Finite Group Databasebeta

← #37559668

Subgroups — #37559668

Order 23872.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C2×#20238599C211yesmaximal, characteristic, complemented
2#26045771C211yesmaximal, characteristic, complemented
3#26045766C211yesmaximal, characteristic
4#26045726C211yesmaximal, characteristic, complemented
5#26045717C211yesmaximal, characteristic, complemented
6#26045714C211yesmaximal, characteristic, complemented
7C2×M16×C373C211yesmaximal, characteristic, complemented
8C2×Dic746C2211yescharacteristic
9C2×D1492C2211yescharacteristic
10C2×C22⋊D373—21no—
11#20238599—22no—
12C4⋊Dic373C2211yescharacteristic
13C4⋊Dic373C2211yescharacteristic
14#20238589—21no—
15C22⋊Dic373—21no—
16#20238587—21no—
17C22×C4×C373C2211yescharacteristic
18M16×C373—22no—
19C2×C8×C373C2211yescharacteristic
20C2×C8×C373C2211yescharacteristic
21C22×D373—21no—
22Dic746—21no—
23Dic746—21no—
24D1492—21no—
25D1492—21no—
26C22⋊D373—42no—
27C2×Dic373—21no—
28C2×Dic373—21no—
29C2×Dic373—21no—
30C23×C373D411yescharacteristic
31C2×C4×C373—21no—
32C2×C4×C373—21no—
33C2×C4×C373D411yescharacteristic
34C2×C4×C373C2311yescharacteristic
35C8×C373—21no—
36C8×C373D412yescomplemented
37C22×C373—21no—
38C22×C373—21no—
39C22×C373—21no—
40C22×C373C2×D411yescharacteristic
41C2×D373—41no—
42C2×D373—41no—
43C2×D373—41no—
44C4×C373—21no—
45C4×C373C2×D411yescharacteristic
46C4×C373#4111yescharacteristic
47Dic373—41no—
48Dic373—41no—
49Dic373—41no—
50C2×C373—41no—
51C2×C373#12111yescharacteristic
52C2×C373AGL(1,ℤ/8)11yescharacteristic
53C2×C373#13711yescharacteristic
54D373—81no—
55C373#46811yescharacteristic, complemented, Sylow
56#468—3731nomaximal, Sylow
57C2×SD16—3731no—
58C2×M16D37311yescharacteristic, complemented
59C22⋊Q8—3731no—
60#121—3731no—
61#107—3731no—
62#103—3731no—
63#102—3731no—
64C2×Q8—3731no—
65C2×D4—3731no—
66C2×D4—7461no—
67C22×C4C2×D37311yescharacteristic
68SD16—7462no—
69M16—22no—
70C2×C8C2×D37311yescharacteristic
71C2×C8C2×D37311yescharacteristic
72C4⋊C4—3731no—
73C4⋊C4—3731no—
74C4⋊C4—7461no—
75C22⋊C4—7461no—
76C22⋊C4—7461no—
77C23—7461no—
78C23D149211yescharacteristic
79Q8—7461no—
80Q8—7461no—
81D4—14922no—
82D4—7461no—
83D4—7461no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—7461no—
87C2×C4—7461no—
88C2×C4—7461no—
89C2×C4D149211yescharacteristic
90C2×C4C22×D37311yescharacteristic
91C8—21no—
92C8C22⋊D37312yescomplemented
93C22—21no—
94C22—21no—
95C22—21no—
96C22—14921no—
97C22—14921no—
98C22—14921no—
99C22C2×D149211yescharacteristic
100C4—21no—
101C4—14921no—
102C4—14921no—
103C4—14921no—
104C4C2×C22⋊D37311yescharacteristic
105C4#2023861011yescharacteristic
106C2—41no—
107C2—29841no—
108C2#2604576611yescharacteristic
109C2#2604582011yescharacteristic
110C2#2604582511yescharacteristic
subgroupG/H
centerC22C2×D1492
commutator, cosocleC2×C4×C373C23
FrattiniC2×C4C22×D373
FittingC2×M16×C373C2
socleC22×C373C2×D4
prime radical#26045766C2
hypercenterC2×M16D373
hypocenterC373#468
LCSG ▷ C2×C4×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#20238599C2semidirect
2. #26045771C2semidirect
4. #26045726C2semidirect
5. #26045717C2semidirect
6. #26045714C2semidirect
7. C2×M16×C373C2semidirect
36. C8×C373D4semidirect
55. C373#468semidirect
58. C2×M16D373semidirect
92. C8C22⋊D373semidirect

Maximal subgroups

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

subgroupindex
C2×#202385992
#260457712
#260457662
#260457262
#260457172
#260457142
C2×M16×C3732
#468373