Finite Group Databasebeta

← #26104118

Subgroups — #26104118

Order 12032.

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

normal subgroups only

This group has 158 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#20241559C211yesmaximal, characteristic, complemented
2#20241421C211yesmaximal, characteristic, complemented
3C47×#1656C211yesmaximal, characteristic
4#15688916—21no—
5#15688901—21no—
6C47×#562C411yescharacteristic, complemented
7C47×#496—21no—
8C47×#490C2211yescharacteristic
9C47×#330C411yescharacteristic, complemented
10C47×#329—21no—
11#13093964—41no—
12#13093939—41no—
13C2×SD16×C47—41no—
14C2×D8×C47—41no—
15#127×C47D411yescharacteristic
16#125×C47C2×C411yescharacteristic
17#124×C47—21no—
18#123×C47—41no—
19C22⋊Q8×C47—41no—
20#121×C47—41no—
21#102×C47—41no—
22#97×C47—21no—
23#97×C47D411yescharacteristic
24C2×Q8×C47—21no—
25C2×D4×C47—21no—
26C2×D4×C47—21no—
27C2×D4×C47—41no—
28C2×D4×C47—41no—
29C22×C4×C47—41no—
30SD16×C47—82no—
31D8×C47—82no—
32C2×C8×C47—21no—
33C2×C8×C47—41no—
34C4⋊C4×C47—21no—
35C4⋊C4×C47—41no—
36C4⋊C4×C47—41no—
37C22⋊C4×C47—41no—
38C22⋊C4×C47—41no—
39C22⋊C4×C47—41no—
40C42×C47C22⋊C411yescharacteristic
41C22⋊Dic47—81no—
42C47⋊C16—42no—
43C23×C47—41no—
44C23×C47—41no—
45Q8×C47—81no—
46D4×C47—81no—
47D4×C47—81no—
48D4×C47—81no—
49D4×C47—81no—
50D4×C47—161no—
51C2×C4×C47—21no—
52C2×C4×C47—41no—
53C2×C4×C47—41no—
54C2×C4×C47—161no—
55C2×C4×C47#9911yescharacteristic
56C2×Dic47—161no—
57C8×C47—41no—
58C8×C47—42no—
59#7523—471nomaximal, Sylow
60C22×C47—81no—
61C22×C47—81no—
62C22×C47—81no—
63C22×C47—81no—
64C22×C47—161no—
65C22×C47AΣL(1,𝔽16)11yescharacteristic
66C4×C47—21no—
67C4×C47—41no—
68C4×C47—81no—
69C4×C47—81no—
70Dic47—321no—
71#1656D4711yescharacteristic
72#1412—471no—
73#1355—471no—
74C2×C47—21no—
75C2×C47—161no—
76C2×C47—161no—
77C2×C47#140411yescharacteristic
78#562Dic4711yescharacteristic, complemented
79#496—21no—
80#490C2×D4711yescharacteristic
81#353—941no—
82#347—941no—
83#330Dic4711yescharacteristic, complemented
84#329—21no—
85C47#752311yescharacteristic, complemented, Sylow
86C2×SD16—41no—
87C2×D8—41no—
88#127C22⋊D4711yescharacteristic
89#125C2×Dic4711yescharacteristic
90#124—21no—
91#123—41no—
92C22⋊Q8—41no—
93#121—41no—
94M32—1881no—
95#102—41no—
96#99—1881no—
97#97—21no—
98#97C22⋊D4711yescharacteristic
99C2×Q8—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—41no—
103C2×D4—41no—
104C22×C4—41no—
105SD16—82no—
106D8—82no—
107C2×C8—21no—
108C2×C8—41no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—3761no—
116C42C22⋊Dic4711yescharacteristic
117C16—1882no—
118C23—41no—
119C23—41no—
120Q8—81no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—81no—
125D4—161no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—161no—
130C2×C4—7521no—
131C2×C4#1309393911yescharacteristic
132C8—41no—
133C8—42no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—161no—
139C22#1568891011yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—81no—
144C4—15041no—
145C2—21no—
146C2—161no—
147C2—161no—
148C2#2024153911yescharacteristic
subgroupG/H
centerC2#20241539
commutator#125×C47C2×C4
Frattini#490C2×D47
Fitting, prime radicalC47×#1656C2
socleC2×C47#1404
cosocleC47×#490C22
hypercenter#1656D47
hypocenterC47#7523
LCSG ▷ #125×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1656

Complements

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

NH ≅ G/Ntype
1. #20241559C2semidirect
2. #20241421C2semidirect
6. C47×#562C4semidirect
9. C47×#330C4semidirect
78. #562Dic47semidirect
83. #330Dic47semidirect
85. C47#7523semidirect

Maximal subgroups

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

subgroupindex
#202415592
#202414212
C47×#16562
#752347