Finite Group Databasebeta

← #26226670

Subgroups — #26226670

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689145C211yesmaximal, characteristic
2#20245793C211yesmaximal, characteristic
3#20245784C211yesmaximal, characteristic
4#20244155C212yesmaximal, complemented
5#20244153C212yesmaximal, complemented
6#20244152C211yesmaximal, characteristic, complemented
7#20244149C211yesmaximal, characteristic, complemented
8#20244146C211yesmaximal, characteristic, complemented
9#20243422C212yesmaximal
10#20243420C212yesmaximal
11C47×#1947C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13C2×#13093982—21no—
14C2×C42×D47C2211yescharacteristic
15C2×#13093957C2212yes—
16C2×#13093956C2212yes—
17C2×#13093955C2212yes—
18#15689145C418yescomplemented
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×Dic47⋊C8C2212yes—
21#15688993C2211yescharacteristic
22#15688990—21no—
23#15688988—21no—
24#15688988C2211yescharacteristic
25#15688988C2211yescharacteristic
26#15688985—21no—
27#15688980—21no—
28#15688980C2211yescharacteristic
29#15688980C2211yescharacteristic
30#15688980C2212yes—
31C4⋊C4×Dic47—21no—
32C4⋊C4×Dic47C2212yes—
33C4×C4⋊Dic47C2211yescharacteristic
34C4×(C2×D47)⋊C4—21no—
35#15688969—21no—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#107×C47C2212yes—
40C2×C4⋊C8×C47C2211yescharacteristic
41C47×#384C2211yescharacteristic
42C4×C4⋊C4×C47C2211yescharacteristic
43C47×#340C2212yes—
44C22×C4×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47C2311yescharacteristic
47C2×#10625324—42no—
48C2×#10625324—42no—
49C2×#10625324C2×C418yes—
50C22×C47⋊C8C2312yes—
51#13093982—44no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2312yes—
57C2×(C2×D47)⋊C4—21no—
58C2×(C2×D47)⋊C4—21no—
59C2×(C2×D47)⋊C4—22no—
60C2×C4×Dic47—21no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47D411yescharacteristic
63C2×C4×Dic47D411yescharacteristic
64C2×C4×Dic47C2311yescharacteristic
65C42×D47C2×C414yes—
66C42×D47Q814yes—
67#13093957—42no—
68#13093956—42no—
69#13093955—42no—
70C4⋊(C47⋊C8)C2×C414yes—
71Dic47⋊C8C2×C418yes—
72C47⋊#95—21no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C22×C8×C47C2312yes—
82C2×C4⋊C4×C47—21no—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47C2312yes—
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C42×C47—21no—
88C2×C42×C47C2311yescharacteristic
89#107×C47—42no—
90C4⋊C8×C47C2×C414yes—
91#95×C47—21no—
92C23×D47C2×D411yescharacteristic
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C22×Dic47—21no—
100C22×Dic47—21no—
101C22×Dic47—22no—
102C22×Dic47C2×D411yescharacteristic
103C22×Dic47C2×D411yescharacteristic
104C22×Dic47C2×D411yescharacteristic
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—21no—
108C2×C4×D47—21no—
109C2×C4×D47—21no—
110C2×C4×D47—21no—
111C2×C4×D47—41no—
112C2×C4×D47—41no—
113C2×C4×D47C4⋊C414yes—
114C2×C4×D47C4⋊C414yes—
115C2×C4×D47C22×C412yes—
116C2×C4×D47C2×Q812yes—
117C2×C8×C47—22no—
118C2×C8×C47—22no—
119C2×C8×C47C22×C414yes—
120C4⋊C4×C47—21no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—22no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—42no—
128C42×C47—42no—
129C42×C47C22×C412yes—
130C42×C47C2×Q812yes—
131(C2×D47)⋊C4—42no—
132(C2×D47)⋊C4—42no—
133(C2×D47)⋊C4—44no—
134C4⋊Dic47—21no—
135C4⋊Dic47—21no—
136C4⋊Dic47—22no—
137C4⋊Dic47—21no—
138C4⋊Dic47—21no—
139C4⋊Dic47—22no—
140C4⋊Dic47—42no—
141C4⋊Dic47—42no—
142C4×Dic47—44no—
143C4×Dic47—42no—
144C4×Dic47C4⋊C414yes—
145C4×Dic47C4⋊C414yes—
146C4×Dic47C22×C412yes—
147C4×Dic47C2×Q812yes—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—22no—
150C2×C47⋊C8C22×C414yes—
151#10625324—48no—
152C23×C47C22×D411yescharacteristic
153C22×D47—21no—
154C22×D47—21no—
155C22×D47—21no—
156C22×D47—21no—
157C22×D47—21no—
158C22×D47—21no—
159C22×D47C2×C4⋊C412yes—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47C2×C4⋊C412yes—
172C2×C4×C47C2×C4⋊C412yes—
173C2×C4×C47AGL(1,ℤ/8)12yes—
174C2×C4×C47#13712yes—
175C2×C4×C47C23×C411yescharacteristic
176C2×C4×C47C22×Q811yescharacteristic
177C2×C4×C47H(5,𝔽2)11yescharacteristic
178C2×C4×C47H(5,𝔽2)11yescharacteristic
179C2×C4×C472−1+411yescharacteristic
180C2×C4×C472−1+411yescharacteristic
181C2×Dic47—21no—
182C2×Dic47—21no—
183C2×Dic47—21no—
184C2×Dic47—21no—
185C2×Dic47—21no—
186C2×Dic47—21no—
187C2×Dic47—41no—
188C2×Dic47—42no—
189C2×Dic47—41no—
190C2×Dic47—42no—
191C2×Dic47—41no—
192C2×Dic47—41no—
193C2×Dic47—42no—
194C2×Dic47—41no—
195C2×Dic47—41no—
196C2×Dic47C2×C4⋊C412yes—
197C2×Dic47C2×C4⋊C412yes—
198C2×Dic47C2×C4⋊C412yes—
199C4×D47—22no—
200C4×D47—22no—
201C4×D47—22no—
202C4×D47—22no—
203C4×D47—28no—
204C4×D47—44no—
205C8×C47—24no—
206C47⋊C8—24no—
207#21963—471nomaximal, Sylow
208C22×C47C22×C4⋊C411yescharacteristic
209C22×C47#53711yescharacteristic
210C22×C47#53711yescharacteristic
211C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
212C22×C47C2×#13711yescharacteristic
213C22×C47#57511yescharacteristic
214C22×C47#57511yescharacteristic
215C2×D47—22no—
216C2×D47—22no—
217C2×D47—22no—
218C2×D47—41no—
219C2×D47—41no—
220C2×D47—41no—
221C2×D47—41no—
222C4×C47—22no—
223C4×C47—82no—
224C4×C47—81no—
225C4×C47—81no—
226C4×C47#42814yes—
227C4×C47#51811yescharacteristic
228C4×C47#51911yescharacteristic
229C4×C47#54411yescharacteristic
230C4×C47#54411yescharacteristic
231Dic47—22no—
232Dic47—22no—
233Dic47—22no—
234Dic47—82no—
235Dic47—81no—
236Dic47—81no—
237C2×#423—471no—
238#2285—471no—
239#2283—471no—
240#1949—472no—
241#1949—472no—
242#1948—471no—
243#1948—471no—
244#1947—471no—
245#1947D4711yescharacteristic, complemented
246#1809—472no—
247#1809—472no—
248C2×C47#231611yescharacteristic
249C2×C47C2×#42811yescharacteristic
250C2×C47#291111yescharacteristic
251C2×C47#325111yescharacteristic
252C2×C47#325111yescharacteristic
253C2×C47#325211yescharacteristic
254C2×C47#325211yescharacteristic
255D47—42no—
256C22×M16—472no—
257C2×#117—941no—
258C22×C42—471no—
259C2×#107—472no—
260C2×#107C2×D4712yes—
261C2×#106—472no—
262C2×#106—472no—
263#423—478no—
264C2×C4⋊C8—472no—
265C2×C4⋊C8—471no—
266C2×C4⋊C8C2×D4711yescharacteristic
267#384—471no—
268#384C2×D4711yescharacteristic
269#383—941no—
270#382—471no—
271#382—471no—
272#382—941no—
273#381—941no—
274#380—471no—
275#380—471no—
276#380—472no—
277#380—941no—
278C4×C4⋊C4—472no—
279C4×C4⋊C4—471no—
280C4×C4⋊C4—941no—
281C4×C4⋊C4C2×D4711yescharacteristic
282C4×C22⋊C4—941no—
283#376—941no—
284#340—472no—
285#340—472no—
286#340—472no—
287#340C2×D4712yes—
288C47#2196311yescharacteristic, complemented, Sylow
289C23×C4—471no—
290C23×C4—471no—
291C23×C4—471no—
292C2×M16—478no—
293C2×M16—1882no—
294C2×M16—1882no—
295C22×C8—472no—
296C22×C8C22×D4712yes—
297#117—1884no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—471no—
301C2×C4⋊C4—471no—
302C2×C4⋊C4—472no—
303C2×C4⋊C4—941no—
304C2×C4⋊C4—941no—
305C2×C4⋊C4C22×D4711yescharacteristic
306C2×C4⋊C4C22×D4712yes—
307C2×C4⋊C4C22×D4711yescharacteristic
308C2×C22⋊C4—942no—
309C2×C22⋊C4—941no—
310C2×C22⋊C4—941no—
311C2×C42—21no—
312C2×C42—471no—
313C2×C42—471no—
314C2×C42—471no—
315C2×C42—474no—
316C2×C42—474no—
317C2×C42—942no—
318C2×C42—941no—
319C2×C42C22×D4711yescharacteristic
320#107—42no—
321#107—1882no—
322#106—1882no—
323#106—1882no—
324C4⋊C8—478no—
325C4⋊C8—474no—
326C4⋊C8C4×D4714yes—
327#95—21no—
328#95—941no—
329#95—941no—
330#95—941no—
331#95—941no—
332#95—941no—
333#95—941no—
334#95—942no—
335#95—942no—
336#95—941no—
337C24—471no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—471no—
342C22×C4—471no—
343C22×C4—471no—
344C22×C4—474no—
345C22×C4—472no—
346C22×C4—472no—
347C22×C4—474no—
348C22×C4—942no—
349C22×C4—942no—
350C22×C4—941no—
351C22×C4—941no—
352C22×C4—941no—
353C22×C4—941no—
354C22×C4—941no—
355C22×C4—941no—
356C22×C4—942no—
357C22×C4—1881no—
358C22×C4—1881no—
359C22×C4D4×D4711yescharacteristic
360C22×C4D4×D4711yescharacteristic
361C22×C4C23×D4711yescharacteristic
362M16—1888no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—474no—
366C2×C8—942no—
367C2×C8—942no—
368C2×C8C2×C4×D4714yes—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—941no—
378C4⋊C4—941no—
379C4⋊C4—942no—
380C4⋊C4—941no—
381C4⋊C4—941no—
382C4⋊C4—942no—
383C4⋊C4—1882no—
384C4⋊C4—1882no—
385C22⋊C4—1882no—
386C22⋊C4—1882no—
387C22⋊C4—1884no—
388C42—42no—
389C42—474no—
390C42—474no—
391C42—472no—
392C42—472no—
393C42—1884no—
394C42—1882no—
395C42C2×C4×D4712yes—
396C42Q8×D4712yes—
397C23—472no—
398C23—941no—
399C23—941no—
400C23—941no—
401C23—941no—
402C23—941no—
403C23—941no—
404C23C2×D4×D4711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—472no—
417C2×C4—472no—
418C2×C4—472no—
419C2×C4—941no—
420C2×C4—941no—
421C2×C4—941no—
422C2×C4—941no—
423C2×C4—941no—
424C2×C4—941no—
425C2×C4—942no—
426C2×C4—942no—
427C2×C4—942no—
428C2×C4—942no—
429C2×C4—948no—
430C2×C4—1884no—
431C2×C4—1882no—
432C2×C4—1881no—
433C2×C4—1881no—
434C2×C4—1881no—
435C2×C4—1881no—
436C2×C4—1882no—
437C2×C4—1881no—
438C2×C4—1882no—
439C2×C4—1881no—
440C2×C4C4⋊C4×D4712yes—
441C2×C4C4⋊C4×D4712yes—
442C2×C4#1309404812yes—
443C2×C4#1309405612yes—
444C2×C4C22×C4×D4711yescharacteristic
445C2×C4C2×Q8×D4711yescharacteristic
446C2×C4#1309406911yescharacteristic
447C2×C4#1309407011yescharacteristic
448C2×C4#1309407111yescharacteristic
449C2×C4#1309407211yescharacteristic
450C8—24no—
451C8—944no—
452C22—942no—
453C22—942no—
454C22—942no—
455C22—1881no—
456C22—1881no—
457C22—1881no—
458C22—1881no—
459C22C2×C4⋊C4×D4711yescharacteristic
460C22#1568970211yescharacteristic
461C22#1568970311yescharacteristic
462C22C2×#1309404811yescharacteristic
463C22C2×#1309405611yescharacteristic
464C22#1568991911yescharacteristic
465C22#1568992211yescharacteristic
466C4—22no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—942no—
471C4—942no—
472C4—942no—
473C4—3762no—
474C4—3761no—
475C4—3761no—
476C4#1568916414yes—
477C4#1568959511yescharacteristic
478C4#1568959911yescharacteristic
479C4#1568975011yescharacteristic
480C4#1568975311yescharacteristic
481C2—1882no—
482C2#2024600311yescharacteristic
483C2C2×#1568916411yescharacteristic
484C2#2025197511yescharacteristic
485C2#2025474611yescharacteristic
486C2#2025475211yescharacteristic
487C2#2025475911yescharacteristic
488C2#2025476411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1947C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1947D47
hypocenterC47#21963
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #20244155C2semidirect
5. #20244153C2semidirect
6. #20244152C2semidirect
7. #20244149C2semidirect
8. #20244146C2semidirect
11. C47×#1947C2semidirect
18. #15689145C4semidirect
245. #1947D47semidirect
288. C47#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#156891452
#202457932
#202457842
#202441552
#202441532
#202441522
#202441492
#202441462
#202434222
#202434202
C47×#19472
#2196347