Finite Group Databasebeta

← #26103820

Subgroups — #26103820

Order 12032.

198 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 37 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#20244675C211yesmaximal, characteristic
2#20241339C211yesmaximal, characteristic, complemented
3C47×#1309C211yesmaximal, characteristic, complemented
4#15689923C411yescharacteristic
5C2×#13093952—21no—
6#15689105—21no—
7#15688905—21no—
8C47⋊#338C411yescharacteristic
9C47×#423—21no—
10C47×#423C2211yescharacteristic
11C2×#10625350—21no—
12#13094057—41no—
13#13094052—41no—
14#13094046—41no—
15C2×#10625329—21no—
16#13093964—41no—
17#13093952—41no—
18#13093952—41no—
19#13093942—41no—
20#13093940—41no—
21C2×M16×C47—22no—
22C2×M16×C47C2×C411yescharacteristic
23C2×M16×C47D411yescharacteristic
24C2×C42×C47D411yescharacteristic
25C4⋊C8×C47—41no—
26C4⋊C8×C47—41no—
27C4⋊C8×C47—41no—
28C22×C4×C47—21no—
29C22×C4×C47C22⋊C411yescharacteristic
30C2×C22⋊D47—41no—
31#10625350—41no—
32#10625350—41no—
33#10625350—41no—
34#10625350—41no—
35C2×D188—21no—
36C2×C4×D47—41no—
37C2×Dic94—21no—
38M16×C47—41no—
39M16×C47—41no—
40M16×C47—82no—
41C2×C8×C47—21no—
42C2×C8×C47—21no—
43C2×C8×C47—42no—
44C42×C47—21no—
45C42×C47D811yescharacteristic
46C42×C47SD1611yescharacteristic
47#10625329—41no—
48#10625329—41no—
49C2×C47⋊C8—41no—
50Dic188—41no—
51D376—41no—
52#10625325—42no—
53C47⋊C16—81no—
54C23×C47#10411yescharacteristic
55C22×D47—41no—
56C2×C4×C47—21no—
57C2×C4×C47—21no—
58C2×C4×C47—21no—
59C2×C4×C47—21no—
60C2×C4×C47—41no—
61C2×C4×C47—41no—
62C2×C4×C47#9911yescharacteristic
63C2×C4×C47#10211yescharacteristic
64C22⋊D47—81no—
65C22⋊D47—81no—
66C2×Dic47—41no—
67D188—41no—
68D188—41no—
69C4×D47—81no—
70C4×D47—81no—
71Dic94—41no—
72Dic94—41no—
73C8×C47—41no—
74C8×C47—41no—
75C8×C47—82no—
76C47⋊C8—81no—
77#7414—471nomaximal, Sylow
78C22×C47—21no—
79C22×C47—41no—
80C22×C47#32711yescharacteristic
81C2×D47—81no—
82C2×D47—161no—
83C4×C47—21no—
84C4×C47—21no—
85C4×C47—21no—
86C4×C47—41no—
87C4×C47AΣL(1,𝔽16)11yescharacteristic
88C4×C47#35511yescharacteristic
89Dic47—81no—
90#2050—471no—
91#1327—471no—
92#1309D4711yescharacteristic, complemented
93C2×C47—21no—
94C2×C47—41no—
95C2×C47#134511yescharacteristic
96D47—161no—
97#575—471no—
98#423—21no—
99#423C2×D4711yescharacteristic
100C2×#104—941no—
101#413—941no—
102#349—941no—
103#338—471no—
104C47#741411yescharacteristic, complemented, Sylow
105C2×#41—941no—
106#137—1881no—
107AGL(1,ℤ/8)—1881no—
108#135—1881no—
109C2×M16—22no—
110C2×M16—941no—
111C2×M16C4×D4711yescharacteristic
112C2×M16D18811yescharacteristic
113C2×C42C22⋊D4711yescharacteristic
114M32—1881no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#104—1881no—
119#104—1881no—
120#101—1881no—
121#100—1881no—
122#41—1881no—
123#41—1881no—
124#41—1881no—
125#41—1881no—
126C2×Q8—941no—
127C2×D4—941no—
128C2×D4—1881no—
129C22×C4—21no—
130C22×C4—1881no—
131C22×C4(C2×D47)⋊C411yescharacteristic
132Q16—1881no—
133SD16—1882no—
134D8—1881no—
135M16—41no—
136M16—41no—
137M16—82no—
138M16—1881no—
139M16—1881no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—42no—
143C2×C8—1881no—
144C42—21no—
145C42#1062533411yescharacteristic
146C42#1062533611yescharacteristic
147C16—3761no—
148C23—1881no—
149C23#1309395211yescharacteristic
150Q8—1881no—
151Q8—1881no—
152D4—3761no—
153D4—3761no—
154D4—1881no—
155D4—1881no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3761no—
163C2×C4—3761no—
164C2×C4—1881no—
165C2×C4#1309393811yescharacteristic
166C2×C4#1309394411yescharacteristic
167C8—41no—
168C8—41no—
169C8—82no—
170C8—3761no—
171C22—21no—
172C22—41no—
173C22—3761no—
174C22—7521no—
175C22#1568885611yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—3761no—
181C4#1568891111yescharacteristic
182C4#1568891911yescharacteristic
183C2—21no—
184C2—41no—
185C2—7521no—
186C2#2024139011yescharacteristic
subgroupG/H
centerC2#20241390
commutatorC2×M16×C47C2×C4
Frattini#423C2×D47
FittingC47×#1309C2
socleC2×C47#1345
prime radical#15689923C4
cosocleC47×#423C22
hypercenter#1309D47
hypocenterC47#7414
LCSG ▷ C2×M16×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #20241339C2semidirect
3. C47×#1309C2semidirect
92. #1309D47semidirect
104. C47#7414semidirect

Maximal subgroups

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

subgroupindex
#202446752
#202413392
C47×#13092
#741447