Finite Group Databasebeta

← #26112029

Subgroups — #26112029

Order 12032.

1088 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×C4×C22⋊Dic47C211yesmaximal, characteristic, complemented
2#20243298C211yesmaximal, characteristic, complemented
3C2×#15688857C212yesmaximal, complemented
4#20241266C212yesmaximal, complemented
5C47×#2847C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C22×C4×Dic47—21no—
8C2×C22⋊(C47⋊C8)—22no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C4×C22⋊Dic47—24no—
11C2×C47⋊#95C2212yes—
12#15688857—24no—
13C24×C4×C47C2211yescharacteristic
14C2×#121×C47C412yescomplemented
15C2×#121×C47C2211yescharacteristic
16C2×#120×C47—21no—
17C47×#386—22no—
18C47×#386—21no—
19C47×#380C412yescomplemented
20C47×#380C2211yescharacteristic
21C23×Dic47—22no—
22C22×C47⋊C8—22no—
23C2×C22⋊Dic47—24no—
24C2×C22⋊Dic47—28no—
25C2×C4×Dic47—24no—
26C2×C4×Dic47—44no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—44no—
30C47⋊#95—22no—
31C47⋊#95D414yescomplemented
32C25×C47D411yescharacteristic
33C22×D4×C47—22no—
34C22×D4×C47—21no—
35C22×D4×C47C2×C411yescharacteristic
36C23×C4×C47—22no—
37C23×C4×C47—22no—
38C23×C4×C47—44no—
39C23×C4×C47C2×C412yes—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D412yes—
42C23×C4×C47C2311yescharacteristic
43#121×C47—22no—
44#121×C47—22no—
45#121×C47—44no—
46#120×C47—44no—
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—22no—
52C2×C22⋊C4×C47—21no—
53#95×C47—22no—
54#95×C47—21no—
55C24×C47—24no—
56C24×C47—21no—
57C24×C47—21no—
58C24×C47—22no—
59C24×C47—22no—
60C24×C47—21no—
61C24×C47—42no—
62C24×C47C22⋊C412yes—
63C24×C47C2×D411yescharacteristic
64C2×D4×C47—22no—
65C2×D4×C47—42no—
66C2×D4×C47—44no—
67C2×D4×C47—44no—
68C2×D4×C47—44no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—42no—
72C2×D4×C47—42no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—24no—
76C22×C4×C47—24no—
77C22×C4×C47—24no—
78C22×C4×C47—24no—
79C22×C4×C47—24no—
80C22×C4×C47—21no—
81C22×C4×C47—22no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47—42no—
85C22×C4×C47—48no—
86C22×C4×C47—48no—
87C22×C4×C47C22⋊C412yes—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47#4112yes—
92C22×Dic47—28no—
93C22×Dic47—24no—
94C22×Dic47—44no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—42no—
99C22⋊C4×C47—44no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—44no—
102C22⋊C4×C47—42no—
103C22⋊Dic47—216no—
104C4×Dic47—48no—
105C2×C47⋊C8—44no—
106C2×C47⋊C8—42no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—24no—
110C23×C47—24no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—22no—
117C23×C47—48no—
118C23×C47—48no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—41no—
124C23×C47—42no—
125C23×C47—41no—
126C23×C47—41no—
127C23×C47—42no—
128C23×C47#10414yes—
129C23×C47C2×C22⋊C411yescharacteristic
130C23×C47#12011yescharacteristic
131C23×C47#12311yescharacteristic
132C23×C47#13512yes—
133C23×C47AGL(1,ℤ/8)12yes—
134D4×C47—82no—
135D4×C47—88no—
136D4×C47—82no—
137D4×C47—82no—
138C2×C4×C47—24no—
139C2×C4×C47—24no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—48no—
147C2×C4×C47—48no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47#9914yes—
155C2×C4×C47C4×D412yes—
156C2×C4×C47#12112yes—
157C2×Dic47—28no—
158C2×Dic47—44no—
159C2×Dic47—48no—
160C2×Dic47—48no—
161C2×Dic47—44no—
162C47⋊C8—82no—
163#9150—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—24no—
169C22×C47—24no—
170C22×C47—24no—
171C22×C47—24no—
172C22×C47—24no—
173C22×C47—48no—
174C22×C47—48no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—81no—
180C22×C47—81no—
181C22×C47—81no—
182C22×C47—82no—
183C22×C47—82no—
184C22×C47#38611yescharacteristic
185C22×C47C2×#9912yes—
186C22×C47#41611yescharacteristic
187C22×C47#41811yescharacteristic
188C22×C47C2×#10412yes—
189C22×C47#44912yes—
190C22×C47#48212yes—
191C4×C47—22no—
192C4×C47—24no—
193C4×C47—22no—
194C4×C47—81no—
195C4×C47—82no—
196Dic47—48no—
197#2847D4711yescharacteristic
198C2×C4×C22⋊C4—471no—
199#1779—471no—
200C2×#327—472no—
201#1294—472no—
202C2×C47—22no—
203C2×C47—22no—
204C2×C47—44no—
205C2×C47—161no—
206C2×C47#150112yes—
207C2×C47#150712yes—
208C2×C47#187511yescharacteristic
209C2×C47#189611yescharacteristic
210C2×C47#189811yescharacteristic
211C24×C4C2×D4711yescharacteristic
212C2×#121Dic4712yescomplemented
213C2×#121C2×D4711yescharacteristic
214C2×#120—21no—
215C22×C22⋊C4—941no—
216C22×C42—941no—
217C2×C22⋊C8—472no—
218C2×C22⋊C8—942no—
219#386—21no—
220#386—22no—
221#380Dic4712yescomplemented
222#380C2×D4711yescharacteristic
223C4×C22⋊C4—944no—
224C2×#95—472no—
225#327—944no—
226C47#915011yescharacteristic, complemented, Sylow
227C25C22⋊D4711yescharacteristic
228C22×D4—22no—
229C22×D4—21no—
230C22×D4C2×Dic4711yescharacteristic
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—44no—
234C23×C4—942no—
235C23×C4C2×Dic4712yes—
236C23×C4C22⋊D4711yescharacteristic
237C23×C4C22⋊D4712yes—
238C23×C4C22×D4711yescharacteristic
239C22×C8—942no—
240#121—22no—
241#121—22no—
242#121—44no—
243#120—44no—
244C2×C4⋊C4C2×Dic4711yescharacteristic
245C2×C4⋊C4C2×Dic4712yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—948no—
251C2×C22⋊C4—944no—
252C2×C42—944no—
253C2×C42—1884no—
254C22⋊C8—942no—
255C22⋊C8—942no—
256C22⋊C8—1884no—
257#95—22no—
258#95—21no—
259#95—474no—
260#95—942no—
261C24—24no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—42no—
268C24C22⋊Dic4712yes—
269C24C2×C22⋊D4711yescharacteristic
270C2×D4—22no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—948no—
294C22×C4—944no—
295C22×C4—1884no—
296C22×C4—1882no—
297C22×C4—1882no—
298C22×C4C22⋊Dic4712yes—
299C22×C4D4×D4712yes—
300C22×C4#1062535212yes—
301C22×C4C22×Dic4711yescharacteristic
302C22×C4C2×C22⋊D4711yescharacteristic
303C2×C8—1884no—
304C2×C8—1882no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—9416no—
312C42—1888no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—48no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23#1309395114yes—
335C23C2×C22⋊Dic4711yescharacteristic
336C23#1309399211yescharacteristic
337C23#1309400511yescharacteristic
338C23#1309404412yes—
339C23#1309405012yes—
340D4—82no—
341D4—88no—
342D4—82no—
343D4—82no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—948no—
361C2×C4—1884no—
362C2×C4—1888no—
363C2×C4—1888no—
364C2×C4—1884no—
365C2×C4#1309393914yes—
366C2×C4D4×Dic4712yes—
367C2×C4#1309399612yes—
368C8—3762no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—48no—
379C22—48no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—82no—
387C22—82no—
388C22—81no—
389C22#1568900311yescharacteristic
390C22C2×#1309393912yes—
391C22#1568911911yescharacteristic
392C22#1568912811yescharacteristic
393C22C2×#1309395112yes—
394C22#1568926612yes—
395C22#1568944412yes—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—82no—
400C4—81no—
401C4—1888no—
402C2—22no—
403C2—22no—
404C2—44no—
405C2—161no—
406C2#2024187912yes—
407C2#2024190412yes—
408C2#2024371111yescharacteristic
409C2#2024383211yescharacteristic
410C2#2024384211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#2847C2
socleC23×C47C2×C22⋊C4
cosocleC23×C4×C47C23
hypercenter#2847D47
hypocenterC47#9150
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊Dic47C2semidirect
2. #20243298C2semidirect
3. C2×#15688857C2semidirect
4. #20241266C2semidirect
14. C2×#121×C47C4semidirect
19. C47×#380C4semidirect
31. C47⋊#95D4semidirect
212. C2×#121Dic47semidirect
221. #380Dic47semidirect
226. C47#9150semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic472
#202432982
C2×#156888572
#202412662
C47×#28472
#915047