Finite Group Databasebeta

← #26127124

Subgroups — #26127124

Order 12032.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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#20254126C211yesmaximal, characteristic
2#20245200C211yesmaximal, characteristic, complemented
3#20241378C212yesmaximal, complemented
4#20241372C212yesmaximal, complemented
5C47×#1841C211yesmaximal, characteristic, complemented
6#15689673—21no—
7#15689521C411yescharacteristic, complemented
8#15689500C411yescharacteristic, complemented
9#15689470—21no—
10#15689376—21no—
11#15689376C2212yes—
12#15689236—21no—
13#15689236C412yescomplemented
14#15688944—21no—
15#15688905C2212yes—
16#15688901—21no—
17C4×M16×C47C2211yescharacteristic
18C47×#344—21no—
19C47×#339—21no—
20C47×#338C2211yescharacteristic
21C47×H(3,ℤ/4)C2211yescharacteristic
22C2×#10625350—41no—
23C2×D376—21no—
24C2×D376C2×C412yes—
25#13094024—21no—
26#13094021—21no—
27#13094012—41no—
28#13093998—22no—
29#13093998—41no—
30C4×D188—22no—
31C4×D188—41no—
32#13093964—22no—
33#13093956—21no—
34#13093956C2×C412yes—
35#13093946—22no—
36#13093946—22no—
37C2×M16×C47—21no—
38C2×M16×C47D411yescharacteristic
39C2×M16×C47C2311yescharacteristic
40#117×C47—21no—
41C2×C42×C47D411yescharacteristic
42#97×C47C2×C411yescharacteristic
43#97×C47D411yescharacteristic
44C4×C8×C47C2×C411yescharacteristic
45C4×C8×C47D411yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47C2×D411yescharacteristic
48C2×C22⋊D47—42no—
49#10625350—84no—
50C2×D188—22no—
51C2×D188—41no—
52C2×D188—41no—
53C2×C4×D47—42no—
54C2×Dic94—41no—
55M16×C47—41no—
56M16×C47—41no—
57M16×C47—41no—
58M16×C47—41no—
59C2×C8×C47—41no—
60C2×C8×C47C22⋊C411yescharacteristic
61C2×C8×C47C22⋊C411yescharacteristic
62C2×C8×C47C22×C411yescharacteristic
63C2×C8×C47C2×D411yescharacteristic
64C4⋊C4×C47—41no—
65C22⋊C4×C47—41no—
66C42×C47—21no—
67C42×C47—41no—
68C42×C47C22⋊C411yescharacteristic
69C42×C47C22⋊C411yescharacteristic
70(C2×D47)⋊C4—42no—
71(C2×D47)⋊C4—42no—
72C4⋊Dic47—22no—
73C4⋊Dic47—41no—
74D376—42no—
75D376—42no—
76C47⋊C16—42no—
77C23×C47AGL(1,ℤ/8)11yescharacteristic
78C22×D47—42no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—42no—
84C2×C4×C47—81no—
85C2×C4×C47C2×C22⋊C411yescharacteristic
86C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
87C22⋊D47—84no—
88C2×Dic47—42no—
89D188—42no—
90D188—42no—
91D188—82no—
92D188—82no—
93C4×D47—84no—
94Dic94—82no—
95C8×C47—21no—
96C8×C47—21no—
97C8×C47—21no—
98C8×C47—21no—
99C8×C47—81no—
100#11837—471nomaximal, Sylow
101C22×C47—21no—
102C22×C47—41no—
103C22×C47#41811yescharacteristic
104C2×D47—82no—
105C2×D47—82no—
106C2×D47—82no—
107C4×C47—22no—
108C4×C47—21no—
109C4×C47—41no—
110C4×C47—161no—
111C4×C47#41011yescharacteristic
112C4×C47#42111yescharacteristic
113Dic47—82no—
114#3184—471no—
115#2165—471no—
116#1841D4711yescharacteristic, complemented
117#1341—472no—
118#1339—472no—
119C2×C47—21no—
120C2×C47—41no—
121C2×C47#151411yescharacteristic
122D47—162no—
123#532—941no—
124#498—471no—
125#493—471no—
126#486—941no—
127#469—472no—
128#469—941no—
129#442—472no—
130#442—941no—
131C4×M16C2×D4711yescharacteristic
132#364—941no—
133#349—472no—
134#347—941no—
135#344—21no—
136#339—21no—
137#338C2×D4711yescharacteristic
138H(3,ℤ/4)C2×D4711yescharacteristic
139C47#1183711yescharacteristic, complemented, Sylow
140C2×#41—1881no—
141C2×D8—472no—
142C2×D8—941no—
143C2×M16—21no—
144C2×M16C22⋊D4711yescharacteristic
145C2×M16C22×D4711yescharacteristic
146C4⋊Q8—941no—
147#127—941no—
148#124—1881no—
149#121—942no—
150#121—1881no—
151C4×D4—942no—
152C4×D4—1881no—
153#117—21no—
154C2×C42C22⋊D4711yescharacteristic
155M32—942no—
156#106—472no—
157#106—941no—
158#102—942no—
159#102—942no—
160#97C4×D4711yescharacteristic
161#97D18811yescharacteristic
162C4×C8C4×D4711yescharacteristic
163C4×C8D18811yescharacteristic
164#41—3764no—
165C2×Q8—1881no—
166C2×D4—942no—
167C2×D4—1881no—
168C2×D4—1881no—
169C2×D4—1882no—
170C22×C4—21no—
171C22×C4—1882no—
172C22×C4C2×C22⋊D4711yescharacteristic
173D8—1882no—
174D8—1882no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—41no—
180C2×C8(C2×D47)⋊C411yescharacteristic
181C2×C8(C2×D47)⋊C411yescharacteristic
182C2×C8C2×C4×D4711yescharacteristic
183C2×C8C2×D18811yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—942no—
186C4⋊C4—1881no—
187C22⋊C4—41no—
188C22⋊C4—1882no—
189C22⋊C4—1882no—
190C42—21no—
191C42—41no—
192C42(C2×D47)⋊C411yescharacteristic
193C42(C2×D47)⋊C411yescharacteristic
194C16—1882no—
195C23—1882no—
196C23#1309404911yescharacteristic
197Q8—3762no—
198D4—3762no—
199D4—3762no—
200D4—3764no—
201D4—1882no—
202D4—1882no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—3764no—
210C2×C4—1882no—
211C2×C4C2×(C2×D47)⋊C411yescharacteristic
212C2×C4#1309404911yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—3762no—
221C22—3762no—
222C22—3762no—
223C22#1568912711yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—161no—
228C4—3762no—
229C4#1568909611yescharacteristic
230C4#1568914111yescharacteristic
231C2—21no—
232C2—41no—
233C2—7522no—
234C2#2024193411yescharacteristic
subgroupG/H
centerC2#20241934
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#1841C2
socleC2×C47#1514
prime radical#20254126C2
cosocleC2×M16×C47C23
hypercenter#1841D47
hypocenterC47#11837
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #20245200C2semidirect
3. #20241378C2semidirect
4. #20241372C2semidirect
5. C47×#1841C2semidirect
7. #15689521C4semidirect
8. #15689500C4semidirect
13. #15689236C4semidirect
116. #1841D47semidirect
139. C47#11837semidirect

Maximal subgroups

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

subgroupindex
#202541262
#202452002
#202413782
#202413722
C47×#18412
#1183747