Finite Group Databasebeta

← #26371073

Subgroups — #26371073

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689161C211yesmaximal, characteristic, complemented
2#20245593C212yesmaximal, complemented
3#20245588C211yesmaximal, characteristic, complemented
4#20245583C211yesmaximal, characteristic, complemented
5#20245576C212yesmaximal, complemented
6#20245571C211yesmaximal, characteristic, complemented
7#20245566C211yesmaximal, characteristic, complemented
8#20245170C212yesmaximal, complemented
9#20245095C212yesmaximal, complemented
10C2×C22⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
11C47×#3050C211yesmaximal, characteristic
12C2×#13093983—21no—
13C22×C47⋊C16C2212yes—
14#15689161—21no—
15#15689161—21no—
16#15689161C2212yes—
17#15689161C2212yes—
18C2×#13093958C2211yescharacteristic
19C2×#13093958C2211yescharacteristic
20C2×#13093956—21no—
21#15688950C2212yescomplemented
22#15688950C2212yescomplemented
23#15688948C2212yescomplemented
24#15688948C2212yescomplemented
25#15688927C2212yescomplemented
26#15688927C2212yescomplemented
27#15688924C2212yescomplemented
28#15688924C2212yescomplemented
29C22⋊(C47⋊C16)C2211yescharacteristic
30C22⋊(C47⋊C16)C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C2×#135×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C47×#525—21no—
35C47×#467C2211yescharacteristic
36C47×#467C2211yescharacteristic
37C47×#466C2211yescharacteristic
38C47×#466C2211yescharacteristic
39C47×#465—21no—
40C47×#465—21no—
41C47×#427C2211yescharacteristic
42C47×#416—21no—
43#13093983—22no—
44#13093983—22no—
45#13093983—41no—
46#13093983—41no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C22⋊Dic47—41no—
50C2×C4×Dic47—41no—
51C2×C47⋊C16—24no—
52C2×C47⋊C16C2312yes—
53C2×C47⋊C16C2312yes—
54#13093958—21no—
55#13093958—21no—
56#13093958C2312yes—
57#13093958C2312yes—
58#13093956—22no—
59#13093956—22no—
60C2×#41×C47—21no—
61C23×C4×C47D411yescharacteristic
62#135×C47—24no—
63C2×Q16×C47C2311yescharacteristic
64C2×SD16×C47—21no—
65C2×D8×C47C2311yescharacteristic
66C22×C8×C47—24no—
67C22×C8×C47D411yescharacteristic
68C22×C8×C47D411yescharacteristic
69C22×C8×C47D411yescharacteristic
70C22×C8×C47C2311yescharacteristic
71C22×C8×C47C2311yescharacteristic
72C22×C8×C47C2311yescharacteristic
73#123×C47—41no—
74C22⋊Q8×C47—21no—
75C22⋊Q8×C47—21no—
76#121×C47—21no—
77#121×C47—21no—
78#120×C47—41no—
79C4×D4×C47—41no—
80C4×D4×C47—41no—
81#117×C47—21no—
82#107×C47C2311yescharacteristic
83#107×C47C2311yescharacteristic
84#106×C47—21no—
85#103×C47—21no—
86#103×C47—21no—
87#102×C47—21no—
88#102×C47—21no—
89C24×C47D811yescharacteristic
90#41×C47—42no—
91#41×C47—42no—
92C2×Q8×C47—21no—
93C2×D4×C47—21no—
94C2×D4×C47—41no—
95C2×D4×C47—41no—
96C2×D4×C47—41no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—41no—
100C22×C4×C47D811yescharacteristic
101C22×C4×C47D811yescharacteristic
102C22×C4×C47D811yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×Dic47—41no—
107C22×Dic47—41no—
108Q16×C47—22no—
109SD16×C47—42no—
110D8×C47—22no—
111C2×C8×C47—22no—
112C2×C8×C47—22no—
113C2×C8×C47—22no—
114C2×C8×C47—22no—
115C2×C8×C47—22no—
116C2×C8×C47C2×D411yescharacteristic
117C2×C8×C47C2×D411yescharacteristic
118C2×C8×C47C2×D411yescharacteristic
119C2×C8×C47#4114yes—
120C2×C8×C47C2411yescharacteristic
121C4⋊C4×C47—21no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—41no—
124C4⋊C4×C47—41no—
125C22⋊C4×C47—41no—
126C22⋊C4×C47—41no—
127C22⋊C4×C47—41no—
128C22⋊C4×C47—41no—
129C22⋊C4×C47—41no—
130C42×C47—41no—
131C22⋊Dic47—42no—
132C22⋊Dic47—42no—
133C4⋊Dic47—22no—
134C4⋊Dic47—22no—
135C4⋊Dic47—41no—
136C4⋊Dic47—41no—
137C4×Dic47—42no—
138C4×Dic47—42no—
139C47⋊C16—22no—
140C47⋊C16—22no—
141C23×C47—22no—
142C23×C47—22no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47C2×D811yescharacteristic
146C23×C47C2×D811yescharacteristic
147C23×C47C2×D811yescharacteristic
148Q8×C47—41no—
149Q8×C47—41no—
150D4×C47—41no—
151D4×C47—41no—
152D4×C47—82no—
153D4×C47—82no—
154D4×C47—82no—
155C2×C4×C47—22no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—22no—
161C2×C4×C47—22no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—82no—
166C2×C4×C47#12312yes—
167C2×C4×C47C2×D811yescharacteristic
168C2×C4×C47C2×D811yescharacteristic
169C2×C4×C47C2×D811yescharacteristic
170C2×C4×C47#13712yes—
171C2×C4×C47C22×D411yescharacteristic
172C2×Dic47—42no—
173C2×Dic47—42no—
174C2×Dic47—82no—
175C2×Dic47—82no—
176C8×C47—21no—
177C8×C47—21no—
178C8×C47#12312yes—
179C8×C47C2×#4112yes—
180#33645—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—44no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47#48012yes—
193C22×C47#50812yes—
194C22×C47C22×D811yescharacteristic
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—81no—
198C4×C47—81no—
199C4×C47—81no—
200C4×C47#48012yes—
201C4×C47C2×#12311yescharacteristic
202C4×C47C2×#13711yescharacteristic
203Dic47—82no—
204Dic47—82no—
205#3050D4711yescharacteristic
206C2×#427—471no—
207#2237—472no—
208#2236—471no—
209#2235—471no—
210#2234—472no—
211#2233—471no—
212#2232—471no—
213#2158—472no—
214#2138—472no—
215C2×C22⋊C16—471no—
216C2×C47—22no—
217C2×C47—22no—
218C2×C47—41no—
219C2×C47—161no—
220C2×C47C2×#48011yescharacteristic
221C2×C47C2×#50811yescharacteristic
222C2×C47#341411yescharacteristic
223C2×#135C2×D4711yescharacteristic
224C23×C8C2×D4711yescharacteristic
225#525—21no—
226C2×#117—941no—
227C22×C16—472no—
228#467C2×D4711yescharacteristic
229#467C2×D4711yescharacteristic
230#466C2×D4711yescharacteristic
231#466C2×D4711yescharacteristic
232#465—21no—
233#465—21no—
234#427—472no—
235#427—472no—
236#427—941no—
237#427—941no—
238#427C2×D4711yescharacteristic
239C2×#107—471no—
240C2×#107—471no—
241C2×#106—941no—
242#416—21no—
243#367—472no—
244#367—472no—
245#366—472no—
246#366—472no—
247#358—472no—
248#358—472no—
249#357—472no—
250#357—472no—
251C22⋊C16—471no—
252C22⋊C16—471no—
253C22⋊C16—472no—
254C47#3364511yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C23×C4C22⋊D4711yescharacteristic
257#135—24no—
258C2×Q16C22×D4711yescharacteristic
259C2×SD16—21no—
260C2×D8C22×D4711yescharacteristic
261C22×C8—24no—
262C22×C8C22⋊D4711yescharacteristic
263C22×C8C22⋊D4711yescharacteristic
264C22×C8C22⋊D4711yescharacteristic
265C22×C8C22×D4711yescharacteristic
266C22×C8C22×D4711yescharacteristic
267C22×C8C22×D4711yescharacteristic
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#120—41no—
274C4×D4—41no—
275C4×D4—41no—
276#117—21no—
277#117—942no—
278#117—942no—
279#117—1881no—
280#117—1881no—
281C2×C4⋊C4—941no—
282C2×C4⋊C4—941no—
283C2×C22⋊C4—1881no—
284C2×C42—1881no—
285C2×C16—472no—
286C2×C16—472no—
287C2×C16—944no—
288#107—472no—
289#107—472no—
290#107—941no—
291#107—941no—
292#107C22×D4711yescharacteristic
293#107C22×D4711yescharacteristic
294#106—21no—
295#106—942no—
296#106—942no—
297#103—21no—
298#103—21no—
299#102—21no—
300#102—21no—
301C24#1062533411yescharacteristic
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—1881no—
313C22×C4—1881no—
314C22×C4#1062533411yescharacteristic
315C22×C4#1062533411yescharacteristic
316C22×C4#1062533411yescharacteristic
317C22×C4C2×C22⋊D4711yescharacteristic
318C22×C4C2×C22⋊D4711yescharacteristic
319C22×C4C2×C22⋊D4711yescharacteristic
320Q16—22no—
321SD16—42no—
322D8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#1062535214yes—
329C2×C8C2×C22⋊D4711yescharacteristic
330C2×C8C2×C22⋊D4711yescharacteristic
331C2×C8C2×C22⋊D4711yescharacteristic
332C2×C8C23×D4711yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—942no—
338C4⋊C4—942no—
339C4⋊C4—1881no—
340C4⋊C4—1881no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—1882no—
347C22⋊C4—1882no—
348C42—41no—
349C42—1882no—
350C42—1882no—
351C16—942no—
352C16—942no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23C2×#1062533411yescharacteristic
358C23C2×#1062533411yescharacteristic
359C23C2×#1062533411yescharacteristic
360Q8—41no—
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—82no—
378C2×C4—3762no—
379C2×C4—3762no—
380C2×C4—1882no—
381C2×C4—1882no—
382C2×C4#1309400512yes—
383C2×C4C2×#1062533411yescharacteristic
384C2×C4C2×#1062533411yescharacteristic
385C2×C4C2×#1062533411yescharacteristic
386C2×C4#1309405412yes—
387C2×C4C22×C22⋊D4711yescharacteristic
388C8—21no—
389C8—21no—
390C8#1309400512yes—
391C8C2×#1062535212yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#1568943412yes—
404C22#1568955312yes—
405C22C22×#1062533411yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—3762no—
412C4—3762no—
413C4#1568943412yes—
414C4C2×#1309400511yescharacteristic
415C4C2×#1309405411yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—161no—
420C2C2×#1568943411yescharacteristic
421C2C2×#1568955311yescharacteristic
422C2#2025652211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
Fitting, prime radicalC47×#3050C2
socleC22×C47C22×D8
hypercenter#3050D47
hypocenterC47#33645
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#15689161C2semidirect
2. #20245593C2semidirect
3. #20245588C2semidirect
4. #20245583C2semidirect
5. #20245576C2semidirect
6. #20245571C2semidirect
7. #20245566C2semidirect
8. #20245170C2semidirect
9. #20245095C2semidirect
10. C2×C22⋊(C47⋊C16)C2semidirect
21. #15688950C22semidirect
22. #15688950C22semidirect
23. #15688948C22semidirect
24. #15688948C22semidirect
25. #15688927C22semidirect
26. #15688927C22semidirect
27. #15688924C22semidirect
28. #15688924C22semidirect
254. C47#33645semidirect

Maximal subgroups

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

subgroupindex
C2×#156891612
#202455932
#202455882
#202455832
#202455762
#202455712
#202455662
#202451702
#202450952
C2×C22⋊(C47⋊C16)2
C47×#30502
#3364547