Finite Group Databasebeta

← #26104117

Subgroups — #26104117

Order 12032.

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

normal subgroups only

This group has 158 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#20242680C211yesmaximal, characteristic
2#20241421C211yesmaximal, characteristic, complemented
3C47×#1412C211yesmaximal, characteristic, complemented
4#15689871C411yescharacteristic, complemented
5#15689510—21no—
6#15688901—21no—
7#15688865C411yescharacteristic, complemented
8#15688862—21no—
9C47×#490C2211yescharacteristic
10C47×#353—21no—
11C2×#10625335—41no—
12C2×#10625334—41no—
13#13094012—21no—
14#13094009—41no—
15#13094002—41no—
16#13093999—41no—
17#13093964—41no—
18#13093935—21no—
19#127×C47D411yescharacteristic
20#125×C47C2×C411yescharacteristic
21#102×C47—41no—
22#99×C47—41no—
23#97×C47D411yescharacteristic
24C2×D4×C47—21no—
25C2×D4×C47—41no—
26C2×D188—21no—
27C2×D188—41no—
28C2×C4×D47—41no—
29C2×Dic94—21no—
30C2×C8×C47—21no—
31C4⋊C4×C47—21no—
32C4⋊C4×C47—41no—
33C22⋊C4×C47—81no—
34C42×C47C22⋊C411yescharacteristic
35#10625335—82no—
36#10625334—82no—
37(C2×D47)⋊C4—41no—
38(C2×D47)⋊C4—41no—
39(C2×D47)⋊C4—41no—
40Dic47⋊C4—41no—
41C2×C47⋊C8—41no—
42C47⋊C16—42no—
43C23×C47—41no—
44C22×D47—41no—
45D4×C47—81no—
46D4×C47—81no—
47D4×C47—81no—
48C2×C4×C47—21no—
49C2×C4×C47—41no—
50C2×C4×C47—161no—
51C2×C4×C47#9911yescharacteristic
52C2×Dic47—41no—
53D188—81no—
54D188—161no—
55C4×D47—161no—
56Dic94—81no—
57C8×C47—41no—
58C47⋊C8—42no—
59#7523—471nomaximal, Sylow
60C22×C47—81no—
61C22×C47—161no—
62C22×C47AΣL(1,𝔽16)11yescharacteristic
63C2×D47—81no—
64C2×D47—81no—
65C2×D47—81no—
66C4×C47—21no—
67C4×C47—41no—
68C4×C47—81no—
69C4×C47—321no—
70Dic47—81no—
71#1656—471no—
72#1412D4711yescharacteristic, complemented
73#1355—471no—
74C2×C47—21no—
75C2×C47—161no—
76C2×C47#140411yescharacteristic
77D47—161no—
78#562—471no—
79#496—941no—
80#490C2×D4711yescharacteristic
81#353—21no—
82#347—941no—
83#330—471no—
84#329—941no—
85C47#752311yescharacteristic, complemented, Sylow
86C2×SD16—1881no—
87C2×D8—1881no—
88#127C22⋊D4711yescharacteristic
89#125C4×D4711yescharacteristic
90#124—941no—
91#123—1881no—
92C22⋊Q8—1881no—
93#121—1881no—
94M32—1881no—
95#102—41no—
96#99—41no—
97#97—941no—
98#97D18811yescharacteristic
99C2×Q8—941no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—941no—
103C2×D4—1881no—
104C22×C4—1881no—
105SD16—3762no—
106D8—3762no—
107C2×C8—21no—
108C2×C8—1881no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—1881no—
112C22⋊C4—81no—
113C22⋊C4—1881no—
114C22⋊C4—1881no—
115C22⋊C4—1881no—
116C42(C2×D47)⋊C411yescharacteristic
117C16—1882no—
118C23—41no—
119C23—1881no—
120Q8—3761no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—3761no—
125D4—7521no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—161no—
129C2×C4—1881no—
130C2×C4—7521no—
131C2×C4#1309393811yescharacteristic
132C8—41no—
133C8—1882no—
134C22—81no—
135C22—161no—
136C22—3761no—
137C22—3761no—
138C22—3761no—
139C22#1568890911yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—321no—
144C4—3761no—
145C2—21no—
146C2—161no—
147C2—7521no—
148C2#2024153811yescharacteristic
subgroupG/H
centerC2#20241538
commutator#125×C47C2×C4
Frattini#490C2×D47
FittingC47×#1412C2
socleC2×C47#1404
prime radical#20242680C2
cosocleC47×#490C22
hypercenter#1412D47
hypocenterC47#7523
LCSG ▷ #125×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1412

Complements

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

NH ≅ G/Ntype
2. #20241421C2semidirect
3. C47×#1412C2semidirect
4. #15689871C4semidirect
7. #15688865C4semidirect
72. #1412D47semidirect
85. C47#7523semidirect

Maximal subgroups

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

subgroupindex
#202426802
#202414212
C47×#14122
#752347