Finite Group Databasebeta

← #26111268

Subgroups — #26111268

Order 12032.

908 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246002C211yesmaximal, characteristic
2#20243323C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241267C212yesmaximal, complemented
5C2×C47×#393C211yesmaximal, characteristic
6C22×C22⋊Dic47—21no—
7C2×#13093948—24no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15688998—24no—
10#15688983—21no—
11#15688983C2212yes—
12#15688847—22no—
13C23×Q8×C47C2211yescharacteristic
14C22×C22⋊C4×C47—21no—
15C22×C22⋊C4×C47C2211yescharacteristic
16C47×#393—22no—
17C47×#393C414yescomplemented
18C2×#95×C47—21no—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47—22no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47D414yes—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—42no—
27C2×C22⋊Dic47—42no—
28#13093948—48no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C25×C47D411yescharacteristic
34C22×Q8×C47—21no—
35C22×Q8×C47—22no—
36C22×Q8×C47—22no—
37C22×Q8×C47—44no—
38C22×Q8×C47C2×C412yescomplemented
39C23×C4×C47—21no—
40C23×C4×C47—21no—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—24no—
46C2×C22⋊C4×C47—44no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49#95×C47—24no—
50#95×C47—21no—
51#95×C47C2×C412yescomplemented
52C24×C47—21no—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—21no—
56C24×C47—42no—
57C24×C47—42no—
58C24×C47C22⋊C412yes—
59C24×C47SD1612yes—
60C24×C47Q1612yes—
61C24×C47C2×D411yescharacteristic
62C2×Q8×C47—28no—
63C2×Q8×C47—24no—
64C2×Q8×C47—44no—
65C2×Q8×C47—44no—
66C2×Q8×C47—44no—
67C2×Q8×C47—48no—
68C2×Q8×C47—42no—
69C22×C4×C47—24no—
70C22×C4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—22no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—42no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47C22⋊C412yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47#4112yes—
88C22×Dic47—24no—
89C22×Dic47—44no—
90C22×Dic47—82no—
91C22×Dic47—82no—
92C22⋊C4×C47—24no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—48no—
95C22⋊Dic47—44no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C4⋊Dic47—44no—
99C4⋊Dic47—82no—
100C2×C47⋊C8—44no—
101C2×C47⋊C8—42no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—44no—
113C23×C47—44no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—82no—
117C23×C47—82no—
118C23×C47—81no—
119C23×C47—81no—
120C23×C47#10314yescomplemented
121C23×C47C2×C22⋊C411yescharacteristic
122C23×C47#12011yescharacteristic
123C23×C47#12311yescharacteristic
124C23×C47C2×SD1611yescharacteristic
125C23×C47C2×Q1611yescharacteristic
126C23×C47AGL(1,ℤ/8)11yescharacteristic
127C23×C47#13711yescharacteristic
128Q8×C47—28no—
129Q8×C47—48no—
130Q8×C47—44no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—22no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—44no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—44no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—81no—
147C2×C4×C47—82no—
148C2×C4×C47—82no—
149C2×C4×C47C4×D412yes—
150C2×C4×C47#12112yes—
151C2×Dic47—44no—
152C2×Dic47—44no—
153C2×Dic47—44no—
154C2×Dic47—88no—
155C47⋊C8—82no—
156#9013—471nomaximal, Sylow
157C22×C47—22no—
158C22×C47—22no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—24no—
166C22×C47—44no—
167C22×C47—44no—
168C22×C47—44no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—82no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47#38611yescharacteristic
175C22×C47C2×#10311yescharacteristic
176C22×C47#41711yescharacteristic
177C22×C47#45011yescharacteristic
178C22×C47#45111yescharacteristic
179C22×C47#45512yes—
180C22×C47B(4,𝔽2)12yes—
181C22×C47#48311yescharacteristic
182C22×C47#48411yescharacteristic
183C4×C47—22no—
184C4×C47—22no—
185C4×C47—42no—
186C4×C47—44no—
187C4×C47—82no—
188Dic47—84no—
189C2×#393D4711yescharacteristic
190#2315—471no—
191#1787—472no—
192C2×#323—471no—
193#1294—472no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—44no—
199C2×C47#160112yes—
200C2×C47#160212yes—
201C2×C47#187211yescharacteristic
202C2×C47#188511yescharacteristic
203C2×C47#188911yescharacteristic
204C23×Q8C2×D4711yescharacteristic
205C22×C22⋊C4—21no—
206C22×C22⋊C4—941no—
207C22×C22⋊C4C2×D4711yescharacteristic
208C2×#103—944no—
209C2×C22⋊C8—472no—
210#393—22no—
211#393Dic4714yescomplemented
212#385—944no—
213#380—472no—
214#380—941no—
215C2×#95—21no—
216C2×#95C2×D4711yescharacteristic
217#323—942no—
218C47#901311yescharacteristic, complemented, Sylow
219C25C22⋊D4711yescharacteristic
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8C2×Dic4712yescomplemented
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—942no—
228C23×C4C22⋊D4711yescharacteristic
229C23×C4C22×D4711yescharacteristic
230C22×C8—942no—
231C2×C4⋊C4—474no—
232C2×C4⋊C4—942no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—942no—
238C2×C22⋊C4—942no—
239C2×C22⋊C4—1882no—
240C2×C22⋊C4—1882no—
241C2×C22⋊C4C2×Dic4712yes—
242C2×C22⋊C4C22⋊D4712yes—
243#103—1888no—
244C22⋊C8—942no—
245C22⋊C8—942no—
246#95—24no—
247#95—21no—
248#95—942no—
249#95—942no—
250#95C2×Dic4712yescomplemented
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—42no—
256C24—42no—
257C24#1062533612yes—
258C24#1062533712yes—
259C24C22⋊Dic4712yes—
260C24C2×C22⋊D4711yescharacteristic
261C2×Q8—28no—
262C2×Q8—24no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—42no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—944no—
283C22×C4—3762no—
284C22×C4—3762no—
285C22×C4—1884no—
286C22×C4C22⋊Dic4712yes—
287C22×C4D4×D4712yes—
288C22×C4#1062535212yes—
289C22×C4C22×Dic4711yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291C2×C8—1884no—
292C2×C8—1882no—
293C4⋊C4—3762no—
294C4⋊C4—1884no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—48no—
298C22⋊C4—1884no—
299C22⋊C4—1882no—
300C22⋊C4—1882no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—82no—
318C23—82no—
319C23#1309394814yescomplemented
320C23C2×C22⋊Dic4711yescharacteristic
321C23#1309399211yescharacteristic
322C23#1309400511yescharacteristic
323C23C2×#1062533611yescharacteristic
324C23C2×#1062533711yescharacteristic
325C23#1309404911yescharacteristic
326C23#1309405411yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—3768no—
349C2×C4—1884no—
350C2×C4—1884no—
351C2×C4—1884no—
352C2×C4D4×Dic4712yes—
353C2×C4#1309399612yes—
354C8—3762no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—24no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—44no—
368C22—44no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22#1568900311yescharacteristic
373C22C2×#1309394811yescharacteristic
374C22#1568912411yescharacteristic
375C22#1568927311yescharacteristic
376C22#1568927811yescharacteristic
377C22#1568930012yes—
378C22#1568930912yes—
379C22#1568945111yescharacteristic
380C22#1568945611yescharacteristic
381C4—22no—
382C4—22no—
383C4—42no—
384C4—44no—
385C4—82no—
386C4—3764no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#2024233912yes—
393C2#2024234412yes—
394C2#2024369611yescharacteristic
395C2#2024377711yescharacteristic
396C2#2024379711yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#393C2
socleC23×C47#120
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#393D47
hypocenterC47#9013
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #20243323C2semidirect
4. #20241267C2semidirect
17. C47×#393C4semidirect
38. C22×Q8×C47C2×C4semidirect
51. #95×C47C2×C4semidirect
120. C23×C47#103semidirect
211. #393Dic47semidirect
218. C47#9013semidirect
224. C22×Q8C2×Dic47semidirect
250. #95C2×Dic47semidirect
319. C23#13093948semidirect

Maximal subgroups

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

subgroupindex
#202460022
#202433232
C2×#156888472
#202412672
C2×C47×#3932
#901347