Finite Group Databasebeta

← #26104124

Subgroups — #26104124

Order 12032.

138 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 27 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#20241561C211yesmaximal, characteristic, complemented
2#20241423C211yesmaximal, characteristic, complemented
3C47×#1662C211yesmaximal, characteristic
4#15688922—21no—
5#15688901—21no—
6C47×#562C411yescharacteristic
7C47×#491C2211yescharacteristic
8C47×#489—21no—
9C47×#330C411yescharacteristic
10C47×#329—21no—
11#13093964—41no—
12#13093943—41no—
13C4⋊Q8×C47D411yescharacteristic
14#125×C47C2×C411yescharacteristic
15#124×C47—21no—
16#123×C47—41no—
17C22⋊Q8×C47—41no—
18#121×C47—41no—
19#103×C47—41no—
20#103×C47—41no—
21#102×C47—41no—
22#97×C47—21no—
23#97×C47D411yescharacteristic
24C2×Q8×C47—21no—
25C2×Q8×C47—21no—
26C2×D4×C47—21no—
27C2×D4×C47—41no—
28C22×C4×C47—41no—
29C2×C8×C47—21no—
30C2×C8×C47—41no—
31C4⋊C4×C47—21no—
32C4⋊C4×C47—41no—
33C4⋊C4×C47—41no—
34C4⋊C4×C47—41no—
35C22⋊C4×C47—41no—
36C22⋊C4×C47—41no—
37C22⋊C4×C47—41no—
38C42×C47C22⋊C411yescharacteristic
39#10625329—81no—
40C47⋊C16—42no—
41C23×C47—41no—
42Q8×C47—81no—
43Q8×C47—81no—
44D4×C47—81no—
45D4×C47—161no—
46C2×C4×C47—21no—
47C2×C4×C47—41no—
48C2×C4×C47—41no—
49C2×C4×C47—41no—
50C2×C4×C47—161no—
51C2×C4×C47#9911yescharacteristic
52C8×C47—41no—
53C8×C47—81no—
54C47⋊C8—161no—
55#7525—471nomaximal, Sylow
56C22×C47—81no—
57C22×C47—81no—
58C22×C47—81no—
59C22×C47AΣL(1,𝔽16)11yescharacteristic
60C4×C47—21no—
61C4×C47—41no—
62C4×C47—81no—
63C4×C47—81no—
64C4×C47—81no—
65#1662D4711yescharacteristic
66#1413—471no—
67#1356—471no—
68C2×C47—21no—
69C2×C47—161no—
70C2×C47#140411yescharacteristic
71#562Dic4711yescharacteristic
72#491C2×D4711yescharacteristic
73#489—21no—
74#356—941no—
75#347—941no—
76#330Dic4711yescharacteristic
77#329—21no—
78C47#752511yescharacteristic, complemented, Sylow
79C4⋊Q8C22⋊D4711yescharacteristic
80#125C2×Dic4711yescharacteristic
81#124—21no—
82#123—41no—
83C22⋊Q8—41no—
84#121—41no—
85M32—1881no—
86#103—41no—
87#103—41no—
88#102—41no—
89#101—1881no—
90#97—21no—
91#97C22⋊D4711yescharacteristic
92C2×Q8—21no—
93C2×Q8—21no—
94C2×D4—21no—
95C2×D4—41no—
96C22×C4—41no—
97M16—3761no—
98C2×C8—21no—
99C2×C8—41no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C22⋊Dic4711yescharacteristic
108C16—1882no—
109C23—41no—
110Q8—81no—
111Q8—81no—
112D4—81no—
113D4—161no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—161no—
119C2×C4#1309393911yescharacteristic
120C8—41no—
121C8—81no—
122C8—7521no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#1568891011yescharacteristic
127C4—21no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—81no—
132C2—21no—
133C2—161no—
134C2#2024153911yescharacteristic
subgroupG/H
centerC2#20241539
commutator#125×C47C2×C4
Frattini#491C2×D47
FittingC47×#1662C2
socleC2×C47#1404
prime radicalC47×#562C4
cosocleC47×#491C22
hypercenter#1662D47
hypocenterC47#7525
LCSG ▷ #125×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1662

Complements

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

NH ≅ G/Ntype
1. #20241561C2semidirect
2. #20241423C2semidirect
78. C47#7525semidirect

Maximal subgroups

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

subgroupindex
#202415612
#202414232
C47×#16622
#752547