Finite Group Databasebeta

← #26115817

Subgroups — #26115817

Order 12032.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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×#15689354C211yesmaximal, characteristic
2#20244224C211yesmaximal, characteristic
3#20243415C211yesmaximal, characteristic, complemented
4#20243166C211yesmaximal, characteristic, complemented
5#20241960C211yesmaximal, characteristic, complemented
6#20241955C211yesmaximal, characteristic
7C47×#1996C211yesmaximal, characteristic, complemented
8C22×#10625336C411yescharacteristic, complemented
9C23×C47⋊C8C2211yescharacteristic
10C2×#13093998C2211yescharacteristic
11#15689354—21no—
12#15689354—21no—
13#15689354—21no—
14#15689354—21no—
15C2×#13093957—21no—
16C2×#13093955C411yescharacteristic, complemented
17C2×#13093948C411yescharacteristic, complemented
18C2×#13093944C411yescharacteristic, complemented
19C2×#13093938—22no—
20C2×C22⋊(C47⋊C8)C2211yescharacteristic
21C2×#13093935—21no—
22#15688983C2211yescharacteristic
23#15688885—21no—
24#15688878—21no—
25#15688860—22no—
26#15688856—22no—
27C2×C22⋊Q8×C47C2211yescharacteristic
28C47×#380C2211yescharacteristic
29C4×C22⋊C4×C47C2211yescharacteristic
30C47×#342—22no—
31C22×D188C2×C411yescharacteristic
32C22×C22⋊D47—21no—
33C2×#10625336—21no—
34C2×#10625336—21no—
35C2×#10625336—42no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—42no—
38C22×C47⋊C8C2×C411yescharacteristic
39C22×C47⋊C8C2×C411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41C22×C47⋊C8D411yescharacteristic
42C22×C47⋊C8D411yescharacteristic
43C22×C47⋊C8D411yescharacteristic
44#13093998—21no—
45#13093998—21no—
46#13093998—21no—
47#13093998—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47C2×C411yescharacteristic
50C2×C22⋊Dic47—21no—
51C2×(C2×D47)⋊C4—21no—
52#13093957—42no—
53#13093955—21no—
54#13093955—21no—
55#13093948—21no—
56#13093948—21no—
57#13093944—21no—
58#13093944—21no—
59#13093938—44no—
60C22⋊(C47⋊C8)—21no—
61C22⋊(C47⋊C8)D412yes—
62#13093935—42no—
63C47⋊#95—21no—
64C22×Q8×C47C2×C411yescharacteristic
65C23×C4×C47C2311yescharacteristic
66C22⋊Q8×C47—21no—
67C22⋊Q8×C47—21no—
68C22⋊Q8×C47—21no—
69C22⋊Q8×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47C2×C411yescharacteristic
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47—21no—
75C2×C22⋊C4×C47D412yes—
76C2×C42×C47—21no—
77#95×C47—21no—
78#95×C47—21no—
79C24×C47C2×D411yescharacteristic
80C23×D47—21no—
81C2×Q8×C47—21no—
82C2×Q8×C47—21no—
83C2×Q8×C47—41no—
84C2×Q8×C47—41no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—41no—
89C22×C4×C47—41no—
90C22×C4×C47—41no—
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47#4111yescharacteristic
96C22×C4×C47#4111yescharacteristic
97C2×C22⋊D47—41no—
98C2×C22⋊D47—41no—
99C2×C22⋊D47—42no—
100C2×C22⋊D47—42no—
101C22×Dic47—21no—
102C22×Dic47—41no—
103C2×D188—21no—
104C2×D188—21no—
105C2×D188—41no—
106C2×D188—41no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C22⋊C4×C47—22no—
114C22⋊C4×C47—22no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C22⋊C4×C47—42no—
118C42×C47—41no—
119C42×C47—41no—
120C22⋊Dic47—42no—
121#10625336—42no—
122#10625336—42no—
123(C2×D47)⋊C4—41no—
124(C2×D47)⋊C4—41no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127C4⋊Dic47—41no—
128C4⋊Dic47—41no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C2×C47⋊C8—41no—
138C2×C47⋊C8C22⋊C411yescharacteristic
139C2×C47⋊C8C22⋊C411yescharacteristic
140C2×C47⋊C8C22⋊C411yescharacteristic
141C2×C47⋊C8C22⋊C411yescharacteristic
142C23×C47—41no—
143C23×C47—41no—
144C23×C47C4×D411yescharacteristic
145C23×C47#12111yescharacteristic
146C23×C47#12711yescharacteristic
147C23×C47AGL(1,ℤ/8)12yes—
148C23×C47#13712yes—
149C22×D47—41no—
150C22×D47—42no—
151C22×D47—81no—
152C22×D47—81no—
153Q8×C47—41no—
154Q8×C47—41no—
155Q8×C47—81no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—42no—
170C2×C4×C47—81no—
171C2×C4×C47—81no—
172C2×C4×C47—81no—
173C2×C4×C47—81no—
174C2×C4×C47—81no—
175C2×C4×C47C2×C22⋊C411yescharacteristic
176C2×C4×C47C4×D411yescharacteristic
177C2×C4×C47#12111yescharacteristic
178C2×C4×C47#12411yescharacteristic
179C22⋊D47—82no—
180C22⋊D47—81no—
181C22⋊D47—81no—
182C2×Dic47—41no—
183C2×Dic47—42no—
184C2×Dic47—81no—
185C2×Dic47—81no—
186C2×Dic47—81no—
187D188—41no—
188D188—41no—
189D188—81no—
190C47⋊C8—21no—
191C47⋊C8—21no—
192C47⋊C8—21no—
193C47⋊C8—42no—
194C47⋊C8#9912yescomplemented
195#9786—471nomaximal, Sylow
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—42no—
203C22×C47—42no—
204C22×C47#39011yescharacteristic
205C22×C47#44012yes—
206C22×C47#46812yes—
207C22×C47#49611yescharacteristic
208C22×C47#49711yescharacteristic
209C2×D47—81no—
210C2×D47—81no—
211C2×D47—81no—
212C2×D47—162no—
213C4×C47—21no—
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—82no—
217C4×C47—81no—
218C4×C47—81no—
219C4×C47—161no—
220C4×C47C2×#9911yescharacteristic
221C4×C47#41011yescharacteristic
222Dic47—81no—
223Dic47—161no—
224C2×#465—471no—
225#1996D4711yescharacteristic, complemented
226#1964—471no—
227#1808—471no—
228#1752—471no—
229#1520—471no—
230#1519—471no—
231C2×C47—21no—
232C2×C47—22no—
233C2×C47—22no—
234C2×C47—21no—
235C2×C47#196411yescharacteristic
236C2×C47#197311yescharacteristic
237C2×C47#197611yescharacteristic
238D47—161no—
239C22×SD16—471no—
240C23×C8—471no—
241C2×C22⋊Q8C2×D4711yescharacteristic
242C2×#121—471no—
243#465—941no—
244#465—941no—
245#465—941no—
246#465—941no—
247C2×#107—941no—
248C2×#106—471no—
249C2×#103—471no—
250C2×#102—471no—
251C2×#99—942no—
252C2×C22⋊C8—471no—
253C2×#97—941no—
254#380—471no—
255#380C2×D4711yescharacteristic
256C4×C22⋊C4C2×D4711yescharacteristic
257#342—22no—
258#340—941no—
259#336—941no—
260#328—942no—
261#327—942no—
262C47#978611yescharacteristic, complemented, Sylow
263C22×Q8C4×D4711yescharacteristic
264C22×D4—471no—
265C22×D4—941no—
266C23×C4C22×D4711yescharacteristic
267C2×SD16—941no—
268C2×SD16—941no—
269C2×SD16—1882no—
270C22×C8—471no—
271C22×C8—471no—
272C22×C8—471no—
273C22×C8—471no—
274C22×C8—471no—
275C22×C8—471no—
276C22×C8—941no—
277C22×C8—1882no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—941no—
283#121—941no—
284#121—941no—
285#121—941no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—471no—
289C2×C4⋊C4—941no—
290C2×C4⋊C4C4×D4711yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—941no—
294C2×C22⋊C4—941no—
295C2×C22⋊C4C22⋊D4712yes—
296C2×C42—21no—
297#107—1882no—
298#106—941no—
299#106—941no—
300#103—941no—
301#103—941no—
302#102—941no—
303#102—941no—
304#99—1884no—
305C22⋊C8—472no—
306C22⋊C8—941no—
307#97—1882no—
308#95—21no—
309#95—21no—
310#95—941no—
311C24—941no—
312C24C2×C22⋊D4711yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—941no—
318C2×D4—941no—
319C2×D4—1881no—
320C2×D4—1881no—
321C2×D4—1881no—
322C2×D4—1881no—
323C2×D4—1882no—
324C2×D4—1882no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—941no—
332C22×C4—1881no—
333C22×C4C2×C4×D4711yescharacteristic
334C22×C4#1062535011yescharacteristic
335C22×C4D4×D4711yescharacteristic
336C22×C4D4×D4711yescharacteristic
337C22×C4D4×D4711yescharacteristic
338C22×C4#1062535211yescharacteristic
339SD16—1882no—
340SD16—1882no—
341C2×C8—471no—
342C2×C8—471no—
343C2×C8—471no—
344C2×C8—471no—
345C2×C8—942no—
346C2×C8—942no—
347C2×C8—942no—
348C2×C8—942no—
349C2×C8—1881no—
350C2×C8—1881no—
351C2×C8—1881no—
352C2×C8—1881no—
353C2×C8—1881no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—941no—
361C4⋊C4—941no—
362C4⋊C4—1881no—
363C4⋊C4—1881no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—1882no—
370C22⋊C4—1881no—
371C22⋊C4—1881no—
372C42—41no—
373C42—41no—
374C23—41no—
375C23—41no—
376C23—3761no—
377C23—3761no—
378C23—1881no—
379C23—1882no—
380C23C4×C22⋊D4711yescharacteristic
381C23#1309399611yescharacteristic
382C23#1309402011yescharacteristic
383C23#1309404912yes—
384C23#1309405512yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—3761no—
389D4—3761no—
390D4—3761no—
391D4—3762no—
392D4—1881no—
393D4—1881no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—3761no—
414C2×C4—3761no—
415C2×C4—3761no—
416C2×C4—1882no—
417C2×C4—1881no—
418C2×C4C22⋊C4×D4711yescharacteristic
419C2×C4(C22⋊D47)⋊C411yescharacteristic
420C2×C4#1309399711yescharacteristic
421C2×C4#1309401011yescharacteristic
422C8—472no—
423C8—941no—
424C8—941no—
425C8—941no—
426C8—1882no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—3761no—
436C22—3761no—
437C22—3761no—
438C22—7522no—
439C22#1568902311yescharacteristic
440C22#1568922112yes—
441C22#1568937112yes—
442C22#1568951311yescharacteristic
443C22#1568951511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—82no—
448C4—81no—
449C4—81no—
450C4—161no—
451C4—3761no—
452C4—7521no—
453C4#99×D4711yescharacteristic
454C4#1568909511yescharacteristic
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—7521no—
460C2#2024422211yescharacteristic
461C2#2024427211yescharacteristic
462C2#2024428311yescharacteristic
subgroupG/H
centerC22#15689023
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1996C2
socleC22×C47#390
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1996D47
hypocenterC47#9786
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
3. #20243415C2semidirect
4. #20243166C2semidirect
5. #20241960C2semidirect
7. C47×#1996C2semidirect
8. C22×#10625336C4semidirect
16. C2×#13093955C4semidirect
17. C2×#13093948C4semidirect
18. C2×#13093944C4semidirect
194. C47⋊C8#99semidirect
225. #1996D47semidirect
262. C47#9786semidirect

Maximal subgroups

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

subgroupindex
C2×#156893542
#202442242
#202434152
#202431662
#202419602
#202419552
C47×#19962
#978647