Finite Group Databasebeta

← #26115266

Subgroups — #26115266

Order 12032.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
2C2×#15688860C212yesmaximal, complemented
3C2×#15688857C212yesmaximal
4#20241615C211yesmaximal, characteristic, complemented
5C47×#2850C211yesmaximal, characteristic
6C2×C22⋊(C47⋊C8)C2212yes—
7C2×C4×C47⋊C8—21no—
8C2×C47⋊#95—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688860—24no—
13#15688857—24no—
14C2×#124×C47C411yescharacteristic, complemented
15C2×C22⋊Q8×C47C2211yescharacteristic
16C2×#121×C47C2211yescharacteristic
17C22×C42×C47C2211yescharacteristic
18C47×#391C411yescharacteristic, complemented
19C47×#390C411yescharacteristic, complemented
20C47×#388—21no—
21C47×#388—21no—
22C47×#386—21no—
23C47×#386—21no—
24C47×#382C411yescharacteristic, complemented
25C23×Dic47—22no—
26C22×C47⋊C8—22no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)D414yes—
29C4×C47⋊C8—42no—
30C47⋊#95—24no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C47⋊#95—42no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×D4×C47—21no—
36C22×D4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40#124×C47—42no—
41C22⋊Q8×C47—22no—
42C22⋊Q8×C47—22no—
43#121×C47—22no—
44#121×C47—22no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C42×C47—42no—
55C2×C42×C47C2×C411yescharacteristic
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C2×C42×C47D411yescharacteristic
59#95×C47—21no—
60#95×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63C24×C47—21no—
64C24×C47C2×D411yescharacteristic
65C2×Q8×C47—22no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C2×D4×C47—22no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—42no—
74C2×D4×C47—42no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22×C411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D412yes—
93C22×C4×C47#4112yes—
94C22×Dic47—24no—
95C22×Dic47—42no—
96C22×Dic47—42no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—42no—
104C22⋊C4×C47—42no—
105C22⋊C4×C47—42no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—42no—
109C22⋊C4×C47—42no—
110C22⋊C4×C47—42no—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47—41no—
114C42×C47—41no—
115C2×C47⋊C8—24no—
116C2×C47⋊C8—42no—
117C2×C47⋊C8—42no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47C2×C22⋊C411yescharacteristic
130C23×C47#12411yescharacteristic
131C23×C47#12711yescharacteristic
132C23×C47#13512yes—
133C23×C47#13512yes—
134Q8×C47—82no—
135D4×C47—82no—
136D4×C47—82no—
137D4×C47—82no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—44no—
155C2×C4×C47—44no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47#9912yes—
166C2×C4×C47#10412yes—
167C2×C4×C47C4×D412yes—
168C2×C4×C47#12112yes—
169C2×Dic47—44no—
170C2×Dic47—44no—
171C2×Dic47—44no—
172C2×Dic47—82no—
173C2×Dic47—82no—
174C2×Dic47—82no—
175C2×Dic47—82no—
176C47⋊C8—44no—
177#9681—471nomaximal, Sylow
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—22no—
183C22×C47—22no—
184C22×C47—22no—
185C22×C47—22no—
186C22×C47—81no—
187C22×C47—81no—
188C22×C47—81no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47—81no—
193C22×C47#39011yescharacteristic
194C22×C47C2×#9911yescharacteristic
195C22×C47#41011yescharacteristic
196C22×C47#41611yescharacteristic
197C22×C47#41611yescharacteristic
198C22×C47C2×#10411yescharacteristic
199C22×C47#42111yescharacteristic
200C22×C47#48812yes—
201C22×C47#49512yes—
202C4×C47—22no—
203C4×C47—22no—
204C4×C47—22no—
205C4×C47—22no—
206C4×C47—41no—
207C4×C47—41no—
208C4×C47—81no—
209C4×C47—81no—
210C4×C47—81no—
211Dic47—84no—
212#2850D4711yescharacteristic
213C4×C22⋊C8—471no—
214C2×#328—472no—
215C2×#327—472no—
216#1434—471no—
217C2×C47—22no—
218C2×C47—22no—
219C2×C47—161no—
220C2×C47#150011yescharacteristic
221C2×C47#150111yescharacteristic
222C2×C47#151611yescharacteristic
223C2×C47#152111yescharacteristic
224C2×C47#195811yescharacteristic
225C2×C47#196411yescharacteristic
226C2×C47#198011yescharacteristic
227C2×#124Dic4711yescharacteristic, complemented
228C2×C22⋊Q8C2×D4711yescharacteristic
229C2×#121C2×D4711yescharacteristic
230C22×C42C2×D4711yescharacteristic
231C2×C22⋊C8—472no—
232C2×C4×C8—941no—
233#391Dic4711yescharacteristic, complemented
234#390Dic4711yescharacteristic, complemented
235#388—21no—
236#388—21no—
237#386—21no—
238#386—21no—
239#382Dic4711yescharacteristic, complemented
240C2×#95—472no—
241C2×#95—941no—
242C2×#95—941no—
243#336—941no—
244#328—944no—
245#327—944no—
246C47#968111yescharacteristic, complemented, Sylow
247C22×Q8C2×Dic4711yescharacteristic
248C22×D4—21no—
249C22×D4C2×Dic4711yescharacteristic
250C23×C4—942no—
251C23×C4C22⋊D4711yescharacteristic
252C23×C4C22⋊D4711yescharacteristic
253C23×C4C22×D4711yescharacteristic
254C22×C8—942no—
255#124—42no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258#121—22no—
259#121—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×Dic4711yescharacteristic
262C2×C4⋊C4C2×Dic4711yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—42no—
270C2×C42C2×Dic4711yescharacteristic
271C2×C42C2×Dic4711yescharacteristic
272C2×C42C22⋊D4711yescharacteristic
273C2×C42C22⋊D4711yescharacteristic
274C22⋊C8—474no—
275C22⋊C8—942no—
276C4×C8—1882no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—944no—
282#95—942no—
283#95—942no—
284#95—1882no—
285C24—21no—
286C24C2×C22⋊D4711yescharacteristic
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—944no—
309C22×C4—1882no—
310C22×C4—1882no—
311C22×C4—1882no—
312C22×C4—1882no—
313C22×C4—1882no—
314C22×C4—1882no—
315C22×C4C22⋊Dic4711yescharacteristic
316C22×C4C22⋊Dic4711yescharacteristic
317C22×C4C22⋊Dic4711yescharacteristic
318C22×C4C22⋊Dic4711yescharacteristic
319C22×C4D4×D4712yes—
320C22×C4#1062535212yes—
321C22×C4C22×Dic4711yescharacteristic
322C22×C4C2×C22⋊D4711yescharacteristic
323C2×C8—944no—
324C2×C8—1882no—
325C2×C8—1882no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊Dic4711yescharacteristic
352C23#1309401111yescharacteristic
353C23#1309402011yescharacteristic
354C23#1309404412yes—
355C23#1309404412yes—
356Q8—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—3762no—
388C2×C4—3762no—
389C2×C4—3762no—
390C2×C4—3762no—
391C2×C4—1884no—
392C2×C4—1884no—
393C2×C4—1884no—
394C2×C4#1309393912yes—
395C2×C4#1309395112yes—
396C2×C4D4×Dic4712yes—
397C2×C4#1309399512yes—
398C8—1884no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22#1568902611yescharacteristic
415C22C2×#1309393911yescharacteristic
416C22#1568909711yescharacteristic
417C22#1568911911yescharacteristic
418C22#1568911911yescharacteristic
419C22C2×#1309395111yescharacteristic
420C22#1568914011yescharacteristic
421C22#1568947712yes—
422C22#1568950512yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—3764no—
433C2—22no—
434C2—22no—
435C2—161no—
436C2#2024187411yescharacteristic
437C2#2024187911yescharacteristic
438C2#2024194411yescharacteristic
439C2#2024196511yescharacteristic
440C2#2024420011yescharacteristic
441C2#2024422511yescharacteristic
442C2#2024430811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2850C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2850D47
hypocenterC47#9681
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C4×C22⋊(C47⋊C8)C2semidirect
2. C2×#15688860C2semidirect
4. #20241615C2semidirect
14. C2×#124×C47C4semidirect
18. C47×#391C4semidirect
19. C47×#390C4semidirect
24. C47×#382C4semidirect
227. C2×#124Dic47semidirect
233. #391Dic47semidirect
234. #390Dic47semidirect
239. #382Dic47semidirect
246. C47#9681semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊(C47⋊C8)2
C2×#156888602
C2×#156888572
#202416152
C47×#28502
#968147