Finite Group Databasebeta

← #26113716

Subgroups — #26113716

Order 12032.

618 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 107 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#20248768C211yesmaximal, characteristic
2#20245745C211yesmaximal, characteristic
3#20241909C212yesmaximal
4#20241273C212yesmaximal
5C47×#1463C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×#13093983—21no—
8#15689053C412yescomplemented
9#15689043C412yescomplemented
10#15689018—22no—
11#15688996—21no—
12#15688996—21no—
13C4⋊C4×Dic47—24no—
14C4×C22⋊Dic47—21no—
15C4×C22⋊Dic47C2211yescharacteristic
16C4×C22⋊Dic47C2212yes—
17#15688861—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C47×#407C2212yes—
20C47×#324—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094003—22no—
23#13094003—22no—
24#13093983—28no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—22no—
34C2×C4×Dic47—22no—
35C2×C4×Dic47—22no—
36C2×C4×Dic47—21no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95C2×C412yes—
41C47⋊#95D414yes—
42C23×C4×C47D411yescharacteristic
43C23×C4×C47D411yescharacteristic
44C23×C4×C47C2311yescharacteristic
45C2×M16×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—22no—
48C2×C4⋊C4×C47C2×C412yes—
49C2×C4⋊C4×C47D412yes—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C24×C47C2×D411yescharacteristic
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—24no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—81no—
59C22×C4×C47—81no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47#4112yes—
66C22×Dic47—22no—
67C22×Dic47—24no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C2×Dic94—22no—
71C2×Dic94—81no—
72M16×C47—82no—
73C2×C8×C47—42no—
74C4⋊C4×C47—24no—
75C4⋊C4×C47—24no—
76C22⋊Dic47—24no—
77C22⋊Dic47—28no—
78C22⋊Dic47—41no—
79C22⋊Dic47—41no—
80C4⋊Dic47—28no—
81C4⋊Dic47—22no—
82Dic47⋊C4—44no—
83Dic47⋊C4—42no—
84Dic47⋊C4—82no—
85C4×Dic47—28no—
86C4×Dic47—82no—
87C4×Dic47—82no—
88C4×Dic47—81no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—41no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47#12011yescharacteristic
94C23×C47#12311yescharacteristic
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—24no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—42no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47—41no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—81no—
108C2×C4×C47#10414yes—
109C2×C4×C47C4×D412yes—
110C2×C4×C47#12112yes—
111C2×Dic47—28no—
112C2×Dic47—24no—
113C2×Dic47—44no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—44no—
117C2×Dic47—42no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—42no—
121Dic94—82no—
122C8×C47—82no—
123#9408—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—44no—
132C22×C47#38611yescharacteristic
133C22×C47C2×#10412yes—
134C22×C47#45612yes—
135C4×C47—24no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—42no—
139Dic47—48no—
140Dic47—44no—
141Dic47—81no—
142Dic47—81no—
143#2583—471no—
144#2274—471no—
145#1508—472no—
146#1463D4711yescharacteristic
147#1297—472no—
148C2×C47—22no—
149C2×C47—22no—
150C2×C47—41no—
151C2×C47#188611yescharacteristic
152C2×C47#189611yescharacteristic
153C2×C47#191311yescharacteristic
154C2×C22⋊Q8—471no—
155C2×#117—941no—
156C22×C4⋊C4C2×D4711yescharacteristic
157#407C2×D4712yes—
158#398—472no—
159#395—472no—
160#389—942no—
161#385—941no—
162#385—941no—
163C4×C4⋊C4—944no—
164C4×C22⋊C4—471no—
165C4×C22⋊C4—472no—
166C4×C22⋊C4—941no—
167#328—944no—
168#324—22no—
169C47#940811yescharacteristic, complemented, Sylow
170C22×Q8—471no—
171C23×C4D18811yescharacteristic
172C23×C4D18811yescharacteristic
173C23×C4C22×D4711yescharacteristic
174C2×M16—22no—
175C22⋊Q8—942no—
176C22⋊Q8—942no—
177#117—948no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—471no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—942no—
183C2×C4⋊C4—942no—
184C2×C4⋊C4—941no—
185C2×C4⋊C4C4×D4712yes—
186C2×C4⋊C4C22⋊D4712yes—
187C2×C22⋊C4—942no—
188C2×C22⋊C4—941no—
189C2×C22⋊C4—941no—
190C2×C42—942no—
191C2×C42—942no—
192C2×C42—942no—
193C2×C42—941no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—472no—
197#95—474no—
198#95—942no—
199#95—942no—
200#95—942no—
201C24C2×D18811yescharacteristic
202C2×Q8—942no—
203C2×Q8—3761no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—942no—
212C22×C4—944no—
213C22×C4—941no—
214C22×C4—941no—
215C22×C4(C2×D47)⋊C412yes—
216C22×C4(C2×D47)⋊C412yes—
217C22×C4C2×C4×D4711yescharacteristic
218C22×C4D4×D4712yes—
219C22×C4#1062535212yes—
220C22×C4C2×C22⋊D4711yescharacteristic
221M16—82no—
222C2×C8—42no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—948no—
226C4⋊C4—942no—
227C4⋊C4—3762no—
228C4⋊C4—1884no—
229C4⋊C4—1882no—
230C22⋊C4—944no—
231C22⋊C4—948no—
232C22⋊C4—1881no—
233C22⋊C4—1881no—
234C42—948no—
235C42—3762no—
236C42—3761no—
237C42—3762no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×(C2×D47)⋊C411yescharacteristic
242C23#1309399411yescharacteristic
243C23#1309400811yescharacteristic
244Q8—3762no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—948no—
259C2×C4—944no—
260C2×C4—1882no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1884no—
265C2×C4—1882no—
266C2×C4—1884no—
267C2×C4—1882no—
268C2×C4#1309395014yes—
269C2×C4(C22⋊D47)⋊C412yes—
270C2×C4#1309399612yes—
271C8—82no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1568900011yescharacteristic
281C22C2×#1309395012yes—
282C22#1568930812yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—42no—
287C4—3761no—
288C4—3761no—
289C4—1888no—
290C4—1884no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2024377911yescharacteristic
295C2#2024382911yescharacteristic
296C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1463C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1463D47
hypocenterC47#9408
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
8. #15689053C4semidirect
9. #15689043C4semidirect
169. C47#9408semidirect

Maximal subgroups

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

subgroupindex
#202487682
#202457452
#202419092
#202412732
C47×#14632
#940847