Finite Group Databasebeta

← #26111503

Subgroups — #26111503

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253943C211yesmaximal, characteristic
2#20252335C211yesmaximal, characteristic, complemented
3#20241886C211yesmaximal, characteristic, complemented
4#20241787C211yesmaximal, characteristic
5#20241229C211yesmaximal, characteristic, complemented
6#20241226C211yesmaximal, characteristic, complemented
7C47×#1980C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689519C411yescharacteristic, complemented
10#15689495C411yescharacteristic, complemented
11#15689475C411yescharacteristic, complemented
12#15689469C411yescharacteristic, complemented
13#15689350—22no—
14#15689253—21no—
15#15689251—22no—
16C8×Dic94—21no—
17C4×#10625335—22no—
18#15689176C2211yescharacteristic
19#15689151—21no—
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21#15689081—21no—
22C2×C4×C47⋊C8C2211yescharacteristic
23#15688859—22no—
24D4⋊(C47⋊C8)—21no—
25D4⋊(C47⋊C8)—21no—
26C47×#532C2211yescharacteristic
27C2×#104×C47—21no—
28C2×#104×C47—21no—
29C4×M16×C47C2211yescharacteristic
30C4×C4⋊C4×C47C2211yescharacteristic
31C2×#10625335—22no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8—21no—
34#13094024C2×C411yescharacteristic
35#13094024C2×C411yescharacteristic
36#13094016—21no—
37#13094003—21no—
38#13094003D412yes—
39C4×Dic94—21no—
40C4×Dic94D412yes—
41#13093983—21no—
42#13093955—22no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)D411yescharacteristic
46C4⋊(C47⋊C8)D411yescharacteristic
47Dic47⋊C8—22no—
48Dic47⋊C8—22no—
49#13093947—22no—
50#13093945—22no—
51C22⋊(C47⋊C8)—21no—
52#13093934—22no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C4×C47⋊C8C2×C411yescharacteristic
56C4×C47⋊C8C2×C411yescharacteristic
57C8×Dic47—22no—
58C2×#41×C47—21no—
59C2×M16×C47D411yescharacteristic
60C2×M16×C47D411yescharacteristic
61C4⋊Q8×C47C2×C411yescharacteristic
62#127×C47C2×C411yescharacteristic
63#124×C47—21no—
64#121×C47—21no—
65#121×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C42×C47—21no—
70C2×C42×C47C2311yescharacteristic
71#104×C47—42no—
72#104×C47—42no—
73#97×C47—21no—
74C4×C8×C47—21no—
75#95×C47—21no—
76#41×C47—41no—
77#41×C47—41no—
78#41×C47—41no—
79#41×C47—41no—
80C2×Q8×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C2×Dic94—21no—
91C2×Dic94C22⋊C412yes—
92M16×C47—21no—
93M16×C47—21no—
94M16×C47—21no—
95M16×C47SD1612yescomplemented
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—41no—
104C42×C47—41no—
105C42×C47—41no—
106C42×C47C22×C411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47#4111yescharacteristic
110C22⋊Dic47—22no—
111#10625335—44no—
112C4⋊Dic47—21no—
113C4⋊Dic47—21no—
114C4⋊Dic47—21no—
115C4⋊Dic47C22⋊C412yes—
116Dic47⋊C4—22no—
117Dic47⋊C4—22no—
118C4×Dic47—22no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—21no—
122C2×C47⋊C8—21no—
123C2×C47⋊C8—42no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C23×C47—41no—
128C23×C47#12111yescharacteristic
129Q8×C47—41no—
130Q8×C47—41no—
131D4×C47—42no—
132D4×C47—41no—
133D4×C47—41no—
134D4×C47—41no—
135D4×C47—41no—
136D4×C47—81no—
137D4×C47—81no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47—81no—
144C2×C4×C47—81no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47C2×C22⋊C411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47C4×D411yescharacteristic
150C2×C4×C47#12011yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47#12311yescharacteristic
153C2×C4×C47C2×SD1611yescharacteristic
154C2×C4×C47#13511yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47#13711yescharacteristic
157C2×Dic47—22no—
158C2×Dic47—22no—
159Dic94—41no—
160Dic94—41no—
161Dic94—42no—
162C8×C47—22no—
163C8×C47—41no—
164C47⋊C8—22no—
165C47⋊C8—41no—
166C47⋊C8—41no—
167C47⋊C8—41no—
168C47⋊C8—81no—
169#9054—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47#38611yescharacteristic
176C22×C47#46011yescharacteristic
177C22×C47#46411yescharacteristic
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—82no—
182C4×C47—81no—
183C4×C47C4×SD1611yescharacteristic
184C4×C47#44011yescharacteristic
185C4×C47#44311yescharacteristic
186C4×C47#44411yescharacteristic
187C4×C47#44811yescharacteristic
188C4×C47#44911yescharacteristic
189Dic47—42no—
190Dic47—42no—
191#3167—471no—
192#2962—471no—
193#1980D4711yescharacteristic
194#1503—471no—
195#1480—471no—
196#1279—471no—
197#1278—471no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—161no—
201C2×C47#187611yescharacteristic
202C2×C47#188011yescharacteristic
203C2×C47#188411yescharacteristic
204#533—471no—
205#532C2×D4711yescharacteristic
206#498—471no—
207#492—471no—
208#487—471no—
209#486—471no—
210#465—942no—
211#446—941no—
212#446—942no—
213C8×Q8—941no—
214C4×SD16—942no—
215#431—471no—
216#424—941no—
217C2×C4⋊C8—471no—
218C2×#104—21no—
219C2×#104—21no—
220#405—941no—
221C4×M16C2×D4711yescharacteristic
222C2×C4×C8—471no—
223C4×C4⋊C4C2×D4711yescharacteristic
224#328—942no—
225D4⋊C8—941no—
226D4⋊C8—941no—
227C47#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—942no—
230C2×M16C22⋊D4711yescharacteristic
231C2×M16C22⋊D4711yescharacteristic
232C22×C8—941no—
233C22×C8—941no—
234C4⋊Q8—471no—
235C4⋊Q8—471no—
236C4⋊Q8C4×D4711yescharacteristic
237#127C4×D4711yescharacteristic
238#125—941no—
239#124—21no—
240C22⋊Q8—472no—
241C22⋊Q8—941no—
242#121—21no—
243#121—21no—
244C4×Q8—472no—
245C4×Q8—941no—
246C4×D4—21no—
247C4×D4—21no—
248#117—941no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C22×D4711yescharacteristic
252#106—942no—
253C4⋊C8—471no—
254C4⋊C8—471no—
255C4⋊C8—942no—
256C4⋊C8—942no—
257C4⋊C8—941no—
258C4⋊C8—941no—
259#104—42no—
260#104—42no—
261#103—942no—
262#102—942no—
263C22⋊C8—941no—
264#97—21no—
265#97—942no—
266C4×C8—21no—
267C4×C8—471no—
268C4×C8—471no—
269C4×C8—942no—
270C4×C8—941no—
271C4×C8—941no—
272#95—21no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—472no—
279C2×Q8—941no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4D4×D4711yescharacteristic
287C22×C4#1062535211yescharacteristic
288C22×C4C2×C22⋊D4711yescharacteristic
289SD16—1884no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1062533512yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—941no—
297C2×C8—941no—
298C2×C8—941no—
299C2×C8—941no—
300C2×C8—1882no—
301C2×C8—1881no—
302C2×C8—1881no—
303C2×C8—1881no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—472no—
309C4⋊C4—941no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—942no—
313C4⋊C4—942no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—942no—
317C42—41no—
318C42—41no—
319C42—942no—
320C42C2×C4×D4711yescharacteristic
321C42#1062535011yescharacteristic
322C42D4×D4711yescharacteristic
323C42D4×D4711yescharacteristic
324C23—41no—
325C23#1309399511yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—1882no—
329Q8—1881no—
330Q8—1881no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—942no—
348C2×C4—942no—
349C2×C4C22⋊C4×D4711yescharacteristic
350C2×C4(C22⋊D47)⋊C411yescharacteristic
351C2×C4C4×C22⋊D4711yescharacteristic
352C2×C4#1309399211yescharacteristic
353C2×C4#1309399711yescharacteristic
354C2×C4#1309400611yescharacteristic
355C2×C4C2×#1062533511yescharacteristic
356C2×C4#1309404411yescharacteristic
357C2×C4#1309405011yescharacteristic
358C2×C4#1309405411yescharacteristic
359C8—22no—
360C8—41no—
361C8—942no—
362C8—3761no—
363C8—1881no—
364C8—1881no—
365C8—1881no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1568899911yescharacteristic
372C22#1568932011yescharacteristic
373C22#1568934211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—81no—
379C4—1882no—
380C4—1882no—
381C4C4×#1062533511yescharacteristic
382C4#1568922311yescharacteristic
383C4#1568923711yescharacteristic
384C4#1568924211yescharacteristic
385C4#1568926111yescharacteristic
386C4#1568926611yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2024371211yescharacteristic
391C2#2024374611yescharacteristic
392C2#2024376811yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1980C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#1980D47
hypocenterC47#9054
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #20252335C2semidirect
3. #20241886C2semidirect
5. #20241229C2semidirect
6. #20241226C2semidirect
9. #15689519C4semidirect
10. #15689495C4semidirect
11. #15689475C4semidirect
12. #15689469C4semidirect
95. M16×C47SD16semidirect
227. C47#9054semidirect
293. M16#10625335semidirect

Maximal subgroups

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

subgroupindex
#202539432
#202523352
#202418862
#202417872
#202412292
#202412262
C47×#19802
#905447