Finite Group Databasebeta

← #26114087

Subgroups — #26114087

Order 12032.

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

normal subgroups only

This group has 734 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#20247662C211yesmaximal, characteristic
2#20243488C212yesmaximal, complemented
3#20241715C211yesmaximal, characteristic
4#20241241C212yesmaximal, complemented
5C47×#2273C211yesmaximal, characteristic
6C2×#13093983C2211yescharacteristic
7#15689087C2212yes—
8#15689059C412yescomplemented
9#15689048C412yescomplemented
10#15689011—22no—
11#15688996—21no—
12#15688986C2211yescharacteristic
13#15688979—21no—
14#15688883—24no—
15#15688847—22no—
16C22×C22⋊C4×C47—21no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C4×C22⋊C4×C47—21no—
19C4×C22⋊C4×C47—24no—
20C4×C22⋊C4×C47C2212yes—
21C2×#10625329—22no—
22#13093983—22no—
23#13093983—22no—
24C2×C4⋊Dic47C2×C411yescharacteristic, complemented
25C2×C4⋊Dic47C2×C411yescharacteristic, complemented
26C2×Dic47⋊C4—22no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—22no—
31C2×C4×Dic47—21no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C47⋊#95—21no—
36C47⋊#95C2×C412yescomplemented
37C25×C47D411yescharacteristic
38C23×C4×C47—22no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C22⋊C4×C47—28no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47C2×C412yes—
47C2×C22⋊C4×C47D412yes—
48C2×C42×C47—22no—
49C2×C42×C47D414yes—
50#95×C47—22no—
51#95×C47—22no—
52C24×C47—21no—
53C24×C47—24no—
54C24×C47—21no—
55C24×C47—42no—
56C24×C47—81no—
57C24×C47C22⋊C412yes—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—24no—
61C22×C4×C47—28no—
62C22×C4×C47—21no—
63C22×C4×C47—42no—
64C22×C4×C47—81no—
65C22×C4×C47—82no—
66C22×C4×C47—82no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D412yes—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4112yes—
72C22×Dic47—22no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22⋊C4×C47—28no—
76C22⋊C4×C47—28no—
77C22⋊C4×C47—24no—
78C22⋊C4×C47—24no—
79C42×C47—44no—
80C42×C47—82no—
81C22⋊Dic47—42no—
82C22⋊Dic47—44no—
83C22⋊Dic47—41no—
84C22⋊Dic47—41no—
85C22⋊Dic47—82no—
86C4⋊Dic47—22no—
87C4⋊Dic47—22no—
88Dic47⋊C4—82no—
89C4×Dic47—42no—
90#10625329—44no—
91C2×C47⋊C8—42no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—24no—
95C23×C47—22no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—41no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—81no—
105C23×C47—81no—
106C23×C47—88no—
107C23×C47#10414yes—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#12011yescharacteristic
110C23×C47#12311yescharacteristic
111C2×C4×C47—24no—
112C2×C4×C47—28no—
113C2×C4×C47—22no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—44no—
118C2×C4×C47—42no—
119C2×C4×C47—44no—
120C2×C4×C47—42no—
121C2×C4×C47—81no—
122C2×C4×C47—88no—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12112yes—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C47⋊C8—82no—
133#9477—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—24no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—44no—
143C22×C47—44no—
144C22×C47—44no—
145C22×C47—44no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—84no—
150C22×C47#38611yescharacteristic
151C22×C47C2×#10412yes—
152C22×C47#45812yes—
153C4×C47—22no—
154C4×C47—22no—
155C4×C47—48no—
156C4×C47—42no—
157Dic47—82no—
158Dic47—81no—
159Dic47—81no—
160#2482—471no—
161#2273D4711yescharacteristic
162#1826—472no—
163#1462—471no—
164#1284—472no—
165C2×C47—22no—
166C2×C47—22no—
167C2×C47—41no—
168C2×C47—44no—
169C2×C47#189011yescharacteristic
170C2×C47#189611yescharacteristic
171C2×C47#191111yescharacteristic
172C2×#117—471no—
173C22×C22⋊C4—21no—
174C22×C22⋊C4C2×D4711yescharacteristic
175#407—472no—
176#399—472no—
177#397—472no—
178#388—942no—
179#385—941no—
180#381—471no—
181#379—941no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—24no—
184C4×C22⋊C4C2×D4712yes—
185#339—944no—
186#323—942no—
187C47#947711yescharacteristic, complemented, Sylow
188C25C22⋊D4711yescharacteristic
189C23×C4—22no—
190C23×C4C22⋊D4711yescharacteristic
191C23×C4C22×D4711yescharacteristic
192C2×M16—942no—
193#117—942no—
194#117—942no—
195C2×C4⋊C4—471no—
196C2×C4⋊C4—471no—
197C2×C4⋊C4—942no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4—941no—
206C2×C22⋊C4—941no—
207C2×C22⋊C4C4×D4712yes—
208C2×C22⋊C4D18812yes—
209C2×C42—22no—
210C2×C42—941no—
211C2×C42C22⋊D4714yes—
212C22⋊C8—942no—
213C22⋊C8—942no—
214#95—22no—
215#95—22no—
216#95—472no—
217#95—942no—
218#95—941no—
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24(C2×D47)⋊C412yes—
225C24C2×C22⋊D4711yescharacteristic
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—942no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4(C2×D47)⋊C412yes—
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×D18811yescharacteristic
240C22×C4#1062535012yes—
241C22×C4C2×C22⋊D4712yes—
242M16—1884no—
243C2×C8—1882no—
244C4⋊C4—942no—
245C4⋊C4—942no—
246C4⋊C4—3762no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—3762no—
252C22⋊C4—1882no—
253C22⋊C4—1884no—
254C22⋊C4—1881no—
255C22⋊C4—1881no—
256C42—44no—
257C42—82no—
258C42—1882no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1309395214yes—
275C23C2×(C2×D47)⋊C411yescharacteristic
276C23#1309399311yescharacteristic
277C23#1309400711yescharacteristic
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—88no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4C4×C22⋊D4712yes—
298C2×C4#1309399812yes—
299C8—3762no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1568900211yescharacteristic
317C22C2×#1309395212yes—
318C22#1568931312yes—
319C4—22no—
320C4—22no—
321C4—48no—
322C4—42no—
323C4—3762no—
324C4—3761no—
325C4—3761no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#2024380111yescharacteristic
331C2#2024383111yescharacteristic
332C2#2024392011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2273C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2273D47
hypocenterC47#9477
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. #20243488C2semidirect
4. #20241241C2semidirect
8. #15689059C4semidirect
9. #15689048C4semidirect
24. C2×C4⋊Dic47C2×C4semidirect
25. C2×C4⋊Dic47C2×C4semidirect
36. C47⋊#95C2×C4semidirect
187. C47#9477semidirect

Maximal subgroups

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

subgroupindex
#202476622
#202434882
#202417152
#202412412
C47×#22732
#947747