Finite Group Databasebeta

← #26113713

Subgroups — #26113713

Order 12032.

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

normal subgroups only

This group has 726 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#20248546C211yesmaximal, characteristic
2#20241907C212yesmaximal, complemented
3#20241715C211yesmaximal, characteristic
4#20241274C212yesmaximal, complemented
5C47×#2273C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689087C2212yes—
8#15689048C412yescomplemented
9#15689035C412yescomplemented
10#15689017—22no—
11#15688986—21no—
12#15688979—21no—
13C4×C22⋊Dic47C2211yescharacteristic
14#15688859—24no—
15#15688847—22no—
16C22×C22⋊C4×C47—21no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C4×C22⋊C4×C47—24no—
19C4×C22⋊C4×C47—21no—
20C4×C22⋊C4×C47C2212yes—
21C22×Dic94C2×C411yescharacteristic, complemented
22C2×#10625329—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic, complemented
26C2×Dic47⋊C4—21no—
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—22no—
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×C47—22no—
50#95×C47—22no—
51#95×C47D414yes—
52C24×C47—21no—
53C24×C47—24no—
54C24×C47—21no—
55C24×C47—42no—
56C24×C47—81no—
57C24×C47C22⋊C412yes—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—24no—
60C22×C4×C47—21no—
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—
75C2×Dic94—22no—
76C2×Dic94—81no—
77C22⋊C4×C47—28no—
78C22⋊C4×C47—28no—
79C22⋊C4×C47—24no—
80C22⋊C4×C47—24no—
81C42×C47—82no—
82C42×C47—82no—
83C22⋊Dic47—42no—
84C22⋊Dic47—44no—
85C22⋊Dic47—41no—
86C22⋊Dic47—41no—
87C22⋊Dic47—82no—
88C4⋊Dic47—22no—
89Dic47⋊C4—42no—
90C4×Dic47—81no—
91#10625329—82no—
92C2×C47⋊C8—42no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—24no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—41no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—81no—
106C23×C47—81no—
107C23×C47—88no—
108C23×C47#10414yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C23×C47#12011yescharacteristic
111C23×C47#12311yescharacteristic
112C2×C4×C47—28no—
113C2×C4×C47—22no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—42no—
120C2×C4×C47—44no—
121C2×C4×C47—42no—
122C2×C4×C47—81no—
123C2×C4×C47—88no—
124C2×C4×C47C4×D412yes—
125C2×C4×C47#12112yes—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133Dic94—82no—
134C47⋊C8—82no—
135#9407—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—24no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—44no—
145C22×C47—44no—
146C22×C47—44no—
147C22×C47—44no—
148C22×C47—42no—
149C22×C47—42no—
150C22×C47—82no—
151C22×C47—84no—
152C22×C47#38611yescharacteristic
153C22×C47C2×#10412yes—
154C22×C47#45512yes—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—48no—
158C4×C47—42no—
159Dic47—82no—
160Dic47—81no—
161Dic47—81no—
162#2561—471no—
163#2273D4711yescharacteristic
164#1508—472no—
165#1462—471no—
166#1297—472no—
167C2×C47—22no—
168C2×C47—22no—
169C2×C47—41no—
170C2×C47—44no—
171C2×C47#188511yescharacteristic
172C2×C47#189611yescharacteristic
173C2×C47#191311yescharacteristic
174C2×C22⋊Q8—471no—
175C22×C22⋊C4—21no—
176C22×C22⋊C4C2×D4711yescharacteristic
177#407—472no—
178#397—472no—
179#393—472no—
180#388—942no—
181#381—941no—
182#379—941no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—471no—
186C4×C22⋊C4C2×D4712yes—
187#328—944no—
188#323—942no—
189C47#940711yescharacteristic, complemented, Sylow
190C25C22⋊D4711yescharacteristic
191C22×Q8—471no—
192C23×C4—22no—
193C23×C4C22⋊D4711yescharacteristic
194C23×C4C22×D4711yescharacteristic
195C2×M16—942no—
196C22⋊Q8—942no—
197C22⋊Q8—942no—
198C2×C4⋊C4—471no—
199C2×C4⋊C4—941no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—942no—
207C2×C22⋊C4—941no—
208C2×C22⋊C4—941no—
209C2×C22⋊C4C4×D4712yes—
210C2×C22⋊C4D18812yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—941no—
214C22⋊C8—942no—
215C22⋊C8—942no—
216#95—22no—
217#95—472no—
218#95—942no—
219#95—942no—
220#95C22⋊D4714yes—
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24(C2×D47)⋊C412yes—
227C24C2×C22⋊D4711yescharacteristic
228C2×Q8—942no—
229C2×Q8—3761no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—942no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4(C2×D47)⋊C412yes—
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4#1062535012yes—
245C22×C4C2×C22⋊D4712yes—
246M16—3762no—
247C2×C8—1882no—
248C4⋊C4—942no—
249C4⋊C4—1882no—
250C22⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—3762no—
255C22⋊C4—1882no—
256C22⋊C4—1884no—
257C22⋊C4—1881no—
258C22⋊C4—1881no—
259C42—82no—
260C42—82no—
261C42—3761no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1309395214yes—
278C23C2×(C2×D47)⋊C411yescharacteristic
279C23#1309399311yescharacteristic
280C23#1309400711yescharacteristic
281Q8—3762no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—88no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4—1882no—
297C2×C4—1881no—
298C2×C4—1881no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4C4×C22⋊D4712yes—
302C2×C4#1309399812yes—
303C8—3762no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1568900211yescharacteristic
321C22C2×#1309395212yes—
322C22#1568930112yes—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—3762no—
328C4—3761no—
329C4—3761no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#2024377611yescharacteristic
335C2#2024383111yescharacteristic
336C2#2024393011yescharacteristic
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#9407
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. #20241907C2semidirect
4. #20241274C2semidirect
8. #15689048C4semidirect
9. #15689035C4semidirect
21. C22×Dic94C2×C4semidirect
25. C2×C4⋊Dic47C2×C4semidirect
36. C47⋊#95C2×C4semidirect
189. C47#9407semidirect

Maximal subgroups

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

subgroupindex
#202485462
#202419072
#202417152
#202412742
C47×#22732
#940747