Finite Group Databasebeta

← #26117601

Subgroups — #26117601

Order 12032.

600 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#15689485C212yesmaximal, complemented
2#20251434C211yesmaximal, characteristic
3C2×#15688862C212yesmaximal, complemented
4#20241269C211yesmaximal, characteristic, complemented
5C47×#1960C211yesmaximal, characteristic, complemented
6C2×#13094012C2212yes—
7#15689485—24no—
8#15689485—24no—
9C2×#13093947—22no—
10C2×#13093944—22no—
11C2×#13093935C2212yes—
12#15689040—22no—
13#15689036—22no—
14#15689033—22no—
15#15688862—24no—
16C2×C4⋊Q8×C47C2211yescharacteristic
17C2×#103×C47—22no—
18C2×#97×C47C2211yescharacteristic
19C47×#382C2211yescharacteristic
20C22×Dic94D412yes—
21C22×D188—22no—
22C22×D188D412yes—
23C22×C47⋊C8—22no—
24#13094012—22no—
25#13094012—22no—
26#13094012—22no—
27#13094012—22no—
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—24no—
31#13093947—28no—
32#13093944—28no—
33#13093935—22no—
34#13093935D414yescomplemented
35C47⋊#95—22no—
36C22×Q8×C47—21no—
37C22×C8×C47—21no—
38C4⋊Q8×C47—22no—
39C4⋊Q8×C47—21no—
40C4⋊Q8×C47D412yescomplemented
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47D412yes—
43C2×C42×C47C2311yescharacteristic
44#103×C47—44no—
45#97×C47—22no—
46#95×C47—22no—
47C23×D47—22no—
48C2×Q8×C47—22no—
49C2×Q8×C47—42no—
50C2×Q8×C47—42no—
51C2×Q8×C47—41no—
52C22×C4×C47—22no—
53C22×C4×C47—41no—
54C22×C4×C47C2×D412yes—
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—22no—
57C2×D188—22no—
58C2×D188—22no—
59C2×D188—44no—
60C2×D188—82no—
61C2×D188—84no—
62C2×Dic94—22no—
63C2×Dic94—22no—
64C2×Dic94—44no—
65C2×C8×C47—22no—
66C2×C8×C47—41no—
67C2×C8×C47—41no—
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—
76(C2×D47)⋊C4—42no—
77(C2×D47)⋊C4—42no—
78(C2×D47)⋊C4—84no—
79C2×C47⋊C8—24no—
80C2×C47⋊C8—44no—
81C23×C47#12011yescharacteristic
82C22×D47—42no—
83C22×D47—42no—
84C22×D47—42no—
85C22×D47—44no—
86C22×D47—44no—
87Q8×C47—44no—
88Q8×C47—82no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—24no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—44no—
95C2×C4×C47—81no—
96C2×C4×C47—81no—
97C2×C4×C47#12012yes—
98C2×C4×C47#12112yes—
99C2×C4×C47#12111yescharacteristic
100C2×C4×C47#12711yescharacteristic
101C2×Dic47—42no—
102C2×Dic47—44no—
103D188—44no—
104D188—82no—
105D188—88no—
106Dic94—44no—
107Dic94—82no—
108C8×C47—42no—
109C47⋊C8—44no—
110#10085—471nomaximal, Sylow
111C22×C47#39211yescharacteristic
112C22×C47#45311yescharacteristic
113C22×C47#45411yescharacteristic
114C22×C47#45514yes—
115C2×D47—84no—
116C2×D47—84no—
117C2×D47—82no—
118C2×D47—82no—
119C2×D47—82no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—82no—
125C4×C47—82no—
126Dic47—82no—
127C2×#489—472no—
128#2819—471no—
129#1960D4711yescharacteristic, complemented
130C2×#329—472no—
131#1295—471no—
132C2×C47#165912yes—
133C2×C47#166212yes—
134C2×C47#200412yes—
135C2×C47#200611yescharacteristic
136D47—162no—
137C2×C4⋊Q8C2×D4711yescharacteristic
138C2×#124—472no—
139#489—944no—
140#489—944no—
141C2×#103—22no—
142C2×#103—942no—
143C2×#102—942no—
144C2×#97—472no—
145C2×#97C2×D4711yescharacteristic
146#394—942no—
147#393—942no—
148#392—942no—
149#382C2×D4711yescharacteristic
150#329—944no—
151C47#1008511yescharacteristic, complemented, Sylow
152C22×Q8—21no—
153C22×Q8—472no—
154C22×D4—472no—
155C22×D4—942no—
156C22×C8—21no—
157C22×C8—942no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8D18812yescomplemented
161#124—942no—
162#124—942no—
163#124—942no—
164#124—942no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊D4712yes—
167C2×C22⋊C4—944no—
168C2×C22⋊C4—942no—
169C2×C22⋊C4—942no—
170C2×C42C22×D4711yescharacteristic
171#103—44no—
172#103—948no—
173#102—948no—
174#97—22no—
175#97—474no—
176#97—942no—
177#95—22no—
178#95—942no—
179C24—942no—
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—942no—
185C2×Q8—942no—
186C2×Q8—1884no—
187C2×D4—942no—
188C2×D4—942no—
189C2×D4—3762no—
190C2×D4—3764no—
191C2×D4—1884no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—942no—
195C22×C4D4×D4712yes—
196C22×C4C2×C22⋊D4711yescharacteristic
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—944no—
201C2×C8—1884no—
202C4⋊C4—24no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C22⋊C4—3764no—
206C22⋊C4—1882no—
207C22⋊C4—1882no—
208C22⋊C4—1882no—
209C22⋊C4—1882no—
210C42C2×D18811yescharacteristic
211C42#1062535011yescharacteristic
212C42D4×D4712yes—
213C23—1882no—
214C23—1882no—
215C23—1882no—
216C23—1884no—
217C23—1884no—
218C23#1309399211yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—3762no—
222Q8—1884no—
223D4—3762no—
224D4—3768no—
225D4—1884no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—24no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—1882no—
235C2×C4—1884no—
236C2×C4#1309399412yes—
237C2×C4#1309399712yes—
238C2×C4#1309399811yescharacteristic
239C2×C4#1309402011yescharacteristic
240C8—42no—
241C8—1884no—
242C22—3764no—
243C22—3764no—
244C22—3762no—
245C22—3762no—
246C22—3762no—
247C22#1568903211yescharacteristic
248C22#1568929011yescharacteristic
249C22#1568929511yescharacteristic
250C22#1568929914yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—82no—
257C4—3762no—
258C2—7522no—
259C2#2024269212yes—
260C2#2024270912yes—
261C2#2024441812yes—
262C2#2024442911yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1960C2
socleC23×C47#120
prime radical#20251434C2
hypercenter#1960D47
hypocenterC47#10085
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. C2×#15689485C2semidirect
3. C2×#15688862C2semidirect
4. #20241269C2semidirect
5. C47×#1960C2semidirect
34. #13093935D4semidirect
40. C4⋊Q8×C47D4semidirect
129. #1960D47semidirect
151. C47#10085semidirect
160. C4⋊Q8D188semidirect

Maximal subgroups

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

subgroupindex
C2×#156894852
#202514342
C2×#156888622
#202412692
C47×#19602
#1008547