Finite Group Databasebeta

← #26113604

Subgroups — #26113604

Order 12032.

548 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 119 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#20252417C211yesmaximal, characteristic
2#20243376C211yesmaximal, characteristic
3#20241960C212yesmaximal, complemented
4#20241267C212yesmaximal, complemented
5C47×#2564C211yesmaximal, characteristic
6#15689195—24no—
7#15689180—21no—
8C2×#13093939—21no—
9#15689087—21no—
10C2×C22⋊(C47⋊C8)C2212yes—
11#15688983C2212yes—
12#15688890—21no—
13#15688860—24no—
14C2×C22⋊Q8×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#400C411yescharacteristic, complemented
17C47×#399C411yescharacteristic, complemented
18C47×#395C411yescharacteristic, complemented
19C47×#393C411yescharacteristic, complemented
20C47×#387—21no—
21C47×#386—21no—
22C47×#385—21no—
23C47×#380—21no—
24C2×#95×C47C2211yescharacteristic
25C2×#10625329—22no—
26C22×C47⋊C8—22no—
27C2×C4⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C4⋊(C47⋊C8)—22no—
30#13093939—42no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)D414yes—
34#13093935—22no—
35C47⋊#95—22no—
36C22×Q8×C47C2×C411yescharacteristic
37C22×D4×C47D411yescharacteristic
38C23×C4×C47—21no—
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41C22⋊Q8×C47—22no—
42C22⋊Q8×C47—22no—
43C4×D4×C47—22no—
44C4×D4×C47—22no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47—21no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C4⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47C2×C411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56#95×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61#95×C47C2×C411yescharacteristic
62#95×C47C2×C411yescharacteristic
63C24×C47C22⋊C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×Q8×C47—22no—
66C2×Q8×C47—81no—
67C2×D4×C47—22no—
68C2×D4×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—42no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47—81no—
81C22×C4×C47—81no—
82C22×C4×C47—81no—
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4112yes—
90C22×Dic47—42no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—81no—
97C22⋊C4×C47—22no—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—42no—
101C22⋊C4×C47—81no—
102C42×C47—21no—
103C42×C47—21no—
104C22⋊Dic47—42no—
105C22⋊Dic47—42no—
106C4⋊Dic47—82no—
107#10625329—82no—
108C2×C47⋊C8—24no—
109C2×C47⋊C8—82no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47C2×C22⋊C411yescharacteristic
118C23×C47#12112yes—
119C23×C47AGL(1,ℤ/8)11yescharacteristic
120C23×C47#13711yescharacteristic
121C23×C47#13712yes—
122Q8×C47—82no—
123D4×C47—42no—
124D4×C47—42no—
125C2×C4×C47—22no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
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—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—81no—
150C2×C4×C47—82no—
151C2×C4×C47—82no—
152C2×C4×C47C4×D412yes—
153C2×C4×C47#12011yescharacteristic
154C2×C4×C47#12311yescharacteristic
155C2×Dic47—82no—
156C2×Dic47—82no—
157C2×Dic47—82no—
158C47⋊C8—44no—
159#9385—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—21no—
162C22×C47—22no—
163C22×C47—21no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—42no—
167C22×C47—42no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—81no—
172C22×C47#38611yescharacteristic
173C22×C47#41711yescharacteristic
174C22×C47#41911yescharacteristic
175C22×C47#46812yes—
176C22×C47#47012yes—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—22no—
180C4×C47—42no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—81no—
185C4×C47—81no—
186C4×C47—81no—
187Dic47—162no—
188#2973—471no—
189#2564D4711yescharacteristic
190#1799—471no—
191#1520—472no—
192#1294—472no—
193C2×C47—22no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—22no—
197C2×C47—81no—
198C2×C47#188411yescharacteristic
199C2×C47#189411yescharacteristic
200C2×C47#191311yescharacteristic
201C2×C22⋊Q8C2×D4711yescharacteristic
202C2×C4×D4C2×D4711yescharacteristic
203#435—944no—
204#432—941no—
205C2×#99—941no—
206#407—941no—
207C2×C22⋊C8—472no—
208#400Dic4711yescharacteristic, complemented
209#399Dic4711yescharacteristic, complemented
210#395Dic4711yescharacteristic, complemented
211#393Dic4711yescharacteristic, complemented
212#387—21no—
213#386—21no—
214#385—21no—
215#380—21no—
216#380—472no—
217C2×#95C2×D4711yescharacteristic
218#342—941no—
219#328—944no—
220C47#938511yescharacteristic, complemented, Sylow
221C22×Q8C2×Dic4711yescharacteristic
222C22×D4C22⋊D4711yescharacteristic
223C23×C4—21no—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22×D4711yescharacteristic
226C2×M16—942no—
227C22×C8—942no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4C2×Dic4711yescharacteristic
237C2×C4⋊C4C2×Dic4711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—942no—
242C2×C22⋊C4C2×Dic4711yescharacteristic
243C2×C22⋊C4C22⋊D4711yescharacteristic
244C2×C42C22⋊D4711yescharacteristic
245C4⋊C8—942no—
246#99—1882no—
247C22⋊C8—474no—
248C22⋊C8—942no—
249C22⋊C8—942no—
250#97—942no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—942no—
257#95C2×Dic4711yescharacteristic
258#95C2×Dic4711yescharacteristic
259C24C22⋊Dic4711yescharacteristic
260C24C2×C22⋊D4711yescharacteristic
261C2×Q8—22no—
262C2×Q8—81no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—1882no—
280C22×C4C22⋊Dic4711yescharacteristic
281C22×C4C22⋊Dic4711yescharacteristic
282C22×C4C22⋊Dic4711yescharacteristic
283C22×C4D4×D4712yes—
284C22×C4#1062535212yes—
285C22×C4C22×Dic4711yescharacteristic
286C22×C4C2×C22⋊D4711yescharacteristic
287M16—3762no—
288C2×C8—944no—
289C2×C8—3762no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—3762no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C22⋊C4—1882no—
303C22⋊C4—1882no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C22⋊Dic4711yescharacteristic
314C23#1309399612yes—
315C23#1309404911yescharacteristic
316C23#1309405411yescharacteristic
317C23#1309405512yes—
318Q8—82no—
319D4—42no—
320D4—42no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
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—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—3762no—
349C2×C4—3762no—
350C2×C4—3762no—
351C2×C4D4×Dic4712yes—
352C2×C4#1309399211yescharacteristic
353C2×C4#1309400511yescharacteristic
354C8—1884no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#1568900311yescharacteristic
368C22#1568912411yescharacteristic
369C22#1568913211yescharacteristic
370C22#1568937112yes—
371C22#1568937912yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—7522no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#2024377211yescharacteristic
389C2#2024382211yescharacteristic
390C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2564C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2564D47
hypocenterC47#9385
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #20241960C2semidirect
4. #20241267C2semidirect
16. C47×#400C4semidirect
17. C47×#399C4semidirect
18. C47×#395C4semidirect
19. C47×#393C4semidirect
208. #400Dic47semidirect
209. #399Dic47semidirect
210. #395Dic47semidirect
211. #393Dic47semidirect
220. C47#9385semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202433762
#202419602
#202412672
C47×#25642
#938547