Finite Group Databasebeta

← #26113603

Subgroups — #26113603

Order 12032.

548 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 115 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#20248564C211yesmaximal, characteristic
2#20243376C211yesmaximal, characteristic
3#20241961C212yesmaximal, complemented
4#20241268C212yesmaximal, complemented
5C47×#2973C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×#13093939—21no—
8#15689062C411yescharacteristic, complemented
9#15689059C411yescharacteristic, complemented
10#15689043C411yescharacteristic, complemented
11#15689035C411yescharacteristic, complemented
12#15689007—21no—
13#15689003—21no—
14#15688996—21no—
15#15688983C2212yes—
16#15688981—21no—
17C2×C47⋊#95C2211yescharacteristic
18#15688890—21no—
19#15688861—24no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#435—24no—
22C47×#432—21no—
23C47×#407—21no—
24C2×C22⋊C8×C47C2212yes—
25C22×Dic94C2×C411yescharacteristic
26C23×Dic47—21no—
27#13094003—22no—
28#13094003—22no—
29C2×C4⋊Dic47—22no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—21no—
38#13093939—42no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—22no—
45C47⋊#95C2×C411yescharacteristic
46C47⋊#95C2×C411yescharacteristic
47C22×D4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C2×M16×C47—22no—
51C22×C8×C47—22no—
52C4×D4×C47—22no—
53C4×D4×C47—22no—
54C2×C4⋊C4×C47C2×C411yescharacteristic
55C2×C22⋊C4×C47C2×C411yescharacteristic
56C2×C22⋊C4×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C4⋊C8×C47—22no—
59C22⋊C8×C47—22no—
60C22⋊C8×C47—22no—
61C22⋊C8×C47D414yes—
62#97×C47—22no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—22no—
66C2×D4×C47—21no—
67C2×D4×C47—21no—
68C2×D4×C47—22no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—81no—
72C22×C4×C47—81no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4112yes—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—42no—
85C22×Dic47—41no—
86C22×Dic47—41no—
87C22×Dic47—42no—
88C22×Dic47—81no—
89C2×Dic94—22no—
90C2×Dic94—81no—
91M16×C47—82no—
92C2×C8×C47—24no—
93C2×C8×C47—82no—
94C4⋊C4×C47—22no—
95C22⋊C4×C47—22no—
96C22⋊C4×C47—22no—
97C42×C47—21no—
98C42×C47—21no—
99C22⋊Dic47—42no—
100C22⋊Dic47—42no—
101C22⋊Dic47—42no—
102C22⋊Dic47—42no—
103C22⋊Dic47—81no—
104C4⋊Dic47—22no—
105C4⋊Dic47—82no—
106Dic47⋊C4—42no—
107Dic47⋊C4—41no—
108Dic47⋊C4—41no—
109Dic47⋊C4—81no—
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—
122D4×C47—42no—
123D4×C47—42no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—81no—
137C2×C4×C47C4×D412yes—
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47#12311yescharacteristic
140C2×Dic47—22no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154C2×Dic47—82no—
155C2×Dic47—82no—
156C2×Dic47—82no—
157Dic94—82no—
158C8×C47—44no—
159#9385—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—22no—
162C22×C47—21no—
163C22×C47—22no—
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—22no—
179C4×C47—21no—
180C4×C47—41no—
181C4×C47—41no—
182C4×C47—41no—
183Dic47—42no—
184Dic47—81no—
185Dic47—81no—
186Dic47—81no—
187Dic47—162no—
188#2973D4711yescharacteristic
189#2564—471no—
190#1799—471no—
191#1520—472no—
192#1294—472no—
193C2×C47—21no—
194C2×C47—22no—
195C2×C47—21no—
196C2×C47—22no—
197C2×C47—81no—
198C2×C47#188411yescharacteristic
199C2×C47#189411yescharacteristic
200C2×C47#191311yescharacteristic
201C2×C22⋊Q8—471no—
202C2×C4×D4C2×D4711yescharacteristic
203#435—24no—
204#432—21no—
205C2×#99—941no—
206#407—21no—
207C2×C22⋊C8C2×D4712yes—
208#400—471no—
209#399—471no—
210#395—471no—
211#393—471no—
212#387—941no—
213#386—941no—
214#385—941no—
215#380—472no—
216#380—941no—
217C2×#95—471no—
218#342—941no—
219#328—944no—
220C47#938511yescharacteristic, complemented, Sylow
221C22×Q8—471no—
222C22×D4D18811yescharacteristic
223C23×C4—941no—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22×D4711yescharacteristic
226C2×M16—22no—
227C22×C8—22no—
228C22⋊Q8—942no—
229C22⋊Q8—942no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—471no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—942no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4C4×D4711yescharacteristic
238C2×C22⋊C4—941no—
239C2×C22⋊C4—941no—
240C2×C22⋊C4—941no—
241C2×C22⋊C4—942no—
242C2×C22⋊C4C4×D4711yescharacteristic
243C2×C22⋊C4D18811yescharacteristic
244C2×C42C22⋊D4711yescharacteristic
245C4⋊C8—22no—
246#99—1882no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8C22⋊D4714yes—
250#97—22no—
251#95—471no—
252#95—471no—
253#95—941no—
254#95—941no—
255#95—941no—
256#95—941no—
257#95—941no—
258#95—942no—
259C24(C2×D47)⋊C411yescharacteristic
260C24C2×D18811yescharacteristic
261C2×Q8—942no—
262C2×Q8—3761no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—941no—
275C22×C4—3761no—
276C22×C4—1882no—
277C22×C4—1882no—
278C22×C4—1881no—
279C22×C4—1881no—
280C22×C4(C2×D47)⋊C411yescharacteristic
281C22×C4(C2×D47)⋊C411yescharacteristic
282C22×C4(C2×D47)⋊C411yescharacteristic
283C22×C4C2×C4×D4711yescharacteristic
284C22×C4#1062535012yes—
285C22×C4C2×C22⋊D4712yes—
286C22×C4C2×C22⋊D4711yescharacteristic
287M16—82no—
288C2×C8—24no—
289C2×C8—82no—
290C4⋊C4—22no—
291C4⋊C4—942no—
292C4⋊C4—3762no—
293C4⋊C4—3761no—
294C4⋊C4—1882no—
295C4⋊C4—1881no—
296C4⋊C4—1881no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—3761no—
300C22⋊C4—1882no—
301C22⋊C4—1882no—
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×(C2×D47)⋊C411yescharacteristic
314C23#1309399812yes—
315C23#1309405211yescharacteristic
316C23#1309405712yes—
317C23#1309405711yescharacteristic
318Q8—3762no—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—942no—
335C2×C4—3762no—
336C2×C4—3762no—
337C2×C4—3762no—
338C2×C4—3762no—
339C2×C4—3762no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4C4×C22⋊D4712yes—
352C2×C4#1309399311yescharacteristic
353C2×C4#1309400711yescharacteristic
354C8—44no—
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#1568900211yescharacteristic
368C22#1568912511yescharacteristic
369C22#1568913311yescharacteristic
370C22#1568936912yes—
371C22#1568938112yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—3761no—
379C4—3761no—
380C4—3761no—
381C4—1882no—
382C4—7522no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#2024377111yescharacteristic
389C2#2024382111yescharacteristic
390C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2973C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2973D47
hypocenterC47#9385
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. #20241961C2semidirect
4. #20241268C2semidirect
8. #15689062C4semidirect
9. #15689059C4semidirect
10. #15689043C4semidirect
11. #15689035C4semidirect
220. C47#9385semidirect

Maximal subgroups

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

subgroupindex
#202485642
#202433762
#202419612
#202412682
C47×#29732
#938547