Finite Group Databasebeta

← #26104108

Subgroups — #26104108

Order 12032.

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

normal subgroups only

This group has 156 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#20242464C211yesmaximal, characteristic
2#20241429C211yesmaximal, characteristic
3C47×#1409C211yesmaximal, characteristic
4#15689853C411yescharacteristic, complemented
5#15689320—21no—
6#15688944—21no—
7#15688872C411yescharacteristic, complemented
8#15688855—21no—
9C47×#492C2211yescharacteristic
10C47×#353—21no—
11C2×#10625335—41no—
12#13094023—41no—
13#13094022—41no—
14C4×Dic94—21no—
15#13093989—41no—
16#13093964—41no—
17C4⋊(C47⋊C8)—21no—
18#13093945—41no—
19C2×SD16×C47—41no—
20C4⋊Q8×C47C2×C411yescharacteristic
21#127×C47D411yescharacteristic
22#99×C47—41no—
23C4×C8×C47D411yescharacteristic
24C2×Q8×C47—21no—
25C2×D4×C47—21no—
26C2×D4×C47—41no—
27C2×Dic94—21no—
28C2×Dic94—41no—
29SD16×C47—81no—
30SD16×C47—81no—
31C2×C8×C47—21no—
32C4⋊C4×C47—41no—
33C22⋊C4×C47—81no—
34C42×C47C22⋊C411yescharacteristic
35#10625335—82no—
36C4⋊Dic47—21no—
37Dic47⋊C4—42no—
38Dic47⋊C4—41no—
39C4×Dic47—41no—
40C4×Dic47—41no—
41C2×C47⋊C8—41no—
42C47⋊C16—81no—
43C23×C47—41no—
44Q8×C47—81no—
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—
53C2×Dic47—41no—
54Dic94—22no—
55Dic94—41no—
56Dic94—161no—
57C8×C47—21no—
58C8×C47—21no—
59C47⋊C8—81no—
60#7520—471nomaximal, Sylow
61C22×C47—81no—
62C22×C47—161no—
63C22×C47AΣL(1,𝔽16)11yescharacteristic
64C4×C47—21no—
65C4×C47—21no—
66C4×C47—21no—
67C4×C47—81no—
68C4×C47—321no—
69Dic47—42no—
70Dic47—81no—
71#1623—471no—
72#1409D4711yescharacteristic
73#1359—471no—
74C2×C47—21no—
75C2×C47—161no—
76C2×C47#140211yescharacteristic
77Q82—471no—
78#492C2×D4711yescharacteristic
79#460—941no—
80#364—941no—
81#353—21no—
82#333—471no—
83Q8⋊C8—941no—
84C47#752011yescharacteristic, complemented, Sylow
85C2×SD16—41no—
86C2×SD16—1881no—
87C4⋊Q8—1881no—
88C4⋊Q8—1881no—
89C4⋊Q8C4×D4711yescharacteristic
90#127C22⋊D4711yescharacteristic
91C4×Q8—941no—
92C4×Q8—1881no—
93M32—1881no—
94C4⋊C8—941no—
95#102—1881no—
96#99—41no—
97C4×C8D18811yescharacteristic
98C2×Q8—21no—
99C2×Q8—941no—
100C2×Q8—1881no—
101C2×D4—21no—
102C2×D4—41no—
103SD16—81no—
104SD16—81no—
105SD16—3762no—
106C2×C8—21no—
107C2×C8—1881no—
108C4⋊C4—41no—
109C4⋊C4—941no—
110C4⋊C4—1882no—
111C4⋊C4—1881no—
112C22⋊C4—81no—
113C42—1881no—
114C42—1881no—
115C42(C2×D47)⋊C411yescharacteristic
116C16—3761no—
117C23—41no—
118Q8—81no—
119Q8—942no—
120Q8—1881no—
121Q8—7521no—
122D4—81no—
123D4—81no—
124D4—81no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—161no—
128C2×C4—1881no—
129C2×C4—1881no—
130C2×C4#1309393811yescharacteristic
131C8—21no—
132C8—21no—
133C8—3761no—
134C22—81no—
135C22—161no—
136C22#1568890911yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—81no—
141C4—321no—
142C4—3761no—
143C4—1882no—
144C2—21no—
145C2—161no—
146C2#2024153211yescharacteristic
subgroupG/H
centerC2#20241532
commutatorC4⋊Q8×C47C2×C4
Frattini#492C2×D47
FittingC47×#1409C2
socleC2×C47#1402
prime radical#127×C47D4
cosocleC47×#492C22
hypercenter#1409D47
hypocenterC47#7520
LCSG ▷ C4⋊Q8×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1409

Complements

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

NH ≅ G/Ntype
4. #15689853C4semidirect
7. #15688872C4semidirect
84. C47#7520semidirect

Maximal subgroups

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

subgroupindex
#202424642
#202414292
C47×#14092
#752047