Finite Group Databasebeta

← #26225775

Subgroups — #26225775

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2C2×#15689174C211yesmaximal, characteristic
3#20245781C211yesmaximal, characteristic
4#20244354C211yesmaximal, characteristic, complemented
5#20244353C211yesmaximal, characteristic, complemented
6#20244352C211yesmaximal, characteristic, complemented
7#20244344C212yesmaximal, complemented
8#20244342C212yesmaximal, complemented
9#20243477C212yesmaximal
10#20243473C212yesmaximal
11C47×#1991C211yesmaximal, 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#15689174—24no—
25C2×Dic47⋊C8C2212yes—
26C2×(C2×D47)⋊C8C2212yes—
27C2×C4×C47⋊C8C2211yescharacteristic
28#15688990—21no—
29#15688989C2211yescharacteristic
30#15688985—21no—
31#15688980C2212yes—
32C4⋊C4×Dic47C2212yes—
33#15688970—21no—
34#15688968—21no—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#125×C47C2211yescharacteristic
42C2×C4×C8×C47C2211yescharacteristic
43C47×#382C2211yescharacteristic
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—22no—
79C2×C4×Dic47D411yescharacteristic
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82C42×D47C2×C414yes—
83C42×D47D414yes—
84Dic47⋊C8—24no—
85(C2×D47)⋊C8—24no—
86C4×C47⋊C8—22no—
87C47⋊#95—21no—
88C47⋊#95—21no—
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×C47C2311yescharacteristic
100C4×C8×C47—22no—
101#95×C47—21no—
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—42no—
122C2×C4×D47—44no—
123C2×C4×D47—44no—
124C2×C4×D47C22⋊C414yes—
125C2×C4×D47C22⋊C414yes—
126C2×C4×D47C22×C412yes—
127C2×C4×D47C2×D412yes—
128C2×C8×C47—22no—
129C2×C8×C47—22no—
130C2×C8×C47—22no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47—24no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—21no—
135C4⋊C4×C47—24no—
136C4⋊C4×C47C22×C412yes—
137C4⋊C4×C47C22×C412yes—
138C42×C47C22×C412yes—
139C42×C47C2×D412yes—
140(C2×D47)⋊C4—44no—
141C4⋊Dic47—21no—
142C4⋊Dic47—21no—
143C4⋊Dic47—24no—
144C4⋊Dic47—24no—
145C4⋊Dic47—24no—
146C4⋊Dic47C22×C412yes—
147C4⋊Dic47C22×C412yes—
148Dic47⋊C4—24no—
149Dic47⋊C4—24no—
150Dic47⋊C4—24no—
151Dic47⋊C4—24no—
152Dic47⋊C4—24no—
153Dic47⋊C4—24no—
154C4×Dic47—44no—
155C4×Dic47C22⋊C414yes—
156C4×Dic47C22⋊C414yes—
157C4×Dic47C22×C412yes—
158C4×Dic47C2×D412yes—
159C2×C47⋊C8—22no—
160C2×C47⋊C8—22no—
161C2×C47⋊C8—22no—
162C23×C47C22×D411yescharacteristic
163C22×D47—21no—
164C22×D47—21no—
165C22×D47—22no—
166C22×D47C2×C22⋊C412yes—
167C22×D47#13514yes—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47C2×C22⋊C412yes—
178C2×C4×C47C2×C22⋊C412yes—
179C2×C4×C47C23×C411yescharacteristic
180C2×C4×C47C22×D411yescharacteristic
181C2×C4×C472−1+412yes—
182C2×C4×C472−1+412yes—
183C2×Dic47—21no—
184C2×Dic47—21no—
185C2×Dic47—22no—
186C2×Dic47—22no—
187C2×Dic47—22no—
188C2×Dic47—22no—
189C2×Dic47—42no—
190C2×Dic47—42no—
191C2×Dic47—42no—
192C2×Dic47—42no—
193C2×Dic47—42no—
194C2×Dic47C2×C22⋊C412yes—
195C2×Dic47C2×C22⋊C412yes—
196C2×Dic47C2×C22⋊C412yes—
197C4×D47—28no—
198C4×D47—24no—
199C4×D47—24no—
200C4×D47—44no—
201C4×D47—48no—
202C4×D47—48no—
203C8×C47—42no—
204C47⋊C8—42no—
205#21836—471nomaximal, Sylow
206C22×C47C22×C22⋊C411yescharacteristic
207C22×C47#53611yescharacteristic
208C22×C47#53611yescharacteristic
209C22×C47C2×#13512yes—
210C22×C47#57512yes—
211C2×D47—22no—
212C2×D47—22no—
213C2×D47—24no—
214C2×D47—24no—
215C2×D47#41614yes—
216C4×C47—22no—
217C4×C47—22no—
218C4×C47—42no—
219C4×C47—42no—
220C4×C47—82no—
221C4×C47#51912yes—
222C4×C47#53612yes—
223Dic47—22no—
224Dic47—22no—
225Dic47—22no—
226Dic47—42no—
227Dic47—42no—
228Dic47—82no—
229C22×#125—471no—
230C2×#431—471no—
231#2283—471no—
232#1991—471no—
233#1991—471no—
234#1991—471no—
235#1991D4711yescharacteristic, complemented
236#1989—472no—
237#1989—472no—
238#1824—472no—
239#1823—472no—
240C2×C47#231011yescharacteristic
241C2×C47C2×#41611yescharacteristic
242C2×C47#289611yescharacteristic
243C2×C47#323212yes—
244C2×C47#323212yes—
245D47—24no—
246C2×#125—478no—
247C2×#125—471no—
248C2×#125—471no—
249C2×#125—471no—
250C2×#125—941no—
251C2×#125—941no—
252C2×#125—944no—
253C2×#125C2×D4711yescharacteristic
254C22×C4⋊C4—471no—
255C22×C4⋊C4—471no—
256C22×C4⋊C4—941no—
257C22×C4⋊C4—941no—
258C22×C42—471no—
259#431—944no—
260C2×C4⋊C8—472no—
261C2×C22⋊C8—472no—
262C2×C4×C8—471no—
263C2×C4×C8C2×D4711yescharacteristic
264#383—941no—
265#382—471no—
266#382C2×D4711yescharacteristic
267#381—941no—
268#380—472no—
269C4×C4⋊C4—472no—
270C4×C22⋊C4—941no—
271#376—941no—
272#340—472no—
273#340—472no—
274#340—472no—
275#340—472no—
276#340—472no—
277#340—472no—
278#340C2×D4712yes—
279#340C2×D4712yes—
280C47#2183611yescharacteristic, complemented, Sylow
281C23×C4—471no—
282C23×C4—471no—
283C23×C4—471no—
284C23×C4—941no—
285C23×C4—941no—
286C22×C8—472no—
287C22×C8C22×D4712yes—
288#125—22no—
289#125—474no—
290#125—474no—
291#125—474no—
292#125—948no—
293#125—948no—
294#125—942no—
295#125—942no—
296#125—942no—
297#125C4×D4714yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—471no—
301C2×C4⋊C4—471no—
302C2×C4⋊C4—474no—
303C2×C4⋊C4—474no—
304C2×C4⋊C4—472no—
305C2×C4⋊C4—942no—
306C2×C4⋊C4—942no—
307C2×C4⋊C4—948no—
308C2×C4⋊C4—948no—
309C2×C4⋊C4—941no—
310C2×C4⋊C4—941no—
311C2×C4⋊C4—941no—
312C2×C4⋊C4—941no—
313C2×C4⋊C4—941no—
314C2×C4⋊C4—941no—
315C2×C4⋊C4—941no—
316C2×C4⋊C4—941no—
317C2×C4⋊C4C22×D4712yes—
318C2×C4⋊C4C22×D4711yescharacteristic
319C2×C4⋊C4C22×D4711yescharacteristic
320C2×C22⋊C4—942no—
321C2×C42—471no—
322C2×C42—471no—
323C2×C42—471no—
324C2×C42—474no—
325C2×C42—474no—
326C2×C42—942no—
327C2×C42C22×D4711yescharacteristic
328C4⋊C8—944no—
329C22⋊C8—944no—
330C4×C8—22no—
331C4×C8—942no—
332#95—21no—
333#95—21no—
334#95—942no—
335#95—942no—
336#95—941no—
337#95—941no—
338C24—471no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—471no—
343C22×C4—471no—
344C22×C4—471no—
345C22×C4—474no—
346C22×C4—472no—
347C22×C4—472no—
348C22×C4—474no—
349C22×C4—944no—
350C22×C4—944no—
351C22×C4—944no—
352C22×C4—942no—
353C22×C4—942no—
354C22×C4—941no—
355C22×C4—941no—
356C22×C4—942no—
357C22×C4—1882no—
358C22×C4—1884no—
359C22×C4—1884no—
360C22×C4D4×D4711yescharacteristic
361C22×C4D4×D4711yescharacteristic
362C22×C4C23×D4711yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—942no—
367C2×C8—942no—
368C2×C8—942no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—24no—
374C4⋊C4—472no—
375C4⋊C4—472no—
376C4⋊C4—944no—
377C4⋊C4—944no—
378C4⋊C4—944no—
379C4⋊C4—944no—
380C4⋊C4—944no—
381C4⋊C4—944no—
382C4⋊C4—944no—
383C4⋊C4—944no—
384C4⋊C4—941no—
385C4⋊C4—941no—
386C4⋊C4—944no—
387C4⋊C4C2×C4×D4712yes—
388C4⋊C4C2×C4×D4712yes—
389C22⋊C4—1884no—
390C42—474no—
391C42—474no—
392C42—472no—
393C42—472no—
394C42—1884no—
395C42C2×C4×D4712yes—
396C42D4×D4712yes—
397C23—472no—
398C23—474no—
399C23—942no—
400C23—941no—
401C23—941no—
402C23C2×D4×D4711yescharacteristic
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—472no—
413C2×C4—472no—
414C2×C4—472no—
415C2×C4—948no—
416C2×C4—942no—
417C2×C4—942no—
418C2×C4—941no—
419C2×C4—941no—
420C2×C4—944no—
421C2×C4—944no—
422C2×C4—942no—
423C2×C4—942no—
424C2×C4—1884no—
425C2×C4—1888no—
426C2×C4—1888no—
427C2×C4—1882no—
428C2×C4—1882no—
429C2×C4—1882no—
430C2×C4—1882no—
431C2×C4—1882no—
432C2×C4C22⋊C4×D4712yes—
433C2×C4C22⋊C4×D4712yes—
434C2×C4C22×C4×D4711yescharacteristic
435C2×C4C2×D4×D4711yescharacteristic
436C2×C4#1309407112yes—
437C2×C4#1309407212yes—
438C8—42no—
439C8—1882no—
440C22—474no—
441C22—944no—
442C22—944no—
443C22—942no—
444C22—942no—
445C22C2×C22⋊C4×D4711yescharacteristic
446C22#1568969311yescharacteristic
447C22#1568969811yescharacteristic
448C22#135×D4712yes—
449C22#1568992012yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—942no—
456C4—942no—
457C4—942no—
458C4—3762no—
459C4—1882no—
460C4—1882no—
461C4#1568959912yes—
462C4#1568969612yes—
463C2—944no—
464C2#2024597111yescharacteristic
465C2#416×D4711yescharacteristic
466C2#2025190311yescharacteristic
467C2#2025455112yes—
468C2#2025455712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1991C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1991D47
hypocenterC47#21836
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. #20244354C2semidirect
5. #20244353C2semidirect
6. #20244352C2semidirect
7. #20244344C2semidirect
8. #20244342C2semidirect
11. C47×#1991C2semidirect
18. #125×D47C4semidirect
235. #1991D47semidirect
280. 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×#156891742
#202457812
#202443542
#202443532
#202443522
#202443442
#202443422
#202434772
#202434732
C47×#19912
#2183647