Finite Group Databasebeta

← #37559521

Subgroups — #37559521

Order 23872.

120 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 49 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
2#26045757C211yesmaximal, characteristic, complemented
3#26045754C211yesmaximal, characteristic
4#26045715C211yesmaximal, characteristic, complemented
5#26045712C211yesmaximal, characteristic
6#26045702C211yesmaximal, characteristic, complemented
7#107×C373C211yesmaximal, characteristic, complemented
8C2×Dic746C2211yescharacteristic
9C2×D1492C2211yescharacteristic
10C2×C4×D373—21no—
11#20238599C414yescomplemented
12C2×#15688412C2211yescharacteristic
13#20238589—21no—
14#20238587—21no—
15C4×Dic373—21no—
16C4×Dic373C2211yescharacteristic
17C2×C8×C373C2211yescharacteristic
18C4⋊C4×C373C2211yescharacteristic
19C4⋊C4×C373C2211yescharacteristic
20C22×D373—21no—
21Dic746—21no—
22Dic746C2×C412yescomplemented
23D1492—21no—
24D1492C2×C412yes—
25C2×Dic373—21no—
26C2×Dic373D411yescharacteristic
27C2×Dic373D411yescharacteristic
28C4×D373—42no—
29#15688412—21no—
30C2×C4×C373—21no—
31C2×C4×C373—21no—
32C2×C4×C373C2311yescharacteristic
33C8×C373C2×C412yescomplemented
34C22×C373C2×D411yescharacteristic
35C2×D373—22no—
36C2×D373—41no—
37C2×D373—41no—
38C4×C373—41no—
39C4×C373—41no—
40C4×C373C22×C411yescharacteristic
41C4×C373#4111yescharacteristic
42Dic373—22no—
43Dic373—21no—
44Dic373—21no—
45C2×C373C4×D411yescharacteristic
46C2×C373AGL(1,ℤ/8)11yescharacteristic
47C2×C373#13711yescharacteristic
48D373—42no—
49C373#44011yescharacteristic, complemented, Sylow
50#440—3731nomaximal, Sylow
51C2×SD16—3731no—
52C4×Q8—3731no—
53C4×D4—3731no—
54#107D37311yescharacteristic, complemented
55#103—3731no—
56#102—3731no—
57#97—3731no—
58C2×Q8—3731no—
59C2×D4—3731no—
60C22×C4—7461no—
61SD16—3734no—
62C2×C8—3731no—
63C2×C8C2×D37311yescharacteristic
64C4⋊C4—7461no—
65C4⋊C4C2×D37311yescharacteristic
66C4⋊C4C2×D37311yescharacteristic
67C22⋊C4—7461no—
68C42—3731no—
69C42—7461no—
70C23—7461no—
71Q8—3732no—
72Q8—7461no—
73D4—3732no—
74D4—7461no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—3731no—
78C2×C4—3731no—
79C2×C4—14922no—
80C2×C4—7461no—
81C2×C4C22×D37311yescharacteristic
82C8—7461no—
83C8C4×D37312yescomplemented
84C22—14921no—
85C22—14921no—
86C22—7462no—
87C22D4×D37311yescharacteristic
88C4—41no—
89C4—41no—
90C4—7462no—
91C4—7461no—
92C4—7461no—
93C4C2×C4×D37311yescharacteristic
94C4#2023861111yescharacteristic
95C2—14922no—
96C2#2604575411yescharacteristic
97C2#2604581611yescharacteristic
98C2#2604582411yescharacteristic
subgroupG/H
centerC22D4×D373
commutatorC4×C373C22×C4
FrattiniC2×C4C22×D373
Fitting#107×C373C2
socleC22×C373C2×D4
prime radicalC2×D1492C22
cosocleC2×C4×C373C23
hypercenter#107D373
hypocenterC373#440
LCSG ▷ C4×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #26045757C2semidirect
4. #26045715C2semidirect
6. #26045702C2semidirect
7. #107×C373C2semidirect
11. #20238599C4semidirect
22. Dic746C2×C4semidirect
33. C8×C373C2×C4semidirect
49. C373#440semidirect
54. #107D373semidirect
83. C8C4×D373semidirect

Maximal subgroups

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

subgroupindex
C2×#202385992
#260457572
#260457542
#260457152
#260457122
#260457022
#107×C3732
#440373