Finite Group Databasebeta

← #26116733

Subgroups — #26116733

Order 12032.

610 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254141C211yesmaximal, characteristic
2C2×#15689441C211yesmaximal, characteristic
3#20244224C211yesmaximal, characteristic
4#20243478C211yesmaximal, characteristic, complemented
5#20241960C211yesmaximal, characteristic, complemented
6#20241236C211yesmaximal, characteristic, complemented
7C47×#2502C211yesmaximal, characteristic, complemented
8C2×#13093998C2211yescharacteristic
9#15689451—21no—
10#15689441—22no—
11#15689441—22no—
12#15689441—21no—
13#15689354—22no—
14#15689273—21no—
15#15689262—21no—
16C2×#13093955—21no—
17C2×#13093944—21no—
18C2×#13093938—22no—
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20C2×C22⋊(C47⋊C8)C2211yescharacteristic
21#15688983C2211yescharacteristic
22#15688885—22no—
23#15688860—22no—
24C2×C22⋊Q8×C47C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#398—22no—
27C47×#391—21no—
28C47×#385—22no—
29C47×#384—21no—
30C4×C22⋊C4×C47C2211yescharacteristic
31C22×D188D411yescharacteristic
32C22×C22⋊D47—21no—
33C2×#10625336—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8D412yes—
36#13093998—21no—
37#13093998—21no—
38#13093998—21no—
39#13093998—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×C22⋊Dic47—21no—
43C2×(C2×D47)⋊C4—21no—
44#13093955—22no—
45#13093955—24no—
46#13093948—22no—
47#13093944—22no—
48#13093944—24no—
49#13093938—44no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)—21no—
53C22⋊(C47⋊C8)D412yes—
54C47⋊#95—21no—
55C22×Q8×C47D411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47C2311yescharacteristic
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—24no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47D411yescharacteristic
71C2×C22⋊C4×C47—21no—
72C2×C22⋊C4×C47D412yes—
73C2×C42×C47—21no—
74#95×C47—24no—
75#95×C47—21no—
76C24×C47C2×D411yescharacteristic
77C23×D47—21no—
78C2×Q8×C47—21no—
79C2×Q8×C47—21no—
80C2×Q8×C47—42no—
81C22×C4×C47—22no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—22no—
85C22×C4×C47—42no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47—82no—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47#4111yescharacteristic
95C2×C22⋊D47—41no—
96C2×C22⋊D47—41no—
97C2×C22⋊D47—42no—
98C2×C22⋊D47—42no—
99C22×Dic47—21no—
100C22×Dic47—41no—
101C2×D188—21no—
102C2×D188—21no—
103C2×D188—42no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—84no—
113C22⋊C4×C47—22no—
114C22⋊C4×C47—41no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47SD1614yescomplemented
117C42×C47—42no—
118C22⋊Dic47—42no—
119#10625336—82no—
120(C2×D47)⋊C4—41no—
121(C2×D47)⋊C4—41no—
122C4⋊Dic47—21no—
123C4⋊Dic47—21no—
124C4⋊Dic47—42no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—42no—
128C2×C47⋊C8—42no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47#12011yescharacteristic
132C23×C47#12111yescharacteristic
133C23×C47#12711yescharacteristic
134C23×C47C2×SD1612yes—
135C23×C47#13712yes—
136C22×D47—41no—
137C22×D47—42no—
138C22×D47—81no—
139C22×D47—81no—
140Q8×C47—42no—
141Q8×C47—81no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—24no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—44no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—42no—
157C2×C4×C47—82no—
158C2×C4×C47—82no—
159C2×C4×C47#12011yescharacteristic
160C2×C4×C47#12011yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C22⋊D47—82no—
164C22⋊D47—81no—
165C22⋊D47—81no—
166C2×Dic47—41no—
167C2×Dic47—42no—
168C2×Dic47—81no—
169C2×Dic47—81no—
170C2×Dic47—81no—
171D188—42no—
172D188—42no—
173C47⋊C8—42no—
174C47⋊C8—42no—
175#9951—471nomaximal, Sylow
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—42no—
183C22×C47—42no—
184C22×C47#39211yescharacteristic
185C22×C47#44812yes—
186C22×C47#46112yes—
187C22×C47#49211yescharacteristic
188C22×C47#49711yescharacteristic
189C2×D47—81no—
190C2×D47—81no—
191C2×D47—81no—
192C2×D47—162no—
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—44no—
197C4×C47—81no—
198C4×C47—82no—
199C4×C47—81no—
200C4×C47#45812yes—
201Dic47—81no—
202Dic47—161no—
203#3187—471no—
204C2×#481—471no—
205#2502D4711yescharacteristic, complemented
206#1964—471no—
207#1824—471no—
208#1520—471no—
209#1282—471no—
210C2×C47—21no—
211C2×C47—22no—
212C2×C47—22no—
213C2×C47—21no—
214C2×C47#200811yescharacteristic
215C2×C47#201311yescharacteristic
216C2×C47#202211yescharacteristic
217D47—161no—
218C2×C22⋊Q8C2×D4711yescharacteristic
219C2×#121—471no—
220C22×C4⋊C4C2×D4711yescharacteristic
221#483—941no—
222#481—942no—
223#481—942no—
224#481—941no—
225#465—942no—
226#450—941no—
227#448—941no—
228C2×#106—941no—
229C2×#102—941no—
230C2×#99—942no—
231C2×C22⋊C8—471no—
232C2×C22⋊C8—471no—
233#398—22no—
234#391—21no—
235#385—22no—
236#384—21no—
237#380—471no—
238C4×C22⋊C4C2×D4711yescharacteristic
239#340—942no—
240#328—942no—
241C47#995111yescharacteristic, complemented, Sylow
242C22×Q8C22⋊D4711yescharacteristic
243C22×D4—471no—
244C22×D4—941no—
245C23×C4—21no—
246C23×C4C22×D4711yescharacteristic
247C2×SD16—942no—
248C22×C8—472no—
249C22×C8—941no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254#121—941no—
255#121—941no—
256#121—941no—
257#121—941no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—471no—
266C2×C4⋊C4—941no—
267C2×C4⋊C4C22⋊D4712yes—
268C2×C4⋊C4C22⋊D4711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—941no—
271C2×C22⋊C4—941no—
272C2×C22⋊C4C22⋊D4712yes—
273C2×C42—21no—
274#106—944no—
275#106—942no—
276#103—942no—
277#102—944no—
278#102—942no—
279#99—1884no—
280C22⋊C8—472no—
281C22⋊C8—941no—
282C22⋊C8—941no—
283C22⋊C8—941no—
284#95—24no—
285#95—21no—
286#95—941no—
287C24—941no—
288C24C2×C22⋊D4711yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—941no—
293C2×D4—941no—
294C2×D4—1882no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1882no—
298C2×D4—1882no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—941no—
308C22×C4—1881no—
309C22×C4D4×D4711yescharacteristic
310C22×C4D4×D4711yescharacteristic
311C22×C4D4×D4711yescharacteristic
312C22×C4#1062535211yescharacteristic
313C22×C4C2×C22⋊D4711yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—3762no—
316C2×C8—942no—
317C2×C8—942no—
318C2×C8—1882no—
319C2×C8—1882no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—84no—
329C4⋊C4—941no—
330C4⋊C4—941no—
331C4⋊C4—1882no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—1882no—
336C22⋊C4—1881no—
337C22⋊C4—1881no—
338C22⋊C4#1062533614yescomplemented
339C42—42no—
340C23—41no—
341C23—41no—
342C23—3761no—
343C23—3761no—
344C23—1881no—
345C23—1882no—
346C23#1309399311yescharacteristic
347C23#1309399511yescharacteristic
348C23#1309402011yescharacteristic
349C23C2×#1062533612yes—
350C23#1309405512yes—
351Q8—42no—
352Q8—81no—
353D4—3761no—
354D4—3761no—
355D4—3762no—
356D4—1882no—
357D4—1882no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—3761no—
376C2×C4—3761no—
377C2×C4—3761no—
378C2×C4—1881no—
379C2×C4—1882no—
380C2×C4#1309399211yescharacteristic
381C2×C4#1309399211yescharacteristic
382C2×C4#1309399611yescharacteristic
383C2×C4#1309399611yescharacteristic
384C8—1882no—
385C8—1882no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—3761no—
395C22—3761no—
396C22—3761no—
397C22—7522no—
398C22#1568903111yescharacteristic
399C22#1568926212yes—
400C22#1568932512yes—
401C22#1568949711yescharacteristic
402C22#1568951511yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—81no—
408C4—82no—
409C4—81no—
410C4—3761no—
411C4—7521no—
412C4#1568931212yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—7521no—
418C2#2024444111yescharacteristic
419C2#2024447111yescharacteristic
420C2#2024451611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2502C2
socleC22×C47#392
prime radicalC2×#13093998C22
hypercenter#2502D47
hypocenterC47#9951
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
4. #20243478C2semidirect
5. #20241960C2semidirect
6. #20241236C2semidirect
7. C47×#2502C2semidirect
116. C22⋊C4×C47SD16semidirect
205. #2502D47semidirect
241. C47#9951semidirect
338. C22⋊C4#10625336semidirect

Maximal subgroups

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

subgroupindex
#202541412
C2×#156894412
#202442242
#202434782
#202419602
#202412362
C47×#25022
#995147