Finite Group Databasebeta

← #26104105

Subgroups — #26104105

Order 12032.

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

normal subgroups only

This group has 162 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#20242469C211yesmaximal, characteristic
2#20241429C211yesmaximal, characteristic, complemented
3C47×#1409C211yesmaximal, characteristic, complemented
4#15689839C411yescharacteristic, complemented
5#15689315—21no—
6#15688944—21no—
7#15688872C411yescharacteristic, complemented
8#15688852—21no—
9C47×#492C2211yescharacteristic
10C47×#353—21no—
11C2×#10625334—41no—
12#13094013—41no—
13#13093999—41no—
14C4×D188—21no—
15#13093982—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×D188—21no—
28C2×D188—41no—
29C2×C4×D47—41no—
30SD16×C47—81no—
31SD16×C47—81no—
32C2×C8×C47—21no—
33C4⋊C4×C47—41no—
34C22⋊C4×C47—81no—
35C42×C47C22⋊C411yescharacteristic
36#10625334—82no—
37(C2×D47)⋊C4—42no—
38(C2×D47)⋊C4—41no—
39C4⋊Dic47—21no—
40C4×Dic47—41no—
41C2×C47⋊C8—41no—
42C47⋊C16—81no—
43C23×C47—41no—
44C22×D47—41no—
45Q8×C47—81no—
46D4×C47—81no—
47D4×C47—81no—
48D4×C47—81no—
49C2×C4×C47—21no—
50C2×C4×C47—41no—
51C2×C4×C47—161no—
52C2×C4×C47#9911yescharacteristic
53C2×Dic47—41no—
54D188—41no—
55D188—41no—
56D188—161no—
57C4×D47—81no—
58C4×D47—81no—
59C8×C47—21no—
60C8×C47—21no—
61C47⋊C8—81no—
62#7519—471nomaximal, Sylow
63C22×C47—81no—
64C22×C47—161no—
65C22×C47AΣL(1,𝔽16)11yescharacteristic
66C2×D47—81no—
67C2×D47—81no—
68C2×D47—81no—
69C4×C47—21no—
70C4×C47—21no—
71C4×C47—21no—
72C4×C47—81no—
73C4×C47—321no—
74Dic47—81no—
75#1624—471no—
76#1409D4711yescharacteristic, complemented
77#1359—471no—
78C2×C47—21no—
79C2×C47—161no—
80C2×C47#140211yescharacteristic
81D47—161no—
82#555—471no—
83#492C2×D4711yescharacteristic
84#459—941no—
85#364—941no—
86#353—21no—
87#333—471no—
88D4⋊C8—941no—
89C47#751911yescharacteristic, complemented, Sylow
90C2×SD16—41no—
91C2×D8—1881no—
92C4⋊Q8C4×D4711yescharacteristic
93#127C22⋊D4711yescharacteristic
94#124—1881no—
95#121—1881no—
96C4×D4—941no—
97#117—1881no—
98M32—1881no—
99C4⋊C8—941no—
100#102—1881no—
101#99—41no—
102C4×C8D18811yescharacteristic
103C2×Q8—21no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—941no—
107C2×D4—1881no—
108C22×C4—1881no—
109SD16—81no—
110SD16—81no—
111D8—3762no—
112C2×C8—21no—
113C2×C8—1881no—
114C4⋊C4—41no—
115C4⋊C4—941no—
116C22⋊C4—81no—
117C22⋊C4—1882no—
118C22⋊C4—1881no—
119C42—1881no—
120C42(C2×D47)⋊C411yescharacteristic
121C16—3761no—
122C23—41no—
123C23—1881no—
124Q8—81no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—1881no—
129D4—1881no—
130D4—7521no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—161no—
134C2×C4—3761no—
135C2×C4—3761no—
136C2×C4—1881no—
137C2×C4#1309393811yescharacteristic
138C8—21no—
139C8—21no—
140C8—3761no—
141C22—81no—
142C22—161no—
143C22—3761no—
144C22—3761no—
145C22—3761no—
146C22#1568890911yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—81no—
151C4—321no—
152C4—3761no—
153C2—21no—
154C2—161no—
155C2—7521no—
156C2#2024153211yescharacteristic
subgroupG/H
centerC2#20241532
commutatorC4⋊Q8×C47C2×C4
Frattini#492C2×D47
FittingC47×#1409C2
socleC2×C47#1402
prime radical#20242469C2
cosocleC47×#492C22
hypercenter#1409D47
hypocenterC47#7519
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
2. #20241429C2semidirect
3. C47×#1409C2semidirect
4. #15689839C4semidirect
7. #15688872C4semidirect
76. #1409D47semidirect
89. C47#7519semidirect

Maximal subgroups

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

subgroupindex
#202424692
#202414292
C47×#14092
#751947