Finite Group Databasebeta

← #26130037

Subgroups — #26130037

Order 12032.

358 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 67 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#20245559C212yesmaximal, complemented
2#20242974C211yesmaximal, characteristic, complemented
3#20241429C211yesmaximal, characteristic, complemented
4#20241361C212yesmaximal, complemented
5C47×#3291C211yesmaximal, characteristic
6#15689521C2211yescharacteristic, complemented
7#15689519—21no—
8#15689495—21no—
9#15688947—22no—
10#15688944C2212yescomplemented
11#15688933—22no—
12C8×C47⋊C8C2211yescharacteristic, complemented
13C47×#577—22no—
14C47×#492C2211yescharacteristic
15C47×#471—22no—
16C47×#456—21no—
17C47×#453—21no—
18C47×#443C2212yes—
19C2×#10625335—42no—
20#13094024—21no—
21#13093964—22no—
22#13093956—22no—
23#13093955—42no—
24C4×C47⋊C8—21no—
252−1+4×C47—21no—
26H(5,𝔽2)×C47—21no—
27#137×C47—22no—
28#137×C47—42no—
29AGL(1,ℤ/8)×C47—22no—
30AGL(1,ℤ/8)×C47—42no—
31#135×C47—42no—
32#135×C47D412yes—
33C2×SD16×C47—22no—
34C2×SD16×C47—42no—
35#131×C47—22no—
36C4⋊Q8×C47D411yescharacteristic
37#127×C47D411yescharacteristic
38#108×C47D412yes—
39#104×C47—22no—
40#101×C47—21no—
41#100×C47—21no—
42C4×C8×C47C2311yescharacteristic
43#41×C47—22no—
44#41×C47—42no—
45#41×C47—42no—
46#41×C47—81no—
47C2×Q8×C47—21no—
48C2×Q8×C47—42no—
49C2×D4×C47—21no—
50C2×D4×C47—41no—
51C2×D4×C47—42no—
52C2×D4×C47—81no—
53C2×Dic94—41no—
54Q16×C47—42no—
55Q16×C47SD1612yescomplemented
56SD16×C47—22no—
57SD16×C47—42no—
58SD16×C47—42no—
59SD16×C47—42no—
60SD16×C47—42no—
61D8×C47—42no—
62D8×C47SD1612yescomplemented
63M16×C47—22no—
64M16×C47—42no—
65C2×C8×C47—42no—
66C2×C8×C47C2×D412yes—
67C4⋊C4×C47—41no—
68C42×C47C2×D411yescharacteristic
69#10625335—84no—
70C4⋊Dic47—41no—
71C4⋊Dic47—41no—
72C2×C47⋊C8—22no—
73C47⋊C16—42no—
74C23×C47—41no—
75C23×C47—81no—
76Q8×C47—22no—
77Q8×C47—42no—
78Q8×C47—81no—
79D4×C47—22no—
80D4×C47—42no—
81D4×C47—42no—
82D4×C47—82no—
83D4×C47—82no—
84D4×C47—81no—
85C2×C4×C47—21no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—81no—
89C2×C4×C47—82no—
90C2×C4×C47#12011yescharacteristic
91C2×Dic47—81no—
92Dic94—82no—
93C8×C47—42no—
94C8×C47C2×SD1612yes—
95C8×C47AGL(1,ℤ/8)12yes—
96C47⋊C8—24no—
97#12364—471nomaximal, Sylow
98C22×C47—42no—
99C22×C47—81no—
100C22×C47—161no—
101C22×C47—321no—
102C22×C47#45311yescharacteristic
103C4×C47—22no—
104C4×C47—42no—
105C4×C47—81no—
106C4×C47#45012yes—
107Dic47—161no—
108#3291D4711yescharacteristic
109#2230—472no—
110#1705—471no—
111#1359—471no—
112#1335—472no—
113C2×C47—21no—
114C2×C47—82no—
115C2×C47—161no—
116C2×C47#162911yescharacteristic
117#577—22no—
118#498—471no—
119#498—941no—
120#492—941no—
121#492C2×D4711yescharacteristic
122#471—22no—
123#456—21no—
124#453—21no—
125#443C2×D4712yes—
126#365—942no—
127#364—472no—
128#360—942no—
129C82—471no—
130C47#1236411yescharacteristic, complemented, Sylow
1312−1+4—21no—
132H(5,𝔽2)—21no—
133#137—22no—
134#137—42no—
135AGL(1,ℤ/8)—22no—
136AGL(1,ℤ/8)—42no—
137#135—42no—
138#135C22⋊D4712yes—
139C2×SD16—22no—
140C2×SD16—42no—
141C2×SD16—1882no—
142#131—22no—
143C4⋊Q8—941no—
144C4⋊Q8C22⋊D4711yescharacteristic
145#127C22⋊D4711yescharacteristic
146M32—942no—
147#108C22⋊D4712yes—
148#106—942no—
149#106—1882no—
150#104—22no—
151#101—21no—
152#100—21no—
153C4×C8—941no—
154C4×C8C22×D4711yescharacteristic
155#41—22no—
156#41—42no—
157#41—42no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—1881no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—81no—
166Q16—42no—
167Q16#1062533512yescomplemented
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—3764no—
174D8—42no—
175D8#1062533512yescomplemented
176M16—22no—
177M16—42no—
178C2×C8—42no—
179C2×C8—942no—
180C2×C8C2×C22⋊D4712yes—
181C4⋊C4—41no—
182C4⋊C4—1881no—
183C4⋊C4—1881no—
184C42C2×C22⋊D4711yescharacteristic
185C16—1882no—
186C23—41no—
187C23—81no—
188Q8—22no—
189Q8—42no—
190Q8—81no—
191Q8—3762no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—3761no—
204C2×C4#1309399311yescharacteristic
205C8—42no—
206C8—944no—
207C8C2×#1062533512yes—
208C8#1309405012yes—
209C22—42no—
210C22—81no—
211C22—161no—
212C22—321no—
213C22#1568929111yescharacteristic
214C4—22no—
215C4—42no—
216C4—81no—
217C4—7521no—
218C4#1568927412yes—
219C2—21no—
220C2—82no—
221C2—161no—
222C2#2024250411yescharacteristic
subgroupG/H
centerC2#20242504
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3291C2
socleC2×C47#1629
hypercenter#3291D47
hypocenterC47#12364
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #20245559C2semidirect
2. #20242974C2semidirect
3. #20241429C2semidirect
4. #20241361C2semidirect
6. #15689521C22semidirect
10. #15688944C22semidirect
12. C8×C47⋊C8C22semidirect
55. Q16×C47SD16semidirect
62. D8×C47SD16semidirect
130. C47#12364semidirect
167. Q16#10625335semidirect
175. D8#10625335semidirect

Maximal subgroups

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

subgroupindex
#202455592
#202429742
#202414292
#202413612
C47×#32912
#1236447