Finite Group Databasebeta

← #26104122

Subgroups — #26104122

Order 12032.

134 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 25 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#20242689C211yesmaximal, characteristic
2#20241559C211yesmaximal, characteristic
3C47×#1355C211yesmaximal, characteristic
4C47⋊#564C411yescharacteristic, complemented
5#15689487—21no—
6#15688916—21no—
7#15688865C411yescharacteristic, complemented
8#15688864—21no—
9C47×#490C2211yescharacteristic
10C47×#347—21no—
11#13094016—21no—
12#13094015—41no—
13#13094014—41no—
14#13094014—41no—
15#13093945—41no—
16#13093945—41no—
17#13093939—41no—
18#13093935—21no—
19#127×C47D411yescharacteristic
20#125×C47C2×C411yescharacteristic
21M32×C47—41no—
22#102×C47—41no—
23#97×C47D411yescharacteristic
24C2×D4×C47—21no—
25C2×D4×C47—41no—
26C2×C8×C47—21no—
27C4⋊C4×C47—21no—
28C4⋊C4×C47—41no—
29C42×C47C22⋊C411yescharacteristic
30C22⋊Dic47—81no—
31C4⋊Dic47—21no—
32C4⋊Dic47—21no—
33C4⋊Dic47—41no—
34Dic47⋊C4—41no—
35Dic47⋊C4—41no—
36Dic47⋊C4—41no—
37Dic47⋊C4—41no—
38C4×Dic47—41no—
39C2×C47⋊C8—41no—
40C16×C47—42no—
41C23×C47—41no—
42D4×C47—81no—
43D4×C47—82no—
44C2×C4×C47—21no—
45C2×C4×C47—41no—
46C2×C4×C47#9911yescharacteristic
47C2×Dic47—41no—
48C2×Dic47—41no—
49C2×Dic47—161no—
50C8×C47—41no—
51C47⋊C8—81no—
52#7524—471nomaximal, Sylow
53C22×C47—81no—
54C22×C47—161no—
55C22×C47AΣL(1,𝔽16)11yescharacteristic
56C4×C47—21no—
57C4×C47—41no—
58C4×C47—81no—
59Dic47—81no—
60Dic47—81no—
61Dic47—321no—
62#1658—471no—
63#1412—471no—
64#1355D4711yescharacteristic
65C2×C47—21no—
66C2×C47—161no—
67C2×C47#140411yescharacteristic
68#564—471no—
69#490—941no—
70#490C2×D4711yescharacteristic
71#353—941no—
72#347—21no—
73#330—471no—
74#330—941no—
75C47#752411yescharacteristic, complemented, Sylow
76#127D18811yescharacteristic
77#125—941no—
78#125—1881no—
79#125—1881no—
80#125—1881no—
81#125C4×D4711yescharacteristic
82M32—41no—
83#102—41no—
84#102—1881no—
85#102—1881no—
86#99—1881no—
87#97—941no—
88#97C22⋊D4711yescharacteristic
89C2×D4—21no—
90C2×D4—41no—
91C2×C8—21no—
92C2×C8—1881no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—941no—
96C4⋊C4—941no—
97C4⋊C4—1881no—
98C4⋊C4—1881no—
99C4⋊C4—1881no—
100C4⋊C4—1881no—
101C4⋊C4—1881no—
102C22⋊C4—3761no—
103C42—1881no—
104C42(C2×D47)⋊C411yescharacteristic
105C16—42no—
106C23—41no—
107D4—81no—
108D4—82no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—1881no—
112C2×C4—1881no—
113C2×C4—7521no—
114C2×C4#1309393811yescharacteristic
115C8—41no—
116C8—3761no—
117C22—81no—
118C22—161no—
119C22#1568891111yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—3761no—
124C4—3761no—
125C4—15041no—
126C2—21no—
127C2—161no—
128C2#2024154011yescharacteristic
subgroupG/H
centerC2#20241540
commutator#125×C47C2×C4
Frattini#490C2×D47
FittingC47×#1355C2
socleC2×C47#1404
prime radical#127×C47D4
cosocleC47×#490C22
hypercenter#1355D47
hypocenterC47#7524
LCSG ▷ #125×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1355

Complements

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

NH ≅ G/Ntype
4. C47⋊#564C4semidirect
7. #15688865C4semidirect
75. C47#7524semidirect

Maximal subgroups

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

subgroupindex
#202426892
#202415592
C47×#13552
#752447