Finite Group Databasebeta

← #37559384

Subgroups — #37559384

Order 23872.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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×C4×D373C211yesmaximal, characteristic
2C2×#20238593C212yesmaximal, complemented
3C22⋊#15688412C211yesmaximal, characteristic, complemented
4#26045704C212yesmaximal, complemented
5C22⋊C8×C373C211yesmaximal, characteristic, complemented
6C23×D373C411yescharacteristic
7C22×Dic373C411yescharacteristic
8C2×C4×D373—24no—
9C2×C4×D373C411yescharacteristic
10C2×C4×D373C411yescharacteristic
11C2×C4×D373C2211yescharacteristic
12C2×C4×D373C2211yescharacteristic
13#20238593—24no—
14C2×#15688412C2212yescomplemented
15C22×C4×C373C2211yescharacteristic
16C2×C8×C373C2212yescomplemented
17C22×D373—22no—
18C22×D373—22no—
19C22×D373—22no—
20C22×D373C2×C411yescharacteristic
21C22×D373C2×C411yescharacteristic
22C2×Dic373—22no—
23C2×Dic373C2×C411yescharacteristic
24C2×Dic373C2×C411yescharacteristic
25C4×D373—22no—
26C4×D373—22no—
27C4×D373—22no—
28C4×D373D414yes—
29#15688412—22no—
30C23×C373C2×C411yescharacteristic
31C2×C4×C373—22no—
32C2×C4×C373C2×C411yescharacteristic
33C2×C4×C373C2311yescharacteristic
34C8×C373—22no—
35C22×C373—21no—
36C22×C373—21no—
37C22×C373M1612yescomplemented
38C22×C373C22×C411yescharacteristic
39C2×D373—28no—
40C2×D373—21no—
41C2×D373—21no—
42C2×D373—21no—
43C2×D373—21no—
44C2×D373C22⋊C412yes—
45C2×D373M1612yescomplemented
46C4×C373—21no—
47C4×C373C2×D412yes—
48Dic373—21no—
49Dic373C22⋊C412yes—
50C2×C373—22no—
51C2×C373C2×C22⋊C411yescharacteristic
52C2×C373C2×M1611yescharacteristic
53C2×C373C2×M1611yescharacteristic
54D373—22no—
55D373—22no—
56C373#40711yescharacteristic, complemented, Sylow
57#407—3731nomaximal, Sylow
58C23×C4—3731no—
59C2×M16—3732no—
60C22⋊C8—3731no—
61C22⋊C8—3732no—
62C22⋊C8D37311yescharacteristic, complemented
63C24—3731no—
64C22×C4—3731no—
65C22×C4—3731no—
66C22×C4—3731no—
67C22×C4—3731no—
68C22×C4—3731no—
69C22×C4—7464no—
70C22×C4C2×D37311yescharacteristic
71M16—7464no—
72C2×C8—3732no—
73C2×C8C2×D37312yescomplemented
74C23—3731no—
75C23—3731no—
76C23—7462no—
77C23—7462no—
78C23—7462no—
79C23C4×D37311yescharacteristic
80C2×C4—22no—
81C2×C4—3731no—
82C2×C4—3734no—
83C2×C4—3731no—
84C2×C4—7462no—
85C2×C4—7462no—
86C2×C4—7462no—
87C2×C4—7462no—
88C2×C4C4×D37311yescharacteristic
89C2×C4C22×D37311yescharacteristic
90C8—22no—
91C8—7462no—
92C22—21no—
93C22—21no—
94C22—3732no—
95C22—3732no—
96C22—7468no—
97C22—7461no—
98C22—7461no—
99C22—7461no—
100C22—7461no—
101C22#2023859312yescomplemented
102C22C2×C4×D37311yescharacteristic
103C4—21no—
104C4—3732no—
105C4—7461no—
106C4D4×D37312yes—
107C2—22no—
108C2—7462no—
109C2—7462no—
110C2C22⋊C4×D37311yescharacteristic
111C2M16×D37311yescharacteristic
112C2C2×#2023859311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D373
commutator, socleC22×C373C22×C4
FittingC22⋊C8×C373C2
prime radicalC23×D373C4
cosocleC2×C4×C373C23
hypercenterC22⋊C8D373
hypocenterC373#407
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#20238593C2semidirect
3. C22⋊#15688412C2semidirect
4. #26045704C2semidirect
5. C22⋊C8×C373C2semidirect
14. C2×#15688412C22semidirect
16. C2×C8×C373C22semidirect
37. C22×C373M16semidirect
45. C2×D373M16semidirect
56. C373#407semidirect
62. C22⋊C8D373semidirect
73. C2×C8C2×D373semidirect
101. C22#20238593semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3732
C2×#202385932
C22⋊#156884122
#260457042
C22⋊C8×C3732
#407373