Finite Group Databasebeta

← C4×#15689441 #26223407

Subgroups — C4×#15689441 #26223407

Order 12032.

1060 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689441C211yesmaximal, characteristic
2C4×#13093998C211yesmaximal, characteristic
3#20244015C212yesmaximal, complemented
4#20243966C212yesmaximal, complemented
5#20243705C211yesmaximal, characteristic
6#20243581C211yesmaximal, characteristic, complemented
7#20243478C211yesmaximal, characteristic, complemented
8C4×#13093955C212yesmaximal, complemented
9C4×#13093944C212yesmaximal, complemented
10C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C2×#13093998C2211yescharacteristic
13C2×C4×D188C2211yescharacteristic
14C2×C4×C22⋊D47—21no—
15#15689441C418yesdirect product factor
16C2×#13093955C2212yes—
17C2×#13093955C2212yes—
18C2×#13093944C2212yes—
19C2×#13093944C2212yes—
20C2×C22⋊(C47⋊C8)C2211yescharacteristic
21C2×C22⋊(C47⋊C8)C2211yescharacteristic
22C2×C4×C47⋊C8C2212yes—
23#15689025C2211yescharacteristic
24#15689021C2211yescharacteristic
25#15689014—21no—
26#15689002—21no—
27#15688983C2211yescharacteristic
28C4×C4⋊Dic47C2211yescharacteristic
29C4×(C2×D47)⋊C4—21no—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688878C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C42×C47—21no—
38C22×C42×C47C2211yescharacteristic
39C4×C4⋊C4×C47—21no—
40C4×C4⋊C4×C47—21no—
41C4×C4⋊C4×C47C2212yes—
42C4×C4⋊C4×C47C2212yes—
43C2×#95×C47—21no—
44C22×D188C2311yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C22×C47⋊C8C2312yes—
48C22×C47⋊C8C2312yes—
49#13093998—22no—
50#13093998C2×C414yes—
51C4×D188—24no—
52C4×C22⋊D47—44no—
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2311yescharacteristic
56C2×Dic47⋊C4—21no—
57C2×C22⋊Dic47—21no—
58C2×(C2×D47)⋊C4—21no—
59C2×(C2×D47)⋊C4—21no—
60C2×(C2×D47)⋊C4—21no—
61C2×(C2×D47)⋊C4—21no—
62C2×C4×Dic47—21no—
63#13093955—24no—
64#13093955C2×C418yescomplemented
65#13093944—24no—
66#13093944C2×C418yescomplemented
67C22⋊(C47⋊C8)—22no—
68C22⋊(C47⋊C8)C2×C414yescomplemented
69C4×C47⋊C8—24no—
70C47⋊#95—21no—
71C47⋊#95—21no—
72C47⋊#95—21no—
73C23×C4×C47—21no—
74C23×C4×C47—21no—
75C23×C4×C47D411yescharacteristic
76C23×C4×C47D411yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—24no—
82C2×C4⋊C4×C47C2×C414yescomplemented
83C2×C4⋊C4×C47C2312yes—
84C2×C4⋊C4×C47C2312yes—
85C2×C42×C47—24no—
86C2×C42×C47—22no—
87C2×C42×C47—22no—
88C2×C42×C47—44no—
89C2×C42×C47D411yescharacteristic
90C2×C42×C47D411yescharacteristic
91C2×C42×C47C2311yescharacteristic
92C2×C42×C47C2311yescharacteristic
93#95×C47—22no—
94#95×C47—22no—
95C24×C47C2×D411yescharacteristic
96C23×D47—21no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—22no—
101C22×C4×C47—24no—
102C22×C4×C47—24no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—41no—
108C22×C4×C47—42no—
109C22×C4×C47—42no—
110C22×C4×C47—42no—
111C22×C4×C47—42no—
112C22×C4×C47SD1614yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C2×D411yescharacteristic
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47#4112yes—
120C22×C4×C47C2411yescharacteristic
121C2×C22⋊D47—24no—
122C2×C22⋊D47—42no—
123C2×C22⋊D47—42no—
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C2×D188—22no—
128C2×D188—22no—
129C2×D188—21no—
130C2×D188C22×C412yes—
131C2×C4×D47—42no—
132C2×C4×D47—42no—
133C2×C4×D47—42no—
134C4⋊C4×C47—24no—
135C4⋊C4×C47—24no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—22no—
138C4⋊C4×C47C22×C414yes—
139C4⋊C4×C47#4114yes—
140C42×C47—22no—
141C42×C47—22no—
142C42×C47—22no—
143C42×C47—44no—
144C42×C47—44no—
145C42×C47#4114yes—
146C22⋊Dic47—42no—
147(C2×D47)⋊C4—24no—
148(C2×D47)⋊C4—42no—
149(C2×D47)⋊C4—42no—
150(C2×D47)⋊C4—42no—
151C4⋊Dic47—21no—
152C4⋊Dic47—22no—
153C4⋊Dic47—22no—
154C4⋊Dic47C22×C412yes—
155Dic47⋊C4—42no—
156C4×Dic47—42no—
157C2×C47⋊C8—22no—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—22no—
160C2×C47⋊C8—22no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8C22×C414yes—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47C4×D412yes—
168C23×C47C2×SD1612yes—
169C23×C47#13512yes—
170C23×C47C22×D411yescharacteristic
171C22×D47—22no—
172C22×D47—41no—
173C22×D47—41no—
174C22×D47—41no—
175C22×D47—41no—
176C22×D47—41no—
177C22×D47—41no—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—24no—
185C2×C4×C47—24no—
186C2×C4×C47—28no—
187C2×C4×C47—24no—
188C2×C4×C47—21no—
189C2×C4×C47—24no—
190C2×C4×C47—21no—
191C2×C4×C47—21no—
192C2×C4×C47—44no—
193C2×C4×C47—48no—
194C2×C4×C47—48no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—42no—
199C2×C4×C47C4×D412yes—
200C2×C4×C47#12312yes—
201C2×C4×C47#12312yes—
202C2×C4×C47C2×SD1612yes—
203C2×C4×C47C2×SD1612yes—
204C2×C4×C47AGL(1,ℤ/8)12yes—
205C2×C4×C47C23×C411yescharacteristic
206C2×C4×C47C2×#4112yes—
207C2×C4×C47C2×#4112yes—
208C2×C4×C47C2×#4111yescharacteristic
209C22⋊D47—48no—
210C2×Dic47—22no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—41no—
215C2×Dic47—41no—
216C2×Dic47—41no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219D188—24no—
220D188—42no—
221C4×D47—84no—
222C47⋊C8—24no—
223C47⋊C8—42no—
224C4×#481—471nomaximal, Sylow
225C22×C47—22no—
226C22×C47—22no—
227C22×C47—22no—
228C22×C47—22no—
229C22×C47—22no—
230C22×C47—22no—
231C22×C47C4×SD1614yes—
232C22×C47C2×C4×D411yescharacteristic
233C22×C47C2×#12311yescharacteristic
234C22×C47#52511yescharacteristic
235C22×C47C22×SD1611yescharacteristic
236C22×C47C2×#13511yescharacteristic
237C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C47#57511yescharacteristic
239C2×D47—42no—
240C2×D47—42no—
241C2×D47—42no—
242C2×D47—81no—
243C2×D47—81no—
244C2×D47—81no—
245C2×D47—81no—
246C4×C47—22no—
247C4×C47—22no—
248C4×C47—22no—
249C4×C47—41no—
250C4×C47—44no—
251C4×C47—44no—
252C4×C47#48114yescomplemented
253C4×C47C4×#4112yes—
254C4×C47#52912yes—
255Dic47—42no—
256Dic47—81no—
257Dic47—81no—
258C2×#481—471no—
259C4×#121—471no—
260C2×C4×C4⋊C4D4711yescharacteristic, complemented
261#1926—472no—
262#1919—472no—
263#1874—471no—
264#1853—471no—
265#1824—471no—
266C4×#106—472no—
267C4×#102—472no—
268C4×C22⋊C8—471no—
269C2×C47—24no—
270C2×C47C4×#12311yescharacteristic
271C2×C47C2×C4×SD1611yescharacteristic
272C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
273C2×C47C2×#48111yescharacteristic
274C2×C47#309111yescharacteristic
275C2×C47#310311yescharacteristic
276C2×C47#310511yescharacteristic
277D47—82no—
278C2×#121—471no—
279C2×C4×D4—471no—
280C2×C4×D4—941no—
281C22×C4⋊C4C2×D4711yescharacteristic
282C22×C4⋊C4C2×D4711yescharacteristic
283C22×C42—21no—
284C22×C42C2×D4711yescharacteristic
285#481—478no—
286C2×#106—472no—
287C2×#106—472no—
288C2×#102—472no—
289C2×#102—472no—
290C2×C22⋊C8—471no—
291C2×C22⋊C8—471no—
292C2×C4×C8—472no—
293#390—471no—
294#389—471no—
295#388—941no—
296#386—941no—
297#380—471no—
298C4×C4⋊C4—21no—
299C4×C4⋊C4—21no—
300C4×C4⋊C4—471no—
301C4×C4⋊C4C2×D4712yes—
302C4×C4⋊C4C2×D4712yes—
303C4×C22⋊C4—941no—
304C2×#95—21no—
305#340—472no—
306#340—472no—
307#340—472no—
308#340—472no—
309#336—472no—
310C47C4×#48111yescharacteristic, complemented, Sylow
311C22×D4—471no—
312C22×D4—941no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—941no—
316C23×C4C22⋊D4711yescharacteristic
317C23×C4C22⋊D4711yescharacteristic
318C23×C4C22×D4711yescharacteristic
319C22×C8—472no—
320C22×C8—472no—
321#121—474no—
322#121—942no—
323C4×D4—944no—
324C4×D4—1884no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—471no—
330C2×C4⋊C4—471no—
331C2×C4⋊C4—471no—
332C2×C4⋊C4—941no—
333C2×C4⋊C4C4×D4714yescomplemented
334C2×C4⋊C4C22×D4712yes—
335C2×C4⋊C4C22×D4712yes—
336C2×C22⋊C4—941no—
337C2×C22⋊C4—941no—
338C2×C22⋊C4—941no—
339C2×C22⋊C4—941no—
340C2×C22⋊C4—941no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—941no—
346C2×C42C22⋊D4711yescharacteristic
347C2×C42C22⋊D4711yescharacteristic
348C2×C42C22×D4711yescharacteristic
349C2×C42C22×D4711yescharacteristic
350#106—478no—
351#106—944no—
352#102—478no—
353#102—944no—
354C22⋊C8—474no—
355C22⋊C8—942no—
356C4×C8—944no—
357#95—22no—
358#95—22no—
359#95—941no—
360#95—941no—
361#95—941no—
362C24—941no—
363C24C2×C22⋊D4711yescharacteristic
364C2×D4—472no—
365C2×D4—944no—
366C2×D4—942no—
367C2×D4—942no—
368C2×D4—941no—
369C2×D4—1882no—
370C2×D4—1882no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—941no—
387C22×C4—941no—
388C22×C4—941no—
389C22×C4—1882no—
390C22×C4—1882no—
391C22×C4—1882no—
392C22×C4#1062533614yes—
393C22×C4C2×C4×D4712yes—
394C22×C4#1062535012yes—
395C22×C4C2×C22⋊D4711yescharacteristic
396C22×C4C2×C22⋊D4711yescharacteristic
397C22×C4C2×C22⋊D4711yescharacteristic
398C22×C4C2×C22⋊D4711yescharacteristic
399C22×C4C2×C22⋊D4711yescharacteristic
400C22×C4C23×D4711yescharacteristic
401C2×C8—474no—
402C2×C8—942no—
403C2×C8—942no—
404C2×C8—942no—
405C2×C8—942no—
406C2×C8—942no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—472no—
412C4⋊C4—941no—
413C4⋊C4—942no—
414C4⋊C4—942no—
415C4⋊C4—1882no—
416C4⋊C4C2×C4×D4714yes—
417C4⋊C4#1062535014yes—
418C22⋊C4—944no—
419C22⋊C4—1882no—
420C22⋊C4—1882no—
421C22⋊C4—1882no—
422C22⋊C4—1882no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—44no—
428C42—1882no—
429C42#1062535014yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—942no—
435C23—1881no—
436C23—1881no—
437C23—1881no—
438C23—1881no—
439C23—1881no—
440C23—1881no—
441C23C4×C22⋊D4712yes—
442C23C2×#1062533612yes—
443C23#1309404412yes—
444C23C22×C22⋊D4711yescharacteristic
445D4—944no—
446D4—1888no—
447D4—1882no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—942no—
470C2×C4—3764no—
471C2×C4—1881no—
472C2×C4—1881no—
473C2×C4—1881no—
474C2×C4—1881no—
475C2×C4—1881no—
476C2×C4—1881no—
477C2×C4—1881no—
478C2×C4—1881no—
479C2×C4C4×C22⋊D4712yes—
480C2×C4#1309400712yes—
481C2×C4#1309400712yes—
482C2×C4C2×#1062533612yes—
483C2×C4C2×#1062533612yes—
484C2×C4#1309405012yes—
485C2×C4C22×C4×D4711yescharacteristic
486C2×C4C2×#1062535012yes—
487C2×C4C2×#1062535012yes—
488C2×C4C2×#1062535011yescharacteristic
489C8—944no—
490C8—1882no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—3761no—
498C22—3761no—
499C22—3761no—
500C22—3761no—
501C22—1882no—
502C22—1882no—
503C22—1882no—
504C22C4×#1062533614yes—
505C22C2×C4×C22⋊D4711yescharacteristic
506C22C2×#1309400711yescharacteristic
507C22#1568963411yescharacteristic
508C22C22×#1062533611yescharacteristic
509C22C2×#1309404411yescharacteristic
510C22C2×#1309405011yescharacteristic
511C22#1568992111yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—44no—
516C4—41no—
517C4—44no—
518C4—3761no—
519C4—3761no—
520C4—1882no—
521C4#1568944114yesdirect product factor
522C4C4×#1062535012yes—
523C4#1568965712yes—
524C2—24no—
525C2—3762no—
526C2C4×#1309400711yescharacteristic
527C2C2×C4×#1062533611yescharacteristic
528C2C4×#1309405011yescharacteristic
529C2C2×#1568944111yescharacteristic
530C2#2025330111yescharacteristic
531C2#2025341911yescharacteristic
532C2#2025343511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenterC2×C4×C4⋊C4D47
hypocenterC47C4×#481
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20244015C2semidirect
4. #20243966C2semidirect
6. #20243581C2semidirect
7. #20243478C2semidirect
8. C4×#13093955C2semidirect
9. C4×#13093944C2semidirect
10. C4×C22⋊(C47⋊C8)C2semidirect
11. C2×C4×C4⋊C4×C47C2semidirect
15. #15689441C4direct (N × H)
64. #13093955C2×C4semidirect
66. #13093944C2×C4semidirect
68. C22⋊(C47⋊C8)C2×C4semidirect
82. C2×C4⋊C4×C47C2×C4semidirect
252. C4×C47#481semidirect
260. C2×C4×C4⋊C4D47semidirect
310. C47C4×#481semidirect
333. C2×C4⋊C4C4×D47semidirect
521. C4#15689441direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894412
C4×#130939982
#202440152
#202439662
#202437052
#202435812
#202434782
C4×#130939552
C4×#130939442
C4×C22⋊(C47⋊C8)2
C2×C4×C4⋊C4×C472
C4×#48147