Finite Group Databasebeta

← #26118557

Subgroups — #26118557

Order 12032.

574 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 103 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#20254509C212yesmaximal, complemented
2#20253834C211yesmaximal, characteristic
3#20241947C212yesmaximal, complemented
4#20241233C211yesmaximal, characteristic, complemented
5C47×#1960C211yesmaximal, characteristic, complemented
6C2×#13094046—22no—
7#15689673C2212yes—
8#15689526—22no—
9#15689517—22no—
10#15689512—22no—
11#15689471—22no—
12#15689436—24no—
13#15689400—24no—
14#15689176C2212yes—
15#15688856—24no—
16C2×C4⋊Q8×C47C2211yescharacteristic
17C2×#103×C47—22no—
18C2×#97×C47C2211yescharacteristic
19C47×#382C2211yescharacteristic
20C2×#10625350—22no—
21#13094046—44no—
22#13094046—44no—
23C2×Dic188—22no—
24C2×#10625325—24no—
25C2×D376—22no—
26#13094024D412yes—
27#13094021D412yes—
28#13094012—22no—
29#13093998—22no—
30#13093998—22no—
31C4×D188—22no—
32C4×D188—22no—
33#13093957—28no—
34C4⋊(C47⋊C8)—22no—
35#13093944—28no—
36C22⋊(C47⋊C8)D414yes—
37C4×C47⋊C8—22no—
38C22×Q8×C47—21no—
39C22×C8×C47D412yes—
40C4⋊Q8×C47—22no—
41C4⋊Q8×C47—21no—
42C4⋊Q8×C47D412yes—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—21no—
45C2×C42×C47C2311yescharacteristic
46#103×C47—44no—
47#97×C47—22no—
48#95×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—41no—
51C2×Q8×C47—42no—
52C2×Q8×C47—42no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C2×D412yes—
56C22×C4×C47C2×D411yescharacteristic
57C2×C22⋊D47—42no—
58#10625350—44no—
59#10625350—44no—
60C2×D188—22no—
61C2×D188—22no—
62C2×D188—22no—
63C2×C4×D47—42no—
64C2×Dic94—22no—
65C2×C8×C47—22no—
66C2×C8×C47—22no—
67C2×C8×C47D814yes—
68C4⋊C4×C47—24no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—42no—
71C42×C47C2×D411yescharacteristic
72C42×C47C2×D412yes—
73C42×C47#4111yescharacteristic
74(C2×D47)⋊C4—42no—
75(C2×D47)⋊C4—42no—
76C4⋊Dic47—22no—
77C4⋊Dic47—22no—
78C2×C47⋊C8—24no—
79Dic188—44no—
80D376—44no—
81#10625325—48no—
82C23×C47#12711yescharacteristic
83C22×D47—42no—
84Q8×C47—44no—
85Q8×C47—82no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—81no—
89C2×C4×C47—81no—
90C2×C4×C47—82no—
91C2×C4×C47—81no—
92C2×C4×C47#12011yescharacteristic
93C2×C4×C47#12012yes—
94C2×C4×C47#12112yes—
95C2×C4×C47#12111yescharacteristic
96C2×C4×C47C2×D812yes—
97C2×C4×C47AGL(1,ℤ/8)12yes—
98C22⋊D47—82no—
99C22⋊D47—82no—
100C2×Dic47—42no—
101D188—44no—
102D188—44no—
103D188—82no—
104C4×D47—84no—
105Dic94—44no—
106C8×C47—24no—
107C47⋊C8—44no—
108#10225—471nomaximal, Sylow
109C22×C47—22no—
110C22×C47#39211yescharacteristic
111C22×C47#49311yescharacteristic
112C22×C47#49611yescharacteristic
113C2×D47—82no—
114C2×D47—82no—
115C2×D47—82no—
116C4×C47—22no—
117C4×C47—22no—
118C4×C47—82no—
119C4×C47—161no—
120C4×C47#44712yes—
121C4×C47#45912yes—
122Dic47—82no—
123#3226—472no—
124#3150—471no—
125#1960D4711yescharacteristic, complemented
126#1517—472no—
127#1281—471no—
128C2×C47—21no—
129C2×C47—21no—
130C2×C47#200912yes—
131C2×C47#201111yescharacteristic
132D47—162no—
133C2×#135—942no—
134#532—472no—
135C2×C4⋊Q8C2×D4711yescharacteristic
136#500—942no—
137#497—942no—
138#496—942no—
139#486—942no—
140#480—944no—
141#474—944no—
142#431—472no—
143C2×#103—22no—
144C2×#97C2×D4711yescharacteristic
145#382C2×D4711yescharacteristic
146#327—944no—
147C47#1022511yescharacteristic, complemented, Sylow
148C2×#41—942no—
149C22×Q8—21no—
150#135—1884no—
151#135—1884no—
152C2×Q16—942no—
153C2×SD16—944no—
154C2×D8—942no—
155C22×C8C22⋊D4712yes—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—472no—
159C4⋊Q8D18812yes—
160#127—472no—
161#124—942no—
162#121—942no—
163#121—942no—
164C4×D4—942no—
165C4×D4—942no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C42C22×D4711yescharacteristic
169#107—948no—
170C4⋊C8—942no—
171#103—44no—
172#102—948no—
173C22⋊C8—474no—
174#97—22no—
175C4×C8—942no—
176#95—22no—
177#41—1884no—
178#41—1884no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—942no—
184C2×D4—942no—
185C2×D4—942no—
186C2×D4—942no—
187C2×D4—1882no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—1882no—
191C22×C4D4×D4712yes—
192C22×C4C2×C22⋊D4711yescharacteristic
193Q16—1884no—
194SD16—1888no—
195D8—1884no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—944no—
199C2×C8#1062533414yes—
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—942no—
204C4⋊C4—942no—
205C22⋊C4—1882no—
206C22⋊C4—1882no—
207C42C2×D18811yescharacteristic
208C42#1062535011yescharacteristic
209C42D4×D4712yes—
210C23—1882no—
211C23#1309402011yescharacteristic
212Q8—44no—
213Q8—82no—
214Q8—1884no—
215D4—3762no—
216D4—3762no—
217D4—3762no—
218D4—1884no—
219D4—1884no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—3764no—
227C2×C4—1882no—
228C2×C4#1309399211yescharacteristic
229C2×C4#1309399412yes—
230C2×C4#1309399712yes—
231C2×C4#1309399811yescharacteristic
232C2×C4C2×#1062533412yes—
233C2×C4#1309404912yes—
234C8—24no—
235C8—1884no—
236C22—22no—
237C22—3762no—
238C22—3762no—
239C22—3762no—
240C22#1568903211yescharacteristic
241C22#1568949911yescharacteristic
242C22#1568951311yescharacteristic
243C4—22no—
244C4—22no—
245C4—82no—
246C4—161no—
247C4—3762no—
248C4#1568925612yes—
249C4#1568931812yes—
250C2—21no—
251C2—21no—
252C2—7522no—
253C2#2024445012yes—
254C2#2024445811yescharacteristic
subgroupG/H
centerC22#15689032
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1960C2
socleC22×C47#392
prime radical#20253834C2
hypercenter#1960D47
hypocenterC47#10225
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. #20254509C2semidirect
3. #20241947C2semidirect
4. #20241233C2semidirect
5. C47×#1960C2semidirect
125. #1960D47semidirect
147. C47#10225semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202538342
#202419472
#202412332
C47×#19602
#1022547