Finite Group Databasebeta

← #26225778

Subgroups — #26225778

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2C2×#15689175C211yesmaximal, characteristic
3#20245784C211yesmaximal, characteristic
4#20244351C212yesmaximal, complemented
5#20244351C212yesmaximal, complemented
6#20244346C211yesmaximal, characteristic, complemented
7#20244345C211yesmaximal, characteristic, complemented
8#20244343C211yesmaximal, characteristic, complemented
9#20243477C212yesmaximal
10#20243473C212yesmaximal
11C47×#1989C211yesmaximal, characteristic, complemented
12C2×#13094016C2211yescharacteristic
13C2×#13094015C2211yescharacteristic
14C2×#13094015C2211yescharacteristic
15C2×#13094014—21no—
16C2×#13094014—21no—
17#125×D47—24no—
18#125×D47C418yescomplemented
19C2×C4⋊C4×D47—21no—
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47C2211yescharacteristic
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C42×D47C2211yescharacteristic
24#15689175—24no—
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×(C2×D47)⋊C8C2212yes—
27C2×C8×Dic47C2212yes—
28#15688990—21no—
29#15688988C2212yes—
30#15688985—21no—
31#15688980C2212yes—
32C4×C4⋊Dic47C2211yescharacteristic
33C4×(C2×D47)⋊C4—21no—
34#15688969—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#125×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C4×C4⋊C4×C47C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C22×C4×D47—21no—
47C22×C4×D47—21no—
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094016—22no—
53#13094016C2×C414yes—
54#13094015—22no—
55#13094015—22no—
56#13094015C2×C414yes—
57#13094015C2×C414yes—
58#13094014—28no—
59#13094014—28no—
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47C2312yes—
63C2×C4⋊Dic47C2311yescharacteristic
64C2×C4⋊Dic47C2311yescharacteristic
65C2×Dic47⋊C4—21no—
66C2×Dic47⋊C4—21no—
67C2×Dic47⋊C4—21no—
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C4⋊C4×D47—28no—
72C4⋊C4×D47—28no—
73C4⋊C4×D47—22no—
74C4⋊C4×D47—22no—
75C4⋊C4×D47C2×C414yes—
76C4⋊C4×D47C2×C414yes—
77C2×(C2×D47)⋊C4—22no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47D411yescharacteristic
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82C42×D47C2×C414yes—
83C42×D47D414yes—
84C4⋊(C47⋊C8)—22no—
85(C2×D47)⋊C8—24no—
86C8×Dic47—24no—
87C47⋊#95—21no—
88C47⋊#95—22no—
89C47⋊#95—22no—
90C47⋊#95—22no—
91C22×C8×C47C2312yes—
92#125×C47—22no—
93#125×C47C2×C414yes—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47C2312yes—
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C42×C47—21no—
100C2×C42×C47C2311yescharacteristic
101C4⋊C8×C47—22no—
102#95×C47—21no—
103C23×D47C2×D411yescharacteristic
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2411yescharacteristic
110C22×Dic47—21no—
111C22×Dic47—21no—
112C22×Dic47—22no—
113C22×Dic47C2×D411yescharacteristic
114C22×Dic47C2×D411yescharacteristic
115C22×Dic47C2×D411yescharacteristic
116C2×C4×D47—24no—
117C2×C4×D47—22no—
118C2×C4×D47—22no—
119C2×C4×D47—24no—
120C2×C4×D47—24no—
121C2×C4×D47—41no—
122C2×C4×D47—41no—
123C2×C4×D47—42no—
124C2×C4×D47—42no—
125C2×C4×D47—44no—
126C2×C4×D47C22⋊C414yes—
127C2×C4×D47C22⋊C414yes—
128C2×C4×D47C22×C412yes—
129C2×C4×D47C2×D412yes—
130C2×C8×C47—22no—
131C2×C8×C47—24no—
132C4⋊C4×C47—24no—
133C4⋊C4×C47—24no—
134C4⋊C4×C47—21no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47C22×C412yes—
139C4⋊C4×C47C22×C412yes—
140C42×C47—42no—
141C42×C47C22×C412yes—
142C42×C47C2×D412yes—
143(C2×D47)⋊C4—44no—
144C4⋊Dic47—21no—
145C4⋊Dic47—21no—
146C4⋊Dic47—24no—
147C4⋊Dic47—24no—
148C4⋊Dic47—22no—
149C4⋊Dic47—22no—
150C4⋊Dic47C22×C412yes—
151C4⋊Dic47C22×C412yes—
152Dic47⋊C4—24no—
153Dic47⋊C4—24no—
154Dic47⋊C4—24no—
155Dic47⋊C4—24no—
156Dic47⋊C4—24no—
157Dic47⋊C4—24no—
158C4×Dic47—42no—
159C4×Dic47C22⋊C414yes—
160C4×Dic47C22⋊C414yes—
161C4×Dic47C22×C412yes—
162C4×Dic47C2×D412yes—
163C2×C47⋊C8—22no—
164C2×C47⋊C8—24no—
165C23×C47C22×D411yescharacteristic
166C22×D47—22no—
167C22×D47—21no—
168C22×D47—21no—
169C22×D47C2×C22⋊C412yes—
170C22×D47#13514yes—
171C2×C4×C47—22no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—42no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47—42no—
182C2×C4×C47C2×C22⋊C412yes—
183C2×C4×C47C2×C22⋊C412yes—
184C2×C4×C47C23×C411yescharacteristic
185C2×C4×C47C22×D411yescharacteristic
186C2×C4×C472−1+412yes—
187C2×C4×C472−1+412yes—
188C2×Dic47—22no—
189C2×Dic47—22no—
190C2×Dic47—21no—
191C2×Dic47—21no—
192C2×Dic47—22no—
193C2×Dic47—22no—
194C2×Dic47—42no—
195C2×Dic47—41no—
196C2×Dic47—41no—
197C2×Dic47—42no—
198C2×Dic47—42no—
199C2×Dic47—42no—
200C2×Dic47C2×C22⋊C412yes—
201C2×Dic47C2×C22⋊C412yes—
202C2×Dic47C2×C22⋊C412yes—
203C4×D47—28no—
204C4×D47—24no—
205C4×D47—22no—
206C4×D47—22no—
207C4×D47—44no—
208C4×D47—48no—
209C4×D47—48no—
210C8×C47—42no—
211C47⋊C8—42no—
212#21836—471nomaximal, Sylow
213C22×C47C22×C22⋊C411yescharacteristic
214C22×C47#53612yes—
215C22×C47C2×#13512yes—
216C22×C47#57511yescharacteristic
217C22×C47#57511yescharacteristic
218C2×D47—22no—
219C2×D47—22no—
220C2×D47—24no—
221C2×D47—24no—
222C2×D47#41614yes—
223C4×C47—22no—
224C4×C47—22no—
225C4×C47—42no—
226C4×C47—42no—
227C4×C47—82no—
228C4×C47#51912yes—
229C4×C47#53611yescharacteristic
230C4×C47#53611yescharacteristic
231Dic47—22no—
232Dic47—22no—
233Dic47—22no—
234Dic47—42no—
235Dic47—42no—
236Dic47—82no—
237C22×#125—471no—
238C2×#431—471no—
239#2283—471no—
240#1991—472no—
241#1991—472no—
242#1989—471no—
243#1989—471no—
244#1989—471no—
245#1989D4711yescharacteristic, complemented
246#1824—472no—
247#1823—472no—
248C2×C47#231011yescharacteristic
249C2×C47C2×#41611yescharacteristic
250C2×C47#289611yescharacteristic
251C2×C47#323212yes—
252C2×C47#323212yes—
253D47—24no—
254C2×#125—478no—
255C2×#125—471no—
256C2×#125—471no—
257C2×#125—471no—
258C2×#125—941no—
259C2×#125—941no—
260C2×#125—944no—
261C2×#125C2×D4711yescharacteristic
262C22×C4⋊C4—471no—
263C22×C4⋊C4—471no—
264C22×C4⋊C4—941no—
265C22×C4⋊C4—941no—
266C22×C42—471no—
267#431—944no—
268C2×C4⋊C8—471no—
269C2×C4⋊C8C2×D4711yescharacteristic
270C2×C22⋊C8—472no—
271C2×C4×C8—472no—
272#383—941no—
273#382—472no—
274#381—941no—
275#380—472no—
276C4×C4⋊C4—471no—
277C4×C4⋊C4C2×D4711yescharacteristic
278C4×C22⋊C4—941no—
279#376—941no—
280#340—472no—
281#340—472no—
282#340—472no—
283#340—472no—
284#340—472no—
285#340—472no—
286#340C2×D4712yes—
287#340C2×D4712yes—
288C47#2183611yescharacteristic, complemented, Sylow
289C23×C4—471no—
290C23×C4—471no—
291C23×C4—471no—
292C23×C4—941no—
293C23×C4—941no—
294C22×C8—472no—
295C22×C8C22×D4712yes—
296#125—22no—
297#125—474no—
298#125—474no—
299#125—474no—
300#125—948no—
301#125—948no—
302#125—942no—
303#125—942no—
304#125—942no—
305#125C4×D4714yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—471no—
309C2×C4⋊C4—471no—
310C2×C4⋊C4—474no—
311C2×C4⋊C4—474no—
312C2×C4⋊C4—472no—
313C2×C4⋊C4—942no—
314C2×C4⋊C4—942no—
315C2×C4⋊C4—948no—
316C2×C4⋊C4—948no—
317C2×C4⋊C4—941no—
318C2×C4⋊C4—941no—
319C2×C4⋊C4—941no—
320C2×C4⋊C4—941no—
321C2×C4⋊C4—941no—
322C2×C4⋊C4—941no—
323C2×C4⋊C4—941no—
324C2×C4⋊C4—941no—
325C2×C4⋊C4C22×D4712yes—
326C2×C4⋊C4C22×D4711yescharacteristic
327C2×C4⋊C4C22×D4711yescharacteristic
328C2×C22⋊C4—942no—
329C2×C42—21no—
330C2×C42—471no—
331C2×C42—471no—
332C2×C42—471no—
333C2×C42—474no—
334C2×C42—474no—
335C2×C42—941no—
336C2×C42C22×D4711yescharacteristic
337C4⋊C8—22no—
338C4⋊C8—942no—
339C22⋊C8—944no—
340C4×C8—944no—
341#95—21no—
342#95—942no—
343#95—942no—
344#95—942no—
345#95—941no—
346C24—471no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—471no—
351C22×C4—471no—
352C22×C4—471no—
353C22×C4—474no—
354C22×C4—472no—
355C22×C4—472no—
356C22×C4—474no—
357C22×C4—944no—
358C22×C4—944no—
359C22×C4—944no—
360C22×C4—942no—
361C22×C4—942no—
362C22×C4—941no—
363C22×C4—941no—
364C22×C4—942no—
365C22×C4—1881no—
366C22×C4—1881no—
367C22×C4—1884no—
368C22×C4—1882no—
369C22×C4—1882no—
370C22×C4D4×D4711yescharacteristic
371C22×C4D4×D4711yescharacteristic
372C22×C4C23×D4711yescharacteristic
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—942no—
376C2×C8—944no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—472no—
384C4⋊C4—472no—
385C4⋊C4—944no—
386C4⋊C4—944no—
387C4⋊C4—944no—
388C4⋊C4—944no—
389C4⋊C4—944no—
390C4⋊C4—944no—
391C4⋊C4—944no—
392C4⋊C4—944no—
393C4⋊C4—941no—
394C4⋊C4—941no—
395C4⋊C4—942no—
396C4⋊C4—942no—
397C4⋊C4C2×C4×D4712yes—
398C4⋊C4C2×C4×D4712yes—
399C22⋊C4—1884no—
400C42—42no—
401C42—474no—
402C42—474no—
403C42—472no—
404C42—472no—
405C42—1882no—
406C42C2×C4×D4712yes—
407C42D4×D4712yes—
408C23—472no—
409C23—474no—
410C23—941no—
411C23—941no—
412C23—942no—
413C23C2×D4×D4711yescharacteristic
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—472no—
426C2×C4—472no—
427C2×C4—472no—
428C2×C4—948no—
429C2×C4—942no—
430C2×C4—941no—
431C2×C4—941no—
432C2×C4—942no—
433C2×C4—944no—
434C2×C4—942no—
435C2×C4—942no—
436C2×C4—942no—
437C2×C4—942no—
438C2×C4—1884no—
439C2×C4—1888no—
440C2×C4—1888no—
441C2×C4—1882no—
442C2×C4—1882no—
443C2×C4—1882no—
444C2×C4—1882no—
445C2×C4—1881no—
446C2×C4—1881no—
447C2×C4C22⋊C4×D4712yes—
448C2×C4C22⋊C4×D4712yes—
449C2×C4C22×C4×D4711yescharacteristic
450C2×C4C2×D4×D4711yescharacteristic
451C2×C4#1309407112yes—
452C2×C4#1309407212yes—
453C8—42no—
454C8—1882no—
455C22—474no—
456C22—942no—
457C22—944no—
458C22—944no—
459C22—942no—
460C22C2×C22⋊C4×D4711yescharacteristic
461C22#1568969612yes—
462C22#135×D4712yes—
463C22#1568991911yescharacteristic
464C22#1568992211yescharacteristic
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—942no—
471C4—942no—
472C4—942no—
473C4—3762no—
474C4—1882no—
475C4—1882no—
476C4#1568959912yes—
477C4#1568969311yescharacteristic
478C4#1568969811yescharacteristic
479C2—944no—
480C2#2024597111yescharacteristic
481C2#416×D4711yescharacteristic
482C2#2025190411yescharacteristic
483C2#2025455412yes—
484C2#2025456012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1989C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1989D47
hypocenterC47#21836
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #20244351C2semidirect
5. #20244351C2semidirect
6. #20244346C2semidirect
7. #20244345C2semidirect
8. #20244343C2semidirect
11. C47×#1989C2semidirect
18. #125×D47C4semidirect
245. #1989D47semidirect
288. C47#21836semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D472
C2×#156891752
#202457842
#202443512
#202443462
#202443452
#202443432
#202434772
#202434732
C47×#19892
#2183647