Finite Group Databasebeta

← #26116457

Subgroups — #26116457

Order 12032.

640 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243368C211yesmaximal, characteristic
2C4×C47⋊#95C211yesmaximal, characteristic
3#20241266C212yesmaximal
4#20241266C212yesmaximal
5C47×#2851C211yesmaximal, characteristic
6C22×C4×Dic47—21no—
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12C2×#125×C47C411yescharacteristic, complemented
13C22×C42×C47C2211yescharacteristic
14C47×#389C411yescharacteristic, complemented
15C47×#389C411yescharacteristic, complemented
16C47×#387—21no—
17C47×#387—21no—
18C47×#385—21no—
19C47×#385—21no—
20C47×#382C411yescharacteristic, complemented
21C47×#380C2211yescharacteristic
22C47×#380C2211yescharacteristic
23C23×Dic47—22no—
24C22×C47⋊C8—22no—
25C2×C4×Dic47—24no—
26C2×C4×Dic47—42no—
27C2×C4×Dic47—42no—
28C4⋊(C47⋊C8)—42no—
29C22⋊(C47⋊C8)—24no—
30C47⋊#95—24no—
31C47⋊#95—22no—
32C47⋊#95Q814yes—
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36#125×C47—42no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×Q812yes—
79C22×C4×C47#4112yes—
80C22×Dic47—24no—
81C22×Dic47—24no—
82C22×Dic47—24no—
83C22×Dic47—42no—
84C22×Dic47—42no—
85C22×Dic47—42no—
86C22×Dic47—42no—
87C4⋊C4×C47—42no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—42no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—42no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C42×C47—22no—
99C42×C47—22no—
100C42×C47—41no—
101C42×C47—41no—
102C4×Dic47—48no—
103C2×C47⋊C8—42no—
104C2×C47⋊C8—42no—
105C2×C47⋊C8—42no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47C2×C22⋊C411yescharacteristic
111C23×C47C22⋊Q812yes—
112C23×C47#13512yes—
113C23×C47AGL(1,ℤ/8)11yescharacteristic
114C23×C47#13711yescharacteristic
115C2×C4×C47—24no—
116C2×C4×C47—24no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
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—42no—
131C2×C4×C47—42no—
132C2×C4×C47—44no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47#9912yes—
146C2×C4×C47#10412yes—
147C2×C4×C47#10412yes—
148C2×C4×C47C4×Q812yes—
149C2×C4×C47#12411yescharacteristic
150C2×C4×C47C4⋊Q811yescharacteristic
151C2×Dic47—28no—
152C2×Dic47—44no—
153C2×Dic47—44no—
154C2×Dic47—44no—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—44no—
158C47⋊C8—82no—
159#9910—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47#39111yescharacteristic
169C22×C47C2×#9911yescharacteristic
170C22×C47#41011yescharacteristic
171C22×C47#41611yescharacteristic
172C22×C47#41711yescharacteristic
173C22×C47C2×#10411yescharacteristic
174C22×C47C2×#10411yescharacteristic
175C22×C47#47812yes—
176C22×C47#47912yes—
177C4×C47—21no—
178C4×C47—22no—
179C4×C47—21no—
180C4×C47—22no—
181C4×C47—22no—
182C4×C47—22no—
183C4×C47—41no—
184C4×C47—81no—
185C4×C47—81no—
186C4×C47—81no—
187C4×C47—81no—
188Dic47—48no—
189#2851D4711yescharacteristic
190#1797—471no—
191C4×#95—471no—
192#1294—472no—
193#1294—472no—
194C2×C47—22no—
195C2×C47—22no—
196C2×C47#150011yescharacteristic
197C2×C47#150111yescharacteristic
198C2×C47#150411yescharacteristic
199C2×C47#150911yescharacteristic
200C2×C47#198911yescharacteristic
201C2×C47#199411yescharacteristic
202C2×C47#199611yescharacteristic
203C2×#125Dic4711yescharacteristic, complemented
204C22×C42—941no—
205C22×C42C2×D4711yescharacteristic
206C2×C4⋊C8—941no—
207C2×C22⋊C8—472no—
208#389Dic4711yescharacteristic, complemented
209#389Dic4711yescharacteristic, complemented
210#387—21no—
211#387—21no—
212#385—21no—
213#385—21no—
214#382Dic4711yescharacteristic, complemented
215#380C2×D4711yescharacteristic
216#380C2×D4711yescharacteristic
217C2×#95—472no—
218C2×#95—941no—
219#336—941no—
220C47#991011yescharacteristic, complemented, Sylow
221C23×C4—942no—
222C23×C4C22⋊D4711yescharacteristic
223C23×C4C22⋊D4711yescharacteristic
224C23×C4C22×D4711yescharacteristic
225C22×C8—942no—
226#125—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C4⋊C4C2×Dic4711yescharacteristic
233C2×C4⋊C4C2×Dic4711yescharacteristic
234C2×C4⋊C4C2×Dic4711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—42no—
238C2×C42—944no—
239C2×C42—1882no—
240C2×C42—1882no—
241C2×C42C2×Dic4711yescharacteristic
242C2×C42C2×Dic4711yescharacteristic
243C2×C42C22⋊D4711yescharacteristic
244C2×C42C22⋊D4711yescharacteristic
245C4⋊C8—1882no—
246C22⋊C8—944no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—474no—
254#95—944no—
255#95—942no—
256C24C2×C22⋊D4711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—944no—
271C22×C4—944no—
272C22×C4—944no—
273C22×C4—1882no—
274C22×C4—1882no—
275C22×C4—1882no—
276C22×C4—1882no—
277C22×C4C22⋊Dic4711yescharacteristic
278C22×C4C22⋊Dic4711yescharacteristic
279C22×C4C22⋊Dic4711yescharacteristic
280C22×C4C22⋊Dic4711yescharacteristic
281C22×C4Q8×D4712yes—
282C22×C4#1062535412yes—
283C22×C4C22×Dic4711yescharacteristic
284C22×C4C2×C22⋊D4711yescharacteristic
285C2×C8—1882no—
286C2×C8—1882no—
287C2×C8—1882no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C42—1888no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×C22⋊Dic4711yescharacteristic
309C23#1309400012yes—
310C23#1309404412yes—
311C23#1309404911yescharacteristic
312C23#1309405411yescharacteristic
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—948no—
344C2×C4—1884no—
345C2×C4—1884no—
346C2×C4—1884no—
347C2×C4—1884no—
348C2×C4—1884no—
349C2×C4—1884no—
350C2×C4#1309393912yes—
351C2×C4#1309395112yes—
352C2×C4#1309395112yes—
353C2×C4Q8×Dic4712yes—
354C2×C4#1309401311yescharacteristic
355C2×C4#1309402211yescharacteristic
356C8—3762no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1568903011yescharacteristic
366C22C2×#1309393911yescharacteristic
367C22#1568909711yescharacteristic
368C22#1568911911yescharacteristic
369C22#1568912411yescharacteristic
370C22C2×#1309395111yescharacteristic
371C22C2×#1309395111yescharacteristic
372C22#1568942012yes—
373C22#1568942712yes—
374C4—21no—
375C4—22no—
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—1888no—
386C2—22no—
387C2—22no—
388C2#2024187411yescharacteristic
389C2#2024187911yescharacteristic
390C2#2024189211yescharacteristic
391C2#2024191311yescharacteristic
392C2#2024434611yescharacteristic
393C2#2024436611yescharacteristic
394C2#2024437411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2851C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2851D47
hypocenterC47#9910
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C47C4semidirect
14. C47×#389C4semidirect
15. C47×#389C4semidirect
20. C47×#382C4semidirect
203. C2×#125Dic47semidirect
208. #389Dic47semidirect
209. #389Dic47semidirect
214. #382Dic47semidirect
220. C47#9910semidirect

Maximal subgroups

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

subgroupindex
#202433682
C4×C47⋊#952
#202412662
C47×#28512
#991047