Finite Group Databasebeta

← #26111836

Subgroups — #26111836

Order 12032.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689176C211yesmaximal, characteristic
2#20251727C211yesmaximal, characteristic
3C2×#15688859C212yesmaximal
4#20241267C212yesmaximal
5C47×#1434C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689176—24no—
8C2×C4⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C2×C4×C47⋊C8—21no—
11#15689029C411yescharacteristic, complemented
12#15689029C411yescharacteristic, complemented
13#15689021C411yescharacteristic, complemented
14#15689021C411yescharacteristic, complemented
15#15689007—21no—
16#15689007—21no—
17#15688996—21no—
18#15688986—21no—
19#15688983C2211yescharacteristic
20#15688859—24no—
21C22×C42×C47C2211yescharacteristic
22C2×#95×C47—21no—
23C2×#95×C47—21no—
24C2×#95×C47C2212yes—
25C22×Dic94C2×C411yescharacteristic
26C22×C47⋊C8—22no—
27#13094003—22no—
28#13094003—22no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C4⋊(C47⋊C8)—24no—
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)D414yes—
40C4×C47⋊C8—24no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C23×C4×C47—22no—
49C23×C4×C47D411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C2×C42×C47—22no—
53C2×C42×C47—22no—
54C2×C42×C47C2×C411yescharacteristic
55C2×C42×C47C2×C411yescharacteristic
56C2×C42×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58#95×C47—24no—
59#95×C47—22no—
60#95×C47—22no—
61#95×C47—42no—
62C24×C47C2×D411yescharacteristic
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—24no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47—42no—
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47SD1612yes—
82C22×C4×C47Q1612yes—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47#4112yes—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C2×Dic94—22no—
92C2×Dic94—41no—
93C2×Dic94—41no—
94C42×C47—22no—
95C42×C47—22no—
96C42×C47—21no—
97C42×C47—21no—
98C42×C47—21no—
99C42×C47—21no—
100C22⋊Dic47—42no—
101C22⋊Dic47—42no—
102C4⋊Dic47—22no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105Dic47⋊C4—42no—
106Dic47⋊C4—42no—
107Dic47⋊C4—41no—
108Dic47⋊C4—41no—
109C2×C47⋊C8—24no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—42no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47C2×C22⋊C411yescharacteristic
117C23×C47#12112yes—
118C23×C47C2×SD1611yescharacteristic
119C23×C47C2×Q1611yescharacteristic
120C23×C47#13512yes—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—44no—
139C2×C4×C47—44no—
140C2×C4×C47—44no—
141C2×C4×C47—44no—
142C2×C4×C47—82no—
143C2×C4×C47—82no—
144C2×C4×C47—82no—
145C2×C4×C47—82no—
146C2×C4×C47#10312yes—
147C2×C4×C47#10312yes—
148C2×C4×C47#10412yes—
149C2×C4×C47C4×D412yes—
150C2×C4×C47#12011yescharacteristic
151C2×C4×C47#12311yescharacteristic
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162C2×Dic47—41no—
163C2×Dic47—41no—
164Dic94—82no—
165C47⋊C8—44no—
166#9112—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—22no—
175C22×C47#38611yescharacteristic
176C22×C47#41011yescharacteristic
177C22×C47#41011yescharacteristic
178C22×C47C2×#10311yescharacteristic
179C22×C47#41611yescharacteristic
180C22×C47C2×#10411yescharacteristic
181C22×C47#42111yescharacteristic
182C22×C47#46512yes—
183C22×C47#46712yes—
184C4×C47—22no—
185C4×C47—22no—
186C4×C47—22no—
187C4×C47—21no—
188C4×C47—22no—
189C4×C47—21no—
190C4×C47—41no—
191C4×C47—84no—
192Dic47—81no—
193Dic47—81no—
194Dic47—81no—
195Dic47—81no—
196C2×#431—471no—
197#2855—471no—
198C2×#328—472no—
199#1434D4711yescharacteristic
200#1294—472no—
201C2×C47—22no—
202C2×C47—22no—
203C2×C47#150011yescharacteristic
204C2×C47#150311yescharacteristic
205C2×C47#151811yescharacteristic
206C2×C47#152111yescharacteristic
207C2×C47#187111yescharacteristic
208C2×C47#189211yescharacteristic
209C2×C47#189911yescharacteristic
210C2×C22⋊Q8—471no—
211C22×C42C2×D4711yescharacteristic
212#431—944no—
213C2×C4⋊C8—941no—
214C2×C22⋊C8—472no—
215C2×C4×C8—941no—
216#391—471no—
217#391—471no—
218#389—471no—
219#389—471no—
220#387—941no—
221#387—941no—
222#385—941no—
223#381—941no—
224#380—471no—
225C2×#95—21no—
226C2×#95—21no—
227C2×#95C2×D4712yes—
228#328—944no—
229C47#911211yescharacteristic, complemented, Sylow
230C22×Q8—471no—
231C23×C4—22no—
232C23×C4C22⋊D4711yescharacteristic
233C23×C4C22⋊D4711yescharacteristic
234C23×C4C22×D4711yescharacteristic
235C22×C8—942no—
236C22⋊Q8—942no—
237C22⋊Q8—942no—
238C2×C4⋊C4—471no—
239C2×C4⋊C4—471no—
240C2×C4⋊C4—471no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C22⋊C4—941no—
245C2×C22⋊C4—941no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C4×D4711yescharacteristic
249C2×C42C4×D4711yescharacteristic
250C2×C42D18811yescharacteristic
251C2×C42D18811yescharacteristic
252C4⋊C8—944no—
253C22⋊C8—474no—
254C22⋊C8—942no—
255C4×C8—944no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—42no—
260#95—941no—
261#95—941no—
262#95—941no—
263#95—941no—
264#95—941no—
265#95—941no—
266#95—941no—
267C24C2×C22⋊D4711yescharacteristic
268C2×Q8—942no—
269C2×Q8—1881no—
270C2×Q8—1881no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4(C2×D47)⋊C411yescharacteristic
290C22×C4(C2×D47)⋊C411yescharacteristic
291C22×C4(C2×D47)⋊C411yescharacteristic
292C22×C4(C2×D47)⋊C411yescharacteristic
293C22×C4#1062533512yes—
294C22×C4#1062533712yes—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4C2×D18811yescharacteristic
297C22×C4D4×D4712yes—
298C22×C4#1062535212yes—
299C2×C8—944no—
300C2×C8—1882no—
301C2×C8—1882no—
302C4⋊C4—942no—
303C4⋊C4—1881no—
304C4⋊C4—1881no—
305C4⋊C4—1882no—
306C4⋊C4—1882no—
307C4⋊C4—1881no—
308C4⋊C4—1881no—
309C22⋊C4—1882no—
310C22⋊C4—1882no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×(C2×D47)⋊C411yescharacteristic
322C23#1309399612yes—
323C23C2×#1062533511yescharacteristic
324C23C2×#1062533711yescharacteristic
325C23#1309404412yes—
326Q8—3762no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—1881no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4—1881no—
356C2×C4—1881no—
357C2×C4—1881no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4#1309394712yes—
365C2×C4#1309394712yes—
366C2×C4#1309395212yes—
367C2×C4(C22⋊D47)⋊C412yes—
368C2×C4#1309399411yescharacteristic
369C2×C4#1309400811yescharacteristic
370C8—1884no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22#1568900011yescharacteristic
380C22#1568909611yescharacteristic
381C22#1568909611yescharacteristic
382C22C2×#1309394711yescharacteristic
383C22#1568911711yescharacteristic
384C22C2×#1309395211yescharacteristic
385C22#1568914111yescharacteristic
386C22#1568935012yes—
387C22#1568936212yes—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—84no—
396C4—3761no—
397C4—3761no—
398C4—3761no—
399C4—3761no—
400C2—22no—
401C2—22no—
402C2#2024187211yescharacteristic
403C2#2024188611yescharacteristic
404C2#2024195111yescharacteristic
405C2#2024196311yescharacteristic
406C2#2024368811yescharacteristic
407C2#2024380911yescharacteristic
408C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1434C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1434D47
hypocenterC47#9112
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
11. #15689029C4semidirect
12. #15689029C4semidirect
13. #15689021C4semidirect
14. #15689021C4semidirect
229. C47#9112semidirect

Maximal subgroups

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

subgroupindex
C2×#156891762
#202517272
C2×#156888592
#202412672
C47×#14342
#911247