Finite Group Databasebeta

← #26104135

Subgroups — #26104135

Order 12032.

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

normal subgroups only

This group has 186 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#20243002C211yesmaximal, characteristic
2#20241551C211yesmaximal, characteristic, complemented
3C47×#1408C211yesmaximal, characteristic, complemented
4#15689500C411yescharacteristic, complemented
5#15689499—21no—
6#15689499—21no—
7#15688916—21no—
8C8×C47⋊C8C411yescharacteristic, complemented
9C47×#493C2211yescharacteristic
10C47×#353—21no—
11C2×D376—41no—
12C2×#10625334—42no—
13C2×#10625334—42no—
14#13094021—21no—
15#13093939—41no—
16C4×C47⋊C8—21no—
17C2×D8×C47—41no—
18#127×C47D411yescharacteristic
19#127×C47D411yescharacteristic
20#99×C47—41no—
21C4×C8×C47C2×C411yescharacteristic
22C2×D4×C47—21no—
23C2×D4×C47—21no—
24C2×D4×C47—41no—
25C2×D4×C47—41no—
26C2×D188—41no—
27C2×D188—41no—
28C2×D188—41no—
29D8×C47—82no—
30C2×C8×C47—21no—
31C22⋊C4×C47—81no—
32C42×C47C22⋊C411yescharacteristic
33C22⋊Dic47—81no—
34#10625334—84no—
35#10625334—84no—
36C2×C47⋊C8—22no—
37D376—82no—
38C23×C47—41no—
39C23×C47—41no—
40C22×D47—81no—
41D4×C47—81no—
42D4×C47—81no—
43D4×C47—82no—
44D4×C47—82no—
45C2×C4×C47—21no—
46C2×C4×C47—161no—
47C2×C4×C47#9911yescharacteristic
48C2×Dic47—161no—
49D188—82no—
50D188—82no—
51D188—82no—
52C8×C47—22no—
53C47⋊C8—24no—
54#7529—471nomaximal, Sylow
55C22×C47—81no—
56C22×C47—81no—
57C22×C47—161no—
58C22×C47—161no—
59C22×C47#35311yescharacteristic
60C2×D47—161no—
61C2×D47—161no—
62C2×D47—161no—
63C4×C47—22no—
64C4×C47—21no—
65C4×C47—321no—
66Dic47—321no—
67#1712—471no—
68#1408—471no—
69#1408D4711yescharacteristic, complemented
70C2×C47—21no—
71C2×C47—161no—
72C2×C47—161no—
73C2×C47#140811yescharacteristic
74D47—321no—
75#493—471no—
76#493—941no—
77#493—941no—
78#493C2×D4711yescharacteristic
79#353—21no—
80#353—941no—
81C82—471no—
82C47#752911yescharacteristic, complemented, Sylow
83C2×D8—41no—
84C2×D8—1881no—
85C2×D8—1882no—
86C2×D8—1882no—
87#127—941no—
88#127D18811yescharacteristic
89#127C22⋊D4711yescharacteristic
90#99—41no—
91#99—1881no—
92C4×C8—941no—
93C4×C8C4×D4711yescharacteristic
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—1881no—
99C2×D4—1881no—
100C2×D4—1881no—
101D8—82no—
102D8—3762no—
103D8—3764no—
104D8—3764no—
105C2×C8—21no—
106C2×C8—942no—
107C22⋊C4—81no—
108C22⋊C4—3761no—
109C42(C2×D47)⋊C411yescharacteristic
110C23—41no—
111C23—41no—
112C23—3761no—
113D4—81no—
114D4—81no—
115D4—82no—
116D4—82no—
117D4—3762no—
118D4—3762no—
119D4—3762no—
120C2×C4—21no—
121C2×C4—161no—
122C2×C4—7521no—
123C2×C4#1309393811yescharacteristic
124C8—22no—
125C8—944no—
126C22—81no—
127C22—81no—
128C22—161no—
129C22—161no—
130C22—7521no—
131C22—7521no—
132C22—7521no—
133C22#1568891511yescharacteristic
134C4—22no—
135C4—21no—
136C4—321no—
137C4—15041no—
138C2—21no—
139C2—161no—
140C2—161no—
141C2—15041no—
142C2#2024155011yescharacteristic
subgroupG/H
centerC2#20241550
commutatorC4×C8×C47C2×C4
Frattini#493C2×D47
FittingC47×#1408C2
socleC2×C47#1408
prime radical#20243002C2
cosocleC47×#493C22
hypercenter#1408D47
hypocenterC47#7529
LCSG ▷ C4×C8×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ #1408

Complements

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

NH ≅ G/Ntype
2. #20241551C2semidirect
3. C47×#1408C2semidirect
4. #15689500C4semidirect
8. C8×C47⋊C8C4semidirect
69. #1408D47semidirect
82. C47#7529semidirect

Maximal subgroups

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

subgroupindex
#202430022
#202415512
C47×#14082
#752947