Finite Group Databasebeta

← #26112321

Subgroups — #26112321

Order 12032.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 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#20249193C211yesmaximal, characteristic
2#20241959C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241255C212yesmaximal, complemented
5C47×#2315C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689059C412yescomplemented
9#15689046—22no—
10#15689035C412yescomplemented
11#15688996—21no—
12#15688983C2211yescharacteristic
13#15688979—21no—
14#15688859—24no—
15#15688847—22no—
16C22×C22⋊C4×C47—21no—
17C22×C22⋊C4×C47C2211yescharacteristic
18C47×#385—24no—
19C47×#380—21no—
20C47×#380C2212yes—
21C22×Dic94C2×C411yescharacteristic, complemented
22C22×C47⋊C8—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic, complemented
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yescomplemented
37C25×C47D411yescharacteristic
38C23×C4×C47—22no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—42no—
48C2×C22⋊C4×C47—42no—
49C2×C22⋊C4×C47C2×C412yes—
50C2×C22⋊C4×C47D412yes—
51#95×C47—22no—
52#95×C47D414yes—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—81no—
58C24×C47—81no—
59C24×C47C22⋊C412yes—
60C24×C47SD1612yes—
61C24×C47Q1612yes—
62C24×C47C2×D411yescharacteristic
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—24no—
66C22×C4×C47—44no—
67C22×C4×C47—81no—
68C22×C4×C47—81no—
69C22×C4×C47—82no—
70C22×C4×C47—82no—
71C22×C4×C47C22⋊C412yes—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47#4112yes—
76C22×Dic47—21no—
77C22×Dic47—22no—
78C22×Dic47—21no—
79C2×Dic94—22no—
80C2×Dic94—81no—
81C4⋊C4×C47—82no—
82C4⋊C4×C47—82no—
83C22⋊C4×C47—24no—
84C22⋊C4×C47—24no—
85C22⋊C4×C47—44no—
86C22⋊C4×C47—42no—
87C22⋊C4×C47—42no—
88C22⋊Dic47—41no—
89C22⋊Dic47—41no—
90C22⋊Dic47—41no—
91C22⋊Dic47—41no—
92C22⋊Dic47—82no—
93C22⋊Dic47—82no—
94C4⋊Dic47—22no—
95C4⋊Dic47—81no—
96Dic47⋊C4—42no—
97Dic47⋊C4—82no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—82no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—81no—
113C23×C47—81no—
114C23×C47—82no—
115C23×C47—82no—
116C23×C47—82no—
117C23×C47—82no—
118C23×C47#10314yes—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47#12011yescharacteristic
121C23×C47#12311yescharacteristic
122C23×C47C2×SD1611yescharacteristic
123C23×C47C2×Q1611yescharacteristic
124C23×C47AGL(1,ℤ/8)11yescharacteristic
125C23×C47#13711yescharacteristic
126C2×C4×C47—22no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—44no—
134C2×C4×C47—44no—
135C2×C4×C47—44no—
136C2×C4×C47—81no—
137C2×C4×C47—88no—
138C2×C4×C47C4×D412yes—
139C2×C4×C47#12112yes—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—42no—
147Dic94—82no—
148C47⋊C8—82no—
149#9199—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—24no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—44no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47#38611yescharacteristic
170C22×C47C2×#10311yescharacteristic
171C22×C47#41711yescharacteristic
172C22×C47#45011yescharacteristic
173C22×C47#45111yescharacteristic
174C22×C47#48311yescharacteristic
175C22×C47#48411yescharacteristic
176C4×C47—22no—
177C4×C47—22no—
178C4×C47—42no—
179C4×C47—84no—
180Dic47—82no—
181Dic47—81no—
182Dic47—81no—
183#2626—471no—
184#2315D4711yescharacteristic
185#1520—472no—
186C2×#323—471no—
187#1289—472no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—44no—
195C2×C47#187211yescharacteristic
196C2×C47#191111yescharacteristic
197C2×C47#191311yescharacteristic
198C2×C22⋊Q8—471no—
199C22×C22⋊C4—21no—
200C22×C22⋊C4C2×D4711yescharacteristic
201C2×C22⋊C8—472no—
202#399—472no—
203#396—942no—
204#393—472no—
205#385—24no—
206#385—941no—
207#380—21no—
208#380—471no—
209#380C2×D4712yes—
210#379—941no—
211#328—944no—
212#323—942no—
213C47#919911yescharacteristic, complemented, Sylow
214C25C22⋊D4711yescharacteristic
215C22×Q8—471no—
216C23×C4—22no—
217C23×C4C22⋊D4711yescharacteristic
218C23×C4C22×D4711yescharacteristic
219C22×C8—942no—
220C22⋊Q8—942no—
221C22⋊Q8—942no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—471no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—941no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—942no—
235C2×C22⋊C4—942no—
236C2×C22⋊C4—941no—
237C2×C22⋊C4—941no—
238C2×C22⋊C4C4×D4712yes—
239C2×C22⋊C4D18812yes—
240C22⋊C8—942no—
241C22⋊C8—942no—
242#95—22no—
243#95—472no—
244#95—942no—
245#95C22⋊D4714yes—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24(C2×D47)⋊C412yes—
253C24#1062533512yes—
254C24#1062533712yes—
255C24C2×C22⋊D4711yescharacteristic
256C2×Q8—942no—
257C2×Q8—3761no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—942no—
267C22×C4—941no—
268C22×C4—941no—
269C22×C4(C2×D47)⋊C412yes—
270C22×C4C2×C4×D4711yescharacteristic
271C22×C4C2×D18811yescharacteristic
272C22×C4#1062535012yes—
273C22×C4C2×C22⋊D4712yes—
274C2×C8—3762no—
275C2×C8—1882no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—942no—
279C4⋊C4—3762no—
280C4⋊C4—3761no—
281C4⋊C4—1882no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—3762no—
288C22⋊C4—3762no—
289C22⋊C4—1881no—
290C22⋊C4—1881no—
291C22⋊C4—1881no—
292C22⋊C4—1881no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1309394714yes—
312C23C2×(C2×D47)⋊C411yescharacteristic
313C23#1309399311yescharacteristic
314C23#1309400711yescharacteristic
315C23C2×#1062533511yescharacteristic
316C23C2×#1062533711yescharacteristic
317C23#1309405011yescharacteristic
318C23#1309405511yescharacteristic
319Q8—3762no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1881no—
337C2×C4—1881no—
338C2×C4—1882no—
339C2×C4C4×C22⋊D4712yes—
340C2×C4#1309399812yes—
341C8—3762no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1568900211yescharacteristic
362C22C2×#1309394711yescharacteristic
363C22#1568912211yescharacteristic
364C22#1568927411yescharacteristic
365C22#1568927911yescharacteristic
366C22#1568945311yescharacteristic
367C22#1568945811yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—3762no—
373C4—3761no—
374C4—3761no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2024369511yescharacteristic
383C2#2024392011yescharacteristic
384C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2315C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2315D47
hypocenterC47#9199
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #20241959C2semidirect
4. #20241255C2semidirect
8. #15689059C4semidirect
10. #15689035C4semidirect
21. C22×Dic94C2×C4semidirect
26. C2×C4⋊Dic47C2×C4semidirect
36. C47⋊#95C2×C4semidirect
213. C47#9199semidirect

Maximal subgroups

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

subgroupindex
#202491932
#202419592
C2×#156888472
#202412552
C47×#23152
#919947