no—
17C22×D373—24no—
18C22×D373—22no—
19C22×D373—42no—
20C22×D373—42no—
21D1492—24no—
22D1492—22no—
23D1492D414yes—
24#15688412—22no—
25C23×C373D411yescharacteristic
26Q8×C373—22no—
27C2×C4×C373—22no—
28C2×C4×C373—21no—
29C2×C4×C373—21no—
30C2×C4×C373D411yescharacteristic
31C2×C4×C373C2311yescharacteristic
32C22×C373—21no—
33C22×C373—21no—
34C22×C373SD1612yescomplemented
35C22×C373C2×D411yescharacteristic
36C2×D373—24no—
37C2×D373—48no—
38C2×D373—42no—
39C2×D373—42no—
40C4×C373—21no—
41C4×C373—41no—
42C4×C373—41no—
43C4×C373C2×D412yes—
44C2×C373—22no—
45C2×C373#12011yescharacteristic
46C2×C373C2×SD1611yescharacteristic
47C2×C373AGL(1,ℤ/8)11yescharacteristic
48D373—44no—
49C373#45011yescharacteristic, complemented, Sylow
50#450—3731nomaximal, Sylow
51C22×D4—3731no—
52C2×SD16—3732no—
53C22⋊Q8D37311yescharacteristic, complemented
54#102—3732no—
55C22⋊C8—3731no—
56C24—7461no—
57C2×Q8C2×D37311yescharacteristic, complemented
58C2×D4—3732no—
59C2×D4—7464no—
60C2×D4—7461no—
61C22×C4C2×D37311yescharacteristic
62SD16—7464no—
63C2×C8—3732no—
64C4⋊C4—21no—
65C4⋊C4C2×D37311yescharacteristic, complemented
66C22⋊C4—21no—
67C23—14922no—
68C23—14922no—
69C23—7464no—
70C23—7462no—
71C23C22⋊D37311yescharacteristic
72Q8—22no—
73D4—3734no—
74D4—7464no—
75D4—7462no—
76C2×C4—22no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4C22⋊D37311yescharacteristic
80C2×C4C22×D37311yescharacteristic
81C8—7462no—
82C22—21no—
83C22—21no—
84C22—14928no—
85C22—14922no—
86C22—14922no—
87C22—7464no—
88C22#2023859712yescomplemented
89C22C2×C22⋊D37311yescharacteristic
90C4—21no—
91C4—41no—
92C4—41no—
93C4D4×D37312yes—
94C2—22no—
95C2—14924no—
96C2#2604576011yescharacteristic
97C2C2×#2023859711yescharacteristic
98C2#2604581711yescharacteristic
subgroupG/H
centerC22C2×C22⋊D373
commutator, cosocleC2×C4×C373C23
FrattiniC2×C4C22×D373
FittingC22⋊Q8×C373C2
socleC22×C373C2×D4
prime radicalC22×D1492C2
hypercenterC22⋊Q8D373
hypocenterC373#450
LCSG ▷ C2×C4×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#20238597C2semidirect
3. #26045712C2semidirect
4. C22⋊#15688412C2semidirect
5. C22⋊Q8×C373C2semidirect
11. C2×#15688412C22semidirect
12. C2×Q8×C373C22semidirect
15. C4⋊C4×C373C22semidirect
34. C22×C373SD16semidirect
49. C373#450semidirect
53. C22⋊Q8D373semidirect
57. C2×Q8C2×D373semidirect
65. C4⋊C4C2×D373semidirect
88. C22#20238597semidirect

Maximal subgroups

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

subgroupindex
C22×D14922
C2×#202385972
#260457122
C22⋊#156884122
C22⋊Q8×C3732
#450373