Finite Group Databasebeta

← #26113728

Subgroups — #26113728

Order 12032.

644 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 111 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#20248868C211yesmaximal, characteristic
2#20241903C212yesmaximal
3#20241718C211yesmaximal, characteristic, complemented
4#20241274C212yesmaximal, complemented
5C47×#2274C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7#15689087C2212yes—
8#15689062C412yescomplemented
9#15689040C412yescomplemented
10#15689013—22no—
11#15689002—21no—
12#15689002—21no—
13C4×C22⋊Dic47C2211yescharacteristic
14#15688856—24no—
15#15688849—22no—
16C2×#117×C47—21no—
17C22×C4⋊C4×C47C2211yescharacteristic
18C4×C4⋊C4×C47—24no—
19C4×C22⋊C4×C47—21no—
20C4×C22⋊C4×C47C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C2×#10625329—22no—
24#13093998—22no—
25#13093998—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C22⋊Dic47—21no—
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×C4×Dic47—21no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40#117×C47—28no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47C2×C412yes—
45C2×C4⋊C4×C47D412yes—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47—22no—
48C2×C42×C47—22no—
49C2×C42×C47—22no—
50#95×C47—22no—
51#95×C47D414yes—
52C24×C47C2×D411yescharacteristic
53C23×D47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—24no—
57C22×C4×C47—24no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—81no—
61C22×C4×C47—81no—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22⋊C412yes—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4112yes—
68C2×C22⋊D47—44no—
69C2×C22⋊D47—41no—
70C2×C22⋊D47—41no—
71C22×Dic47—21no—
72C22×Dic47—22no—
73C2×D188—22no—
74C2×D188—81no—
75C4⋊C4×C47—28no—
76C4⋊C4×C47—24no—
77C4⋊C4×C47—24no—
78C22⋊C4×C47—28no—
79C42×C47—28no—
80C42×C47—82no—
81C42×C47—82no—
82C22⋊Dic47—42no—
83(C2×D47)⋊C4—44no—
84(C2×D47)⋊C4—42no—
85(C2×D47)⋊C4—82no—
86C4⋊Dic47—22no—
87C4×Dic47—81no—
88#10625329—82no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—41no—
99C22×D47—42no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—24no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—28no—
106C2×C4×C47—22no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—44no—
112C2×C4×C47—42no—
113C2×C4×C47—44no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47#10414yes—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12112yes—
119C22⋊D47—82no—
120C22⋊D47—82no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47—42no—
126D188—82no—
127C47⋊C8—82no—
128#9410—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—44no—
137C22×C47#38611yescharacteristic
138C22×C47C2×#10412yes—
139C22×C47#45412yes—
140C2×D47—84no—
141C2×D47—81no—
142C2×D47—81no—
143C2×D47—81no—
144C4×C47—24no—
145C4×C47—22no—
146C4×C47—22no—
147C4×C47—48no—
148C4×C47—42no—
149Dic47—82no—
150Dic47—81no—
151#2591—471no—
152#2274D4711yescharacteristic, complemented
153#1507—472no—
154#1463—471no—
155#1297—472no—
156C2×C47—22no—
157C2×C47—22no—
158C2×C47—41no—
159C2×C47#188811yescharacteristic
160C2×C47#189611yescharacteristic
161C2×C47#191311yescharacteristic
162D47—161no—
163C2×#121—471no—
164C2×#117—21no—
165C22×C4⋊C4C2×D4711yescharacteristic
166#407—472no—
167#400—472no—
168#394—472no—
169#388—942no—
170#386—941no—
171#386—941no—
172C4×C4⋊C4—24no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—471no—
175C4×C22⋊C4C2×D4712yes—
176#327—944no—
177#324—942no—
178C47#941011yescharacteristic, complemented, Sylow
179C22×D4—471no—
180C22×D4—941no—
181C23×C4C22⋊D4711yescharacteristic
182C23×C4C22⋊D4711yescharacteristic
183C23×C4C22×D4711yescharacteristic
184C2×M16—942no—
185#121—942no—
186#121—942no—
187#117—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—471no—
192C2×C4⋊C4C4×D4712yes—
193C2×C4⋊C4D18812yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—942no—
196C2×C22⋊C4—941no—
197C2×C22⋊C4—942no—
198C2×C22⋊C4—941no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—941no—
203C22⋊C8—942no—
204C22⋊C8—942no—
205#95—22no—
206#95—472no—
207#95—942no—
208#95—942no—
209#95C22⋊D4714yes—
210C24—941no—
211C24C2×C22⋊D4711yescharacteristic
212C2×D4—942no—
213C2×D4—3761no—
214C2×D4—1884no—
215C2×D4—1881no—
216C2×D4—1881no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—942no—
226C22×C4—941no—
227C22×C4(C2×D47)⋊C412yes—
228C22×C4(C2×D47)⋊C412yes—
229C22×C4C2×C4×D4711yescharacteristic
230C22×C4C2×D18811yescharacteristic
231C22×C4#1062535012yes—
232C22×C4C2×C22⋊D4712yes—
233M16—3762no—
234C2×C8—1882no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—942no—
239C22⋊C4—28no—
240C22⋊C4—3762no—
241C22⋊C4—1882no—
242C22⋊C4—1884no—
243C22⋊C4—1882no—
244C42—28no—
245C42—82no—
246C42—82no—
247C42—3761no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—1882no—
252C23—1882no—
253C23—1881no—
254C23—1882no—
255C23C2×(C2×D47)⋊C411yescharacteristic
256C23#1309399311yescharacteristic
257C23#1309400711yescharacteristic
258D4—3762no—
259D4—3762no—
260D4—3762no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1881no—
281C2×C4—1882no—
282C2×C4#1309395214yes—
283C2×C4C4×C22⋊D4712yes—
284C2×C4#1309399812yes—
285C8—3762no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—3764no—
295C22—3761no—
296C22—3761no—
297C22—3761no—
298C22#1568900211yescharacteristic
299C22C2×#1309395212yes—
300C22#1568929612yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—48no—
305C4—42no—
306C4—3762no—
307C4—3761no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—7521no—
312C2#2024379111yescharacteristic
313C2#2024383111yescharacteristic
314C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2274C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#2274D47
hypocenterC47#9410
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #20241718C2semidirect
4. #20241274C2semidirect
5. C47×#2274C2semidirect
8. #15689062C4semidirect
9. #15689040C4semidirect
152. #2274D47semidirect
178. C47#9410semidirect

Maximal subgroups

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

subgroupindex
#202488682
#202419032
#202417182
#202412742
C47×#22742
#941047