Finite Group Databasebeta

← #26225792

Subgroups — #26225792

Order 12032.

1066 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic
3#20245792C211yesmaximal, characteristic
4#20244355C212yesmaximal, complemented
5#20244355C212yesmaximal, 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×#13093981—21no—
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C42×D47C2211yescharacteristic
25#15689179—24no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×(C2×D47)⋊C8C2212yes—
28C2×#13093934C2212yes—
29#15688992C2212yes—
30#15688987—21no—
31#15688982—21no—
32#15688980C2212yes—
33C4×Dic47⋊C4—21no—
34C4×C4⋊Dic47C2211yescharacteristic
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—
60#13093981—44no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2311yescharacteristic
65C2×C4⋊Dic47C2311yescharacteristic
66C2×Dic47⋊C4—21no—
67C2×Dic47⋊C4—21no—
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—28no—
75C4⋊C4×D47—28no—
76C4⋊C4×D47—22no—
77C4⋊C4×D47—22no—
78C4⋊C4×D47C2×C414yes—
79C4⋊C4×D47C2×C414yes—
80C2×(C2×D47)⋊C4—22no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47D414yes—
87C4⋊(C47⋊C8)—22no—
88(C2×D47)⋊C8—24no—
89#13093934—24no—
90C47⋊#95—21no—
91C47⋊#95—22no—
92C22×C8×C47C2312yes—
93#125×C47—22no—
94#125×C47C2×C414yes—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47C2312yes—
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C42×C47—21no—
101C2×C42×C47C2311yescharacteristic
102C4⋊C8×C47—22no—
103#95×C47—21no—
104C23×D47C2×D411yescharacteristic
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2411yescharacteristic
111C22×Dic47—21no—
112C22×Dic47—21no—
113C22×Dic47—22no—
114C22×Dic47C2×D411yescharacteristic
115C22×Dic47C2×D411yescharacteristic
116C22×Dic47C2×D411yescharacteristic
117C2×C4×D47—22no—
118C2×C4×D47—22no—
119C2×C4×D47—22no—
120C2×C4×D47—22no—
121C2×C4×D47—24no—
122C2×C4×D47—24no—
123C2×C4×D47—42no—
124C2×C4×D47—44no—
125C2×C4×D47—42no—
126C2×C4×D47—42no—
127C2×C4×D47C22⋊C414yes—
128C2×C4×D47C22⋊C414yes—
129C2×C4×D47C22×C412yes—
130C2×C4×D47C2×D412yes—
131C2×C8×C47—22no—
132C2×C8×C47—24no—
133C4⋊C4×C47—24no—
134C4⋊C4×C47—24no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47C22×C412yes—
140C4⋊C4×C47C22×C412yes—
141C42×C47—42no—
142C42×C47C22×C412yes—
143C42×C47C2×D412yes—
144(C2×D47)⋊C4—44no—
145C4⋊Dic47—21no—
146C4⋊Dic47—21no—
147C4⋊Dic47—24no—
148C4⋊Dic47—24no—
149C4⋊Dic47—22no—
150C4⋊Dic47—22no—
151C4⋊Dic47C22×C412yes—
152C4⋊Dic47C22×C412yes—
153Dic47⋊C4—24no—
154Dic47⋊C4—24no—
155Dic47⋊C4—24no—
156Dic47⋊C4—24no—
157Dic47⋊C4—24no—
158Dic47⋊C4—24no—
159Dic47⋊C4—44no—
160Dic47⋊C4—44no—
161C4×Dic47—42no—
162C4×Dic47C22⋊C414yes—
163C4×Dic47C22⋊C414yes—
164C4×Dic47C22×C412yes—
165C4×Dic47C2×D412yes—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—24no—
168C23×C47C22×D411yescharacteristic
169C22×D47—22no—
170C22×D47—22no—
171C22×D47C2×C22⋊C412yes—
172C22×D47AGL(1,ℤ/8)12yes—
173C22×D47#13712yes—
174C2×C4×C47—21no—
175C2×C4×C47—21no—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—41no—
180C2×C4×C47—41no—
181C2×C4×C47—42no—
182C2×C4×C47—42no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47C2×C22⋊C412yes—
186C2×C4×C47C2×C22⋊C412yes—
187C2×C4×C47C23×C411yescharacteristic
188C2×C4×C47C22×D411yescharacteristic
189C2×C4×C472−1+412yes—
190C2×C4×C472−1+412yes—
191C2×Dic47—22no—
192C2×Dic47—21no—
193C2×Dic47—21no—
194C2×Dic47—22no—
195C2×Dic47—22no—
196C2×Dic47—22no—
197C2×Dic47—41no—
198C2×Dic47—41no—
199C2×Dic47—42no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—42no—
203C2×Dic47C2×C22⋊C412yes—
204C2×Dic47C2×C22⋊C412yes—
205C2×Dic47C2×C22⋊C412yes—
206C4×D47—28no—
207C4×D47—24no—
208C4×D47—22no—
209C4×D47—22no—
210C4×D47—44no—
211C4×D47—48no—
212C4×D47—48no—
213C8×C47—42no—
214C47⋊C8—42no—
215#21838—471nomaximal, Sylow
216C22×C47C22×C22⋊C411yescharacteristic
217C22×C47#53612yes—
218C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
219C22×C47C2×#13711yescharacteristic
220C22×C47#57512yes—
221C2×D47—22no—
222C2×D47—22no—
223C2×D47—24no—
224C2×D47—24no—
225C2×D47#41714yes—
226C4×C47—22no—
227C4×C47—22no—
228C4×C47—42no—
229C4×C47—42no—
230C4×C47—82no—
231C4×C47#51912yes—
232C4×C47#53611yescharacteristic
233C4×C47#53611yescharacteristic
234Dic47—22no—
235Dic47—22no—
236Dic47—22no—
237Dic47—42no—
238Dic47—42no—
239Dic47—82no—
240C22×#125—471no—
241C2×#432—471no—
242#2285—471no—
243#1992—472no—
244#1992—472no—
245#1989—471no—
246#1989—471no—
247#1989—471no—
248#1989D4711yescharacteristic, complemented
249#1824—472no—
250#1823—472no—
251C2×C47#231011yescharacteristic
252C2×C47C2×#41711yescharacteristic
253C2×C47#289611yescharacteristic
254C2×C47#323412yes—
255C2×C47#323612yes—
256D47—24no—
257C2×#125—478no—
258C2×#125—471no—
259C2×#125—471no—
260C2×#125—471no—
261C2×#125—941no—
262C2×#125—941no—
263C2×#125—944no—
264C2×#125C2×D4711yescharacteristic
265C2×#117—941no—
266C22×C4⋊C4—471no—
267C22×C4⋊C4—471no—
268C22×C4⋊C4—941no—
269C22×C4⋊C4—941no—
270C22×C42—471no—
271#432—944no—
272C2×C4⋊C8—471no—
273C2×C4⋊C8C2×D4711yescharacteristic
274C2×C22⋊C8—472no—
275C2×#97—472no—
276#384—472no—
277#382—941no—
278#380—472no—
279#380—941no—
280C4×C4⋊C4—471no—
281C4×C4⋊C4—941no—
282C4×C4⋊C4C2×D4711yescharacteristic
283#340—472no—
284#340—472no—
285#340—472no—
286#340—472no—
287#340—472no—
288#340—472no—
289#340C2×D4712yes—
290#340C2×D4712yes—
291C47#2183811yescharacteristic, complemented, Sylow
292C23×C4—471no—
293C23×C4—471no—
294C23×C4—471no—
295C23×C4—941no—
296C23×C4—941no—
297C22×C8—472no—
298C22×C8C22×D4712yes—
299#125—22no—
300#125—474no—
301#125—474no—
302#125—474no—
303#125—948no—
304#125—948no—
305#125—942no—
306#125—942no—
307#125—942no—
308#125C4×D4714yes—
309#117—1884no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—471no—
313C2×C4⋊C4—471no—
314C2×C4⋊C4—474no—
315C2×C4⋊C4—474no—
316C2×C4⋊C4—472no—
317C2×C4⋊C4—942no—
318C2×C4⋊C4—942no—
319C2×C4⋊C4—948no—
320C2×C4⋊C4—948no—
321C2×C4⋊C4—941no—
322C2×C4⋊C4—941no—
323C2×C4⋊C4—941no—
324C2×C4⋊C4—941no—
325C2×C4⋊C4—941no—
326C2×C4⋊C4—941no—
327C2×C4⋊C4—941no—
328C2×C4⋊C4—941no—
329C2×C4⋊C4—942no—
330C2×C4⋊C4—942no—
331C2×C4⋊C4C22×D4712yes—
332C2×C4⋊C4C22×D4711yescharacteristic
333C2×C4⋊C4C22×D4711yescharacteristic
334C2×C22⋊C4—942no—
335C2×C42—21no—
336C2×C42—471no—
337C2×C42—471no—
338C2×C42—471no—
339C2×C42—474no—
340C2×C42—474no—
341C2×C42—941no—
342C2×C42C22×D4711yescharacteristic
343C4⋊C8—22no—
344C4⋊C8—942no—
345C22⋊C8—944no—
346#97—944no—
347#95—21no—
348#95—942no—
349#95—941no—
350C24—471no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—471no—
355C22×C4—471no—
356C22×C4—471no—
357C22×C4—474no—
358C22×C4—472no—
359C22×C4—472no—
360C22×C4—474no—
361C22×C4—942no—
362C22×C4—942no—
363C22×C4—944no—
364C22×C4—944no—
365C22×C4—942no—
366C22×C4—942no—
367C22×C4—941no—
368C22×C4—941no—
369C22×C4—942no—
370C22×C4—1882no—
371C22×C4—1882no—
372C22×C4—1882no—
373C22×C4—1884no—
374C22×C4D4×D4711yescharacteristic
375C22×C4D4×D4711yescharacteristic
376C22×C4C23×D4711yescharacteristic
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—942no—
380C2×C8—944no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—472no—
388C4⋊C4—472no—
389C4⋊C4—944no—
390C4⋊C4—944no—
391C4⋊C4—944no—
392C4⋊C4—944no—
393C4⋊C4—944no—
394C4⋊C4—944no—
395C4⋊C4—944no—
396C4⋊C4—944no—
397C4⋊C4—941no—
398C4⋊C4—941no—
399C4⋊C4—942no—
400C4⋊C4—942no—
401C4⋊C4—1884no—
402C4⋊C4—1884no—
403C4⋊C4C2×C4×D4712yes—
404C4⋊C4C2×C4×D4712yes—
405C22⋊C4—1884no—
406C42—42no—
407C42—474no—
408C42—474no—
409C42—472no—
410C42—472no—
411C42—1882no—
412C42C2×C4×D4712yes—
413C42D4×D4712yes—
414C23—472no—
415C23—472no—
416C23—472no—
417C23—942no—
418C23—942no—
419C23C2×D4×D4711yescharacteristic
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—472no—
432C2×C4—472no—
433C2×C4—472no—
434C2×C4—948no—
435C2×C4—941no—
436C2×C4—941no—
437C2×C4—942no—
438C2×C4—942no—
439C2×C4—944no—
440C2×C4—942no—
441C2×C4—942no—
442C2×C4—942no—
443C2×C4—942no—
444C2×C4—1884no—
445C2×C4—1888no—
446C2×C4—1888no—
447C2×C4—1882no—
448C2×C4—1882no—
449C2×C4—1882no—
450C2×C4—1881no—
451C2×C4—1881no—
452C2×C4—1882no—
453C2×C4C22⋊C4×D4712yes—
454C2×C4C22⋊C4×D4712yes—
455C2×C4C22×C4×D4711yescharacteristic
456C2×C4C2×D4×D4711yescharacteristic
457C2×C4#1309407112yes—
458C2×C4#1309407212yes—
459C8—42no—
460C8—1882no—
461C22—474no—
462C22—944no—
463C22—944no—
464C22—942no—
465C22—942no—
466C22C2×C22⋊C4×D4711yescharacteristic
467C22#1568969612yes—
468C22AGL(1,ℤ/8)×D4711yescharacteristic
469C22#137×D4711yescharacteristic
470C22#1568992012yes—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—942no—
477C4—942no—
478C4—942no—
479C4—3762no—
480C4—1882no—
481C4—1882no—
482C4#1568959912yes—
483C4#1568969311yescharacteristic
484C4#1568969811yescharacteristic
485C2—944no—
486C2#2024597111yescharacteristic
487C2#417×D4711yescharacteristic
488C2#2025190311yescharacteristic
489C2#2025457612yes—
490C2#2025459212yes—
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#21838
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. #20244355C2semidirect
5. #20244355C2semidirect
6. #20244346C2semidirect
7. #20244345C2semidirect
8. #20244343C2semidirect
11. C47×#1989C2semidirect
18. #125×D47C4semidirect
248. #1989D47semidirect
291. C47#21838semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D472
C2×#156891792
#202457922
#202443552
#202443462
#202443452
#202443432
#202434772
#202434732
C47×#19892
#2183847