Finite Group Databasebeta

← #26104111

Subgroups — #26104111

Order 12032.

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

normal subgroups only

This group has 148 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#20242531C211yesmaximal, characteristic
2#20241427C211yesmaximal, characteristic, complemented
3C47×#1411C211yesmaximal, characteristic, complemented
4#15689839C411yescharacteristic
5#15689410—21no—
6#15688944—21no—
7#15688872C411yescharacteristic
8#15688852—21no—
9C47×#494C2211yescharacteristic
10C47×#356—21no—
11#13094013—41no—
12#13093999—41no—
13C4×D188—21no—
14#13093982—41no—
15#13093964—41no—
16#13093955—41no—
17C4⋊(C47⋊C8)—21no—
18#13093944—41no—
19C2×Q16×C47—41no—
20C4⋊Q8×C47C2×C411yescharacteristic
21C4⋊Q8×C47D411yescharacteristic
22#101×C47—41no—
23C4×C8×C47D411yescharacteristic
24C2×Q8×C47—21no—
25C2×Q8×C47—21no—
26C2×D188—21no—
27C2×D188—41no—
28C2×C4×D47—41no—
29Q16×C47—81no—
30Q16×C47—81no—
31M16×C47—81no—
32C2×C8×C47—21no—
33C4⋊C4×C47—41no—
34C4⋊C4×C47—41no—
35C42×C47C22⋊C411yescharacteristic
36(C2×D47)⋊C4—42no—
37(C2×D47)⋊C4—41no—
38C4⋊Dic47—21no—
39C4×Dic47—41no—
40C2×C47⋊C8—41no—
41C47⋊C16—81no—
42C22×D47—41no—
43Q8×C47—81no—
44Q8×C47—81no—
45C2×C4×C47—21no—
46C2×C4×C47—41no—
47C2×C4×C47—41no—
48C2×C4×C47#9911yescharacteristic
49C2×Dic47—41no—
50D188—41no—
51D188—41no—
52D188—161no—
53C4×D47—81no—
54C4×D47—81no—
55C8×C47—21no—
56C8×C47—21no—
57C8×C47—161no—
58C47⋊C8—81no—
59#7521—471nomaximal, Sylow
60C22×C47AΣL(1,𝔽16)11yescharacteristic
61C2×D47—81no—
62C2×D47—81no—
63C2×D47—81no—
64C4×C47—21no—
65C4×C47—21no—
66C4×C47—21no—
67C4×C47—81no—
68C4×C47—81no—
69Dic47—81no—
70#1634—471no—
71#1411D4711yescharacteristic, complemented
72#1358—471no—
73C2×C47—21no—
74C2×C47#140211yescharacteristic
75D47—161no—
76#555—471no—
77#494C2×D4711yescharacteristic
78#476—941no—
79#364—941no—
80#356—21no—
81#333—471no—
82D4⋊C8—941no—
83C47#752111yescharacteristic, complemented, Sylow
84C2×Q16—41no—
85C4⋊Q8C4×D4711yescharacteristic
86C4⋊Q8C22⋊D4711yescharacteristic
87#124—1881no—
88#121—1881no—
89C4×D4—941no—
90#117—1881no—
91M32—1881no—
92#106—1881no—
93C4⋊C8—941no—
94#102—1881no—
95#101—41no—
96C4×C8D18811yescharacteristic
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—941no—
100C2×D4—1881no—
101C22×C4—1881no—
102Q16—81no—
103Q16—81no—
104M16—81no—
105C2×C8—21no—
106C2×C8—1881no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—941no—
110C22⋊C4—1882no—
111C22⋊C4—1881no—
112C42—1881no—
113C42(C2×D47)⋊C411yescharacteristic
114C16—3761no—
115C23—1881no—
116Q8—81no—
117Q8—81no—
118D4—1881no—
119D4—1881no—
120D4—7521no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—3761no—
125C2×C4—3761no—
126C2×C4—1881no—
127C2×C4#1309393811yescharacteristic
128C8—21no—
129C8—21no—
130C8—161no—
131C8—3761no—
132C22—3761no—
133C22—3761no—
134C22—3761no—
135C22#1568890911yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—3761no—
142C2—21no—
143C2—7521no—
144C2#2024153211yescharacteristic
subgroupG/H
centerC2#20241532
commutatorC4⋊Q8×C47C2×C4
Frattini#494C2×D47
FittingC47×#1411C2
socleC2×C47#1402
prime radical#15689839C4
cosocleC47×#494C22
hypercenter#1411D47
hypocenterC47#7521
LCSG ▷ C4⋊Q8×C47 ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ #1411

Complements

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

NH ≅ G/Ntype
2. #20241427C2semidirect
3. C47×#1411C2semidirect
71. #1411D47semidirect
83. C47#7521semidirect

Maximal subgroups

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

subgroupindex
#202425312
#202414272
C47×#14112
#752147