Finite Group Databasebeta

← #26116731

Subgroups — #26116731

Order 12032.

610 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#15689439C211yesmaximal, characteristic
2#20247867C211yesmaximal, characteristic, complemented
3#20244225C211yesmaximal, characteristic
4#20243479C211yesmaximal, characteristic, complemented
5#20241960C211yesmaximal, characteristic, complemented
6#20241235C211yesmaximal, characteristic, complemented
7C47×#3187C211yesmaximal, characteristic
8C22×C4⋊Dic47C2211yescharacteristic
9#15689439—22no—
10#15689439—22no—
11C2×#13093955—21no—
12C2×#13093945—21no—
13C2×#13093939—22no—
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15#15689053—22no—
16#15689030—21no—
17#15688995—22no—
18#15688992—21no—
19#15688983C2211yescharacteristic
20C4×C22⋊Dic47C2211yescharacteristic
21#15688886—22no—
22#15688860—22no—
23C2×C22⋊Q8×C47C2211yescharacteristic
24C2×#121×C47C2211yescharacteristic
25C47×#483—21no—
26C47×#481—21no—
27C47×#465—22no—
28C47×#450—21no—
29C47×#448—21no—
30C2×C22⋊C8×C47C2211yescharacteristic
31C23×Dic47—21no—
32C22×C47⋊C8—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47D412yes—
40C2×Dic47⋊C4—24no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47D412yes—
43C2×C4×Dic47—21no—
44#13093955—24no—
45#13093945—24no—
46#13093939—44no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)D412yes—
49C47⋊#95—24no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C22×Q8×C47D411yescharacteristic
53C22×D4×C47—21no—
54C22×D4×C47D411yescharacteristic
55C23×C4×C47C2311yescharacteristic
56C2×SD16×C47—22no—
57C22×C8×C47D412yes—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62#121×C47—21no—
63#121×C47—21no—
64#121×C47—21no—
65#121×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47D411yescharacteristic
68C2×C4⋊C4×C47D411yescharacteristic
69C2×C22⋊C4×C47—21no—
70C2×C22⋊C4×C47—21no—
71#106×C47—22no—
72#103×C47—22no—
73#102×C47—22no—
74C22⋊C8×C47—21no—
75C22⋊C8×C47—21no—
76C24×C47—21no—
77C24×C47C2×D411yescharacteristic
78C2×Q8×C47—21no—
79C2×Q8×C47—21no—
80C2×Q8×C47—42no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—42no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—42no—
87C2×D4×C47—42no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—42no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47#4111yescharacteristic
98C22×Dic47—22no—
99C22×Dic47—22no—
100C22×Dic47—41no—
101C22×Dic47—41no—
102C22×Dic47—41no—
103C22×Dic47—82no—
104SD16×C47—82no—
105C2×C8×C47—22no—
106C2×C8×C47—42no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C22⋊C4×C47—41no—
114C22⋊C4×C47—41no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C22⋊Dic47—22no—
118C22⋊Dic47—42no—
119C22⋊Dic47SD1614yescomplemented
120C4⋊Dic47—22no—
121C4⋊Dic47—22no—
122C4⋊Dic47—41no—
123C4⋊Dic47—41no—
124C4⋊Dic47—42no—
125Dic47⋊C4—84no—
126C4×Dic47—42no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—42no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—42no—
133C23×C47—81no—
134C23×C47—81no—
135C23×C47#12011yescharacteristic
136C23×C47#12111yescharacteristic
137C23×C47#12711yescharacteristic
138C23×C47C2×SD1612yes—
139C23×C47#13712yes—
140Q8×C47—42no—
141Q8×C47—81no—
142D4×C47—42no—
143D4×C47—42no—
144D4×C47—82no—
145D4×C47—81no—
146D4×C47—81no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47#12011yescharacteristic
160C2×C4×C47#12011yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×Dic47—24no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—44no—
167C2×Dic47—42no—
168C2×Dic47—82no—
169C2×Dic47—82no—
170C2×Dic47—81no—
171C2×Dic47—81no—
172C2×Dic47—81no—
173C8×C47—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—81no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47—162no—
188C22×C47#39211yescharacteristic
189C22×C47#44812yes—
190C22×C47#46112yes—
191C22×C47#49211yescharacteristic
192C22×C47#49711yescharacteristic
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—81no—
197C4×C47—81no—
198C4×C47—81no—
199C4×C47#45812yes—
200Dic47—44no—
201Dic47—82no—
202Dic47—161no—
203#3187D4711yescharacteristic
204C2×#481—471no—
205#2502—471no—
206#1964—471no—
207#1824—471no—
208#1520—471no—
209#1282—471no—
210C2×C47—22no—
211C2×C47—21no—
212C2×C47—22no—
213C2×C47—21no—
214C2×C47—161no—
215C2×C47#200811yescharacteristic
216C2×C47#201311yescharacteristic
217C2×C47#202211yescharacteristic
218C2×C22⋊Q8C2×D4711yescharacteristic
219C2×#121C2×D4711yescharacteristic
220C22×C4⋊C4—471no—
221#483—21no—
222#481—21no—
223#481—942no—
224#481—942no—
225#465—22no—
226#450—21no—
227#448—21no—
228C2×#106—941no—
229C2×#102—941no—
230C2×#99—942no—
231C2×C22⋊C8—471no—
232C2×C22⋊C8C2×D4711yescharacteristic
233#398—942no—
234#391—941no—
235#385—942no—
236#384—941no—
237#380—471no—
238C4×C22⋊C4—471no—
239#340—942no—
240#328—942no—
241C47#995111yescharacteristic, complemented, Sylow
242C22×Q8C22⋊D4711yescharacteristic
243C22×D4—21no—
244C22×D4C22⋊D4711yescharacteristic
245C23×C4—941no—
246C23×C4C22×D4711yescharacteristic
247C2×SD16—22no—
248C22×C8—941no—
249C22×C8C22⋊D4712yes—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—472no—
260C2×C4⋊C4—941no—
261C2×C4⋊C4—941no—
262C2×C4⋊C4—941no—
263C2×C4⋊C4—941no—
264C2×C4⋊C4—944no—
265C2×C4⋊C4—941no—
266C2×C4⋊C4—941no—
267C2×C4⋊C4C22⋊D4711yescharacteristic
268C2×C4⋊C4C22⋊D4711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—472no—
272C2×C22⋊C4—941no—
273C2×C42—941no—
274#106—22no—
275#106—944no—
276#103—22no—
277#102—22no—
278#102—944no—
279#99—1884no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—472no—
283C22⋊C8—941no—
284#95—944no—
285#95—941no—
286#95—941no—
287C24—21no—
288C24D4×D4711yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—942no—
304C22×C4—942no—
305C22×C4—3762no—
306C22×C4—1881no—
307C22×C4—1881no—
308C22×C4—1881no—
309C22×C4D4×D4711yescharacteristic
310C22×C4D4×D4711yescharacteristic
311C22×C4#1062535211yescharacteristic
312C22×C4C2×C22⋊D4711yescharacteristic
313C22×C4C2×C22⋊D4711yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315SD16—82no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—942no—
319C2×C8—1882no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—942no—
327C4⋊C4—942no—
328C4⋊C4—3764no—
329C4⋊C4—1882no—
330C4⋊C4—1881no—
331C4⋊C4—1881no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—474no—
337C22⋊C4—942no—
338C22⋊C4—1882no—
339C42—1882no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23#1309399211yescharacteristic
347C23#1309399611yescharacteristic
348C23#1309402011yescharacteristic
349C23SD16×D4712yes—
350C23#1309405312yes—
351Q8—42no—
352Q8—81no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—944no—
371C2×C4—3762no—
372C2×C4—3762no—
373C2×C4—3761no—
374C2×C4—3761no—
375C2×C4—3761no—
376C2×C4—1884no—
377C2×C4—1882no—
378C2×C4—1882no—
379C2×C4—1882no—
380C2×C4#1309399211yescharacteristic
381C2×C4#1309399311yescharacteristic
382C2×C4#1309399511yescharacteristic
383C2×C4#1309399611yescharacteristic
384C8—42no—
385C8—1882no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—162no—
398C22#1568903111yescharacteristic
399C22#1568926012yes—
400C22#1568932612yes—
401C22#1568949411yescharacteristic
402C22#1568951411yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—3762no—
410C4—1884no—
411C4—7521no—
412C4#1568931312yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2#2024443811yescharacteristic
419C2#2024446911yescharacteristic
420C2#2024451711yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#3187C2
socleC22×C47#392
prime radicalC2×#121×C47C22
hypercenter#3187D47
hypocenterC47#9951
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #20247867C2semidirect
4. #20243479C2semidirect
5. #20241960C2semidirect
6. #20241235C2semidirect
119. C22⋊Dic47SD16semidirect
241. C47#9951semidirect

Maximal subgroups

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

subgroupindex
C2×#156894392
#202478672
#202442252
#202434792
#202419602
#202412352
C47×#31872
#995147