Finite Group Databasebeta

← #26113898

Subgroups — #26113898

Order 12032.

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

normal subgroups only

This group has 490 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#20246500C211yesmaximal, characteristic
2#20242012C211yesmaximal, characteristic
3#20241900C212yesmaximal, complemented
4#20241282C212yesmaximal, complemented
5C47×#3140C211yesmaximal, characteristic
6#15689180C2212yes—
7#15689030—21no—
8#15689016—22no—
9#15689003—21no—
10C22⋊C4×Dic47—22no—
11#15688969C2212yes—
12#15688863—22no—
13#15688860—22no—
14#15688860—22no—
15C47×#533C2211yescharacteristic
16C2×#124×C47C2211yescharacteristic
17C47×#489C414yescomplemented
18C47×#488—22no—
19C47×#419—21no—
20C47×#418—21no—
21C4×M16×C47C2211yescharacteristic
22C23×Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—21no—
26C2×C4×Dic47—22no—
27C2×C4×Dic47—21no—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30#13093935—22no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C47⋊#95D412yes—
34C47⋊#95D412yes—
35C22×Q8×C47D411yescharacteristic
36C22×D4×C47D411yescharacteristic
37C2×M16×C47—21no—
38C4⋊Q8×C47C2×C412yescomplemented
39#125×C47—21no—
40#124×C47—22no—
41#124×C47C2×C412yes—
42#124×C47D412yes—
43C22⋊Q8×C47—22no—
44C4×Q8×C47—22no—
45#117×C47—21no—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47C2311yescharacteristic
48#103×C47—24no—
49#102×C47—24no—
50#97×C47C2×C412yescomplemented
51C4×C8×C47—21no—
52C24×C47—21no—
53C2×Q8×C47—21no—
54C2×Q8×C47—22no—
55C2×Q8×C47—41no—
56C2×Q8×C47—41no—
57C2×Q8×C47C22⋊C412yes—
58C2×D4×C47—21no—
59C2×D4×C47—41no—
60C2×D4×C47—41no—
61C2×D4×C47C22⋊C412yes—
62C22×C4×C47—21no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—44no—
68C22×Dic47—81no—
69M16×C47—41no—
70M16×C47—41no—
71C2×C8×C47—22no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—42no—
75C22⋊C4×C47—24no—
76C22⋊C4×C47—24no—
77C22⋊C4×C47—41no—
78C42×C47—41no—
79C42×C47C22×C411yescharacteristic
80C42×C47C2×D411yescharacteristic
81C42×C47#4112yes—
82C22⋊Dic47—44no—
83C22⋊Dic47—81no—
84C4⋊Dic47—81no—
85C4×Dic47—44no—
86C4×Dic47—81no—
87C2×C47⋊C8—42no—
88C23×C47—22no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47#12011yescharacteristic
94Q8×C47—42no—
95Q8×C47—81no—
96Q8×C47—82no—
97D4×C47—42no—
98D4×C47—81no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—42no—
106C2×C4×C47C2×C22⋊C411yescharacteristic
107C2×C4×C47C4×D412yes—
108C2×C4×C47#12112yes—
109C2×C4×C47#12311yescharacteristic
110C2×Dic47—24no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—84no—
117C8×C47—42no—
118C47⋊C8—82no—
119#9439—471nomaximal, Sylow
120C22×C47—22no—
121C22×C47—42no—
122C22×C47—44no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47—82no—
126C22×C47#38611yescharacteristic
127C22×C47#45511yescharacteristic
128C22×C47#45511yescharacteristic
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—42no—
134C4×C47—82no—
135Dic47—44no—
136Dic47—82no—
137#3140D4711yescharacteristic
138#2372—471no—
139#1533—471no—
140#1506—472no—
141#1300—472no—
142C2×C47—21no—
143C2×C47—21no—
144C2×C47—82no—
145C2×C47#188511yescharacteristic
146C2×C47#190612yes—
147#533C2×D4711yescharacteristic
148C2×#124C2×D4711yescharacteristic
149#489Dic4714yescomplemented
150#488—22no—
151#432—472no—
152#419—21no—
153#418—21no—
154C4×M16C2×D4711yescharacteristic
155#391—941no—
156#388—942no—
157#386—941no—
158C4×C22⋊C4—942no—
159#376—472no—
160#329—942no—
161#328—942no—
162#328—942no—
163C47#943911yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D4711yescharacteristic
165C22×D4C22⋊D4711yescharacteristic
166C23×C4—941no—
167C2×M16—21no—
168C4⋊Q8C2×Dic4712yescomplemented
169#125—21no—
170#124—22no—
171#124C2×Dic4712yes—
172#124C22⋊D4712yes—
173C22⋊Q8—22no—
174C4×Q8—22no—
175#117—21no—
176C2×C4⋊C4—941no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—941no—
179C2×C22⋊C4—942no—
180C2×C42—941no—
181C2×C42—942no—
182C2×C42C22×D4711yescharacteristic
183C4⋊C8—942no—
184#103—24no—
185#102—24no—
186C22⋊C8—942no—
187#97—942no—
188#97C2×Dic4712yescomplemented
189C4×C8—21no—
190#95—472no—
191#95—472no—
192#95—942no—
193#95—942no—
194C24—21no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C22⋊Dic4712yes—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C22⋊Dic4712yes—
204C22×C4—21no—
205C22×C4—942no—
206C22×C4—942no—
207C22×C4—3761no—
208C22×C4—1884no—
209C22×C4D4×D4712yes—
210C22×C4C2×C22⋊D4711yescharacteristic
211M16—41no—
212M16—41no—
213C2×C8—22no—
214C2×C8—1882no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—3761no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—41no—
222C22⋊C4—3761no—
223C22⋊C4—1884no—
224C42—41no—
225C42—3761no—
226C42—1884no—
227C42#1062535212yes—
228C42C22×Dic4711yescharacteristic
229C42C2×C22⋊D4711yescharacteristic
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#1309399211yescharacteristic
236Q8—42no—
237Q8—81no—
238Q8—82no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—944no—
249C2×C4—3764no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4C2×C22⋊Dic4711yescharacteristic
256C2×C4D4×Dic4712yes—
257C2×C4#1309399612yes—
258C2×C4#1309400511yescharacteristic
259C8—42no—
260C8—3762no—
261C22—22no—
262C22—42no—
263C22—44no—
264C22—81no—
265C22—82no—
266C22—81no—
267C22#1568900311yescharacteristic
268C22#1568930011yescharacteristic
269C22#1568930011yescharacteristic
270C4—21no—
271C4—21no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—82no—
276C4—3762no—
277C4—1884no—
278C2—21no—
279C2—21no—
280C2—82no—
281C2#2024377711yescharacteristic
282C2#2024389112yes—
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3140C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#3140D47
hypocenterC47#9439
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
3. #20241900C2semidirect
4. #20241282C2semidirect
17. C47×#489C4semidirect
38. C4⋊Q8×C47C2×C4semidirect
50. #97×C47C2×C4semidirect
149. #489Dic47semidirect
163. C47#9439semidirect
168. C4⋊Q8C2×Dic47semidirect
188. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465002
#202420122
#202419002
#202412822
C47×#31402
#943947