Finite Group Databasebeta

← C4×#65343 #12105529

Subgroups — C4×#65343 #12105529

Order 1280.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#124C211yesmaximal, characteristic
2C2×#65343C211yesmaximal, characteristic
3C2×C4×#689C211yesmaximal, characteristic, complemented
4C2×C4×#687C211yesmaximal, characteristic
5C42×Dic10C211yesmaximal, characteristic, complemented
6#10602514C211yesmaximal, characteristic
7C4×#3967C211yesmaximal, characteristic, complemented
8C4×#3964C211yesmaximal, characteristic
9#10600483C211yesmaximal, characteristic, complemented
10#10600459C211yesmaximal, characteristic, complemented
11#10600453C211yesmaximal, characteristic
12#10600429C211yesmaximal, characteristic
13#10600352C211yesmaximal, characteristic, complemented
14#10600333C211yesmaximal, characteristic, complemented
15C4×C4⋊#168C211yesmaximal, characteristic, complemented
16C2×C5×#124C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×D4C2211yescharacteristic
19C22×#689C2211yescharacteristic
20C22×#687C2211yescharacteristic
21C2×C4×Dic10C2211yescharacteristic
22C2×C4×Dic10C2211yescharacteristic
23C2×C4×Dic10C2211yescharacteristic
24C5×#391C2211yescharacteristic
25C5×#390C2211yescharacteristic
26C5×#388—21no—
27C5×#388—21no—
28C5×#382C2211yescharacteristic
29C4×C5×C22⋊C4—21no—
30C4×C5×C22⋊C4—21no—
31C43×C5C2211yescharacteristic
32C2×#3967C2211yescharacteristic
33C2×#3967C2211yescharacteristic
34C2×#3967C2211yescharacteristic
35C2×#3964C2211yescharacteristic
36C2×#3964C2211yescharacteristic
37C2×#3964C2211yescharacteristic
38#65343C418yesdirect product factor
39C4×#689—24no—
40C4×#687—24no—
41C2×#3942C2211yescharacteristic
42C2×#3942C2211yescharacteristic
43C2×#3940C2211yescharacteristic
44C2×#3939C2211yescharacteristic
45C4×C4⋊Dic5C2211yescharacteristic
46C4×C4⋊Dic5C2211yescharacteristic
47C4×C4⋊Dic5C2211yescharacteristic
48C4×#683—21no—
49C4×#683—21no—
50C4×#683—21no—
51C42×Dic5—21no—
52C2×C4⋊#168C2211yescharacteristic
53C2×C4⋊#168C2211yescharacteristic
54C2×C4×#168C2211yescharacteristic
55C2×C4×#168C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#64744C2211yescharacteristic
61#64744C2211yescharacteristic
62C4×#464—51nomaximal, Sylow
63C22×C5×Q8C2311yescharacteristic
64C22×C5×D4C2311yescharacteristic
65C23×C4×C5—21no—
66C22×Dic10C2311yescharacteristic
67C5×#124—22no—
68C5×#124C2×C414yes—
69C4×C5×Q8—22no—
70C4×C5×Q8—22no—
71C4×C5×D4—22no—
72C4×C5×D4—22no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C5×C22⋊C4—21no—
78C2×C5×C22⋊C4—21no—
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4—21no—
81C2×C42×C5—21no—
82C2×C42×C5D411yescharacteristic
83C2×C42×C5D411yescharacteristic
84C2×C42×C5D411yescharacteristic
85C2×C42×C5D411yescharacteristic
86C2×C42×C5C2311yescharacteristic
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2311yescharacteristic
89C2×#689—22no—
90C2×#689—22no—
91C2×#689C2×C414yescomplemented
92C2×#687—22no—
93C2×#687—22no—
94C2×#687C2×C414yes—
95C2×C4⋊Dic5C2311yescharacteristic
96C2×C4⋊Dic5C2311yescharacteristic
97C2×C4⋊Dic5C2311yescharacteristic
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5—21no—
104C2×C4×Dic5—21no—
105C2×C4×Dic5—21no—
106C2×C4×Dic5—21no—
107C22×#168C2311yescharacteristic
108C22×#168C2311yescharacteristic
109C22×#168C2311yescharacteristic
110C22×#168C2311yescharacteristic
111C4×Dic10—22no—
112C4×Dic10—22no—
113C4×Dic10—22no—
114C4×Dic10—22no—
115C4×Dic10C2×C414yescomplemented
116C4×Dic10D414yes—
117C5×#95—21no—
118C5×#95—21no—
119C5×#95—21no—
120C5×#95—21no—
121#3967—22no—
122#3967—22no—
123#3967C2×C414yescomplemented
124#3964—22no—
125#3964—22no—
126#3964C2×C414yes—
127#3963—21no—
128#3963—21no—
129#3963—21no—
130#3942—22no—
131#3942—22no—
132#3940—22no—
133#3939—22no—
134C4⋊#168—22no—
135C4⋊#168C2×C414yescomplemented
136C4×#168—22no—
137C4×#168—22no—
138C2×#464—51no—
139C2×C4×Q16—51no—
140C2×C4×SD16—51no—
141C4×#124D511yescharacteristic
142C42×Q8—51no—
143#1983—51no—
144#1949—51no—
145#1927—51no—
146#1925—51no—
147#1880—51no—
148#1878—51no—
149#1805—51no—
150C4×C4⋊C8—51no—
151C4×#103—51no—
152C4×#102—51no—
153C24×C5—21no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8C22×C412yes—
160C2×C5×D4—21no—
161C2×C5×D4—21no—
162C2×C5×D4—21no—
163C2×C5×D4—21no—
164C2×C5×D4—21no—
165C2×C5×D4C22×C412yes—
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—42no—
170C22×C4×C5—42no—
171C22×C4×C5—42no—
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2×D411yescharacteristic
178C22×C4×C5C2411yescharacteristic
179C22×Dic5—21no—
180C22×Dic5—21no—
181C22×Dic5—21no—
182C22×Dic5—21no—
183C2×Dic10—21no—
184C2×Dic10—21no—
185C2×Dic10—21no—
186C2×Dic10C22×C412yes—
187C2×Dic10C2×D412yes—
188C2×Dic10#4112yes—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—42no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—42no—
197C5×C22⋊C4—42no—
198C5×C22⋊C4—42no—
199C42×C5—22no—
200C42×C5—22no—
201C42×C5—22no—
202C42×C5—21no—
203C42×C5—21no—
204C42×C5—41no—
205C42×C5—42no—
206C42×C5C22×C412yes—
207C42×C5C2×D412yes—
208C42×C5#4112yes—
209C42×C5#4112yes—
210#689—28no—
211#687—28no—
212C4⋊Dic5—21no—
213C4⋊Dic5—21no—
214C4⋊Dic5—21no—
215C4⋊Dic5C22×C412yes—
216C4⋊Dic5C2×D412yes—
217C4⋊Dic5#4112yes—
218#683—24no—
219#683—24no—
220#683—24no—
221C4×Dic5—24no—
222C4×Dic5—24no—
223C4×Dic5—24no—
224C4×Dic5—42no—
225C4×Dic5—42no—
226C4×Dic5—42no—
227C2×#168—21no—
228C2×#168—21no—
229C2×#168—21no—
230C2×#168—21no—
231C2×#168—21no—
232C2×#168—21no—
233C2×#168—21no—
234C2×#168—21no—
235C2×#168—21no—
236C2×#168—21no—
237C2×#168C22×C412yes—
238C2×#168C22×C412yes—
239C22×Q16—51no—
240C22×SD16—51no—
241C2×#124C2×D511yescharacteristic
242C2×C4×Q8—51no—
243C2×C4×Q8—51no—
244C2×C4×Q8—51no—
245C2×C4×Q8C2×D511yescharacteristic
246C2×C4×D4C2×D511yescharacteristic
247#464—58no—
248C4×Q16—104no—
249C4×SD16—104no—
250C2×#107—51no—
251C2×#106—51no—
252C2×C4⋊C8—51no—
253C2×C4⋊C8—51no—
254C2×#103—51no—
255C2×#103—51no—
256C2×#103—51no—
257C2×#103—51no—
258C2×#103—51no—
259C2×#102—51no—
260C2×#102—51no—
261C2×#102—51no—
262C2×C4×C8—51no—
263C2×C4×C8—51no—
264#391C2×D511yescharacteristic
265#390C2×D511yescharacteristic
266#388—21no—
267#388—21no—
268#382C2×D511yescharacteristic
269C4×C4⋊C4—51no—
270C4×C4⋊C4—51no—
271C4×C4⋊C4—51no—
272C4×C4⋊C4—101no—
273C4×C4⋊C4—101no—
274C4×C4⋊C4—101no—
275C4×C22⋊C4—21no—
276C4×C22⋊C4—21no—
277C43—101no—
278C43C2×D511yescharacteristic
279#340—51no—
280#340—51no—
281#340—51no—
282#340—51no—
283#336—51no—
284#336—51no—
285C23×C5—22no—
286C23×C5—41no—
287C23×C5—41no—
288C23×C5—41no—
289C23×C5—41no—
290C23×C5—41no—
291C23×C5—41no—
292C23×C5C22×D411yescharacteristic
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—41no—
296C5×Q8—41no—
297C5×D4—22no—
298C5×D4—22no—
299C5×D4—41no—
300C5×D4—41no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—22no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5—84no—
322C2×C4×C5C4×D412yes—
323C2×C4×C5C4×D412yes—
324C2×C4×C5#12112yes—
325C2×C4×C5#12112yes—
326C2×C4×C5#13512yes—
327C2×C4×C5#13512yes—
328C2×C4×C5#13712yes—
329C2×C4×C5C23×C411yescharacteristic
330C2×C4×C5C22×D411yescharacteristic
331C2×C4×C5C2×#4111yescharacteristic
332C2×C4×C5C2×#4111yescharacteristic
333C2×C4×C5C2×#4111yescharacteristic
334C2×C4×C5C2×#4111yescharacteristic
335C2×Dic5—22no—
336C2×Dic5—22no—
337C2×Dic5—22no—
338C2×Dic5—22no—
339C2×Dic5—22no—
340C2×Dic5—22no—
341C2×Dic5—41no—
342C2×Dic5—41no—
343C2×Dic5—41no—
344C2×Dic5—41no—
345C2×Dic5—41no—
346C2×Dic5—41no—
347Dic10—22no—
348Dic10—22no—
349Dic10—22no—
350Dic10C4×D414yes—
351#168—22no—
352#168—22no—
353#168—41no—
354#168—41no—
355C22×Q8—51no—
356C22×Q8C22×D511yescharacteristic
357C22×D4C22×D511yescharacteristic
358C23×C4—21no—
359C2×Q16—54no—
360C2×Q16—102no—
361C2×Q16—102no—
362C2×SD16—54no—
363C2×SD16—102no—
364C2×SD16—102no—
365C22×C8—51no—
366C22×C8—51no—
367C22×C8—51no—
368C22×C8—51no—
369#124—22no—
370#124C4×D514yes—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—54no—
374C4×Q8—54no—
375C4×Q8—102no—
376C4×Q8—102no—
377C4×Q8—102no—
378C4×Q8—102no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—51no—
383C2×C4⋊C4—51no—
384C2×C4⋊C4—51no—
385C2×C4⋊C4—101no—
386C2×C4⋊C4—101no—
387C2×C4⋊C4—101no—
388C2×C4⋊C4C22×D511yescharacteristic
389C2×C4⋊C4C22×D511yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—101no—
397C2×C42—101no—
398C2×C42—101no—
399C2×C42—101no—
400C2×C42—101no—
401C2×C42—101no—
402C2×C42C22⋊D511yescharacteristic
403C2×C42C22⋊D511yescharacteristic
404C2×C42C22⋊D511yescharacteristic
405C2×C42C22⋊D511yescharacteristic
406C2×C42C22×D511yescharacteristic
407C2×C42C22×D511yescharacteristic
408C2×C42C22×D511yescharacteristic
409#107—102no—
410#106—102no—
411C4⋊C8—54no—
412C4⋊C8—102no—
413#103—54no—
414#103—102no—
415#103—102no—
416#103—102no—
417#103—102no—
418#102—54no—
419#102—102no—
420#102—102no—
421C4×C8—102no—
422C4×C8—102no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—101no—
428#95—101no—
429#95—101no—
430C22×C5—42no—
431C22×C5—42no—
432C22×C5—42no—
433C22×C5—81no—
434C22×C5—81no—
435C22×C5—81no—
436C22×C5—81no—
437C22×C5C2×C4×D411yescharacteristic
438C22×C5C2×#12111yescharacteristic
439C22×C5#52511yescharacteristic
440C22×C5C2×#13511yescharacteristic
441C22×C5C2×#13511yescharacteristic
442C22×C5C2×#13711yescharacteristic
443C22×C5#57511yescharacteristic
444C4×C5—22no—
445C4×C5—22no—
446C4×C5—22no—
447C4×C5—22no—
448C4×C5—41no—
449C4×C5—42no—
450C4×C5—81no—
451C4×C5—81no—
452C4×C5#46414yescomplemented
453C4×C5C2×C4×D411yescharacteristic
454C4×C5C4×#4111yescharacteristic
455C4×C5#52911yescharacteristic
456C4×C5#53211yescharacteristic
457Dic5—24no—
458Dic5—42no—
459Dic5—42no—
460Dic5—42no—
461C24—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—52no—
468C2×Q8—52no—
469C2×Q8—52no—
470C2×Q8—101no—
471C2×Q8—101no—
472C2×Q8—101no—
473C2×Q8C2×C4×D512yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4C2×C4×D512yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—101no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—101no—
490C22×C4C2×C22⋊D511yescharacteristic
491C22×C4C2×C22⋊D511yescharacteristic
492C22×C4C2×C22⋊D511yescharacteristic
493C22×C4C2×C22⋊D511yescharacteristic
494C22×C4C2×C22⋊D511yescharacteristic
495C22×C4C2×C22⋊D511yescharacteristic
496C22×C4C23×D511yescharacteristic
497Q16—108no—
498SD16—108no—
499C2×C8—52no—
500C2×C8—52no—
501C2×C8—101no—
502C2×C8—101no—
503C2×C8—101no—
504C2×C8—101no—
505C2×C8—101no—
506C2×C8—101no—
507C2×C8—101no—
508C2×C8—101no—
509C2×C8—101no—
510C2×C8—101no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—52no—
517C4⋊C4—52no—
518C4⋊C4—52no—
519C4⋊C4—104no—
520C4⋊C4—101no—
521C4⋊C4—101no—
522C4⋊C4—101no—
523C4⋊C4—104no—
524C4⋊C4—104no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—41no—
536C42—42no—
537C42—104no—
538C42—104no—
539C42—104no—
540C42—202no—
541C42—202no—
542C42—202no—
543C42C2×C4×D512yes—
544C42#70912yes—
545C42D4×D512yes—
546C42#71112yes—
547C2×C5—82no—
548C2×C5C4×#12111yescharacteristic
549C2×C5C4×#13511yescharacteristic
550C2×C5C4×#13711yescharacteristic
551C2×C5C2×#46411yescharacteristic
552C2×C5#303511yescharacteristic
553C2×C5#310211yescharacteristic
554C2×C5#310811yescharacteristic
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C22×C22⋊D511yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—54no—
568Q8—102no—
569Q8—102no—
570Q8—102no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—102no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—102no—
600C2×C4—102no—
601C2×C4—102no—
602C2×C4—201no—
603C2×C4—201no—
604C2×C4—201no—
605C2×C4—201no—
606C2×C4—201no—
607C2×C4—201no—
608C2×C4C4×C22⋊D512yes—
609C2×C4C4×C22⋊D512yes—
610C2×C4#408412yes—
611C2×C4#408412yes—
612C2×C4#409612yes—
613C2×C4#409612yes—
614C2×C4#409712yes—
615C2×C4C22×C4×D511yescharacteristic
616C2×C4C2×#70911yescharacteristic
617C2×C4C2×D4×D511yescharacteristic
618C2×C4C2×#71111yescharacteristic
619C2×C4D5×#4111yescharacteristic
620C2×C4D5×#4111yescharacteristic
621C8—102no—
622C8—102no—
623C8—201no—
624C8—201no—
625C5C4×#46411yescharacteristic, complemented, Sylow
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×C4×C22⋊D511yescharacteristic
634C22C2×#408411yescharacteristic
635C22C2×#409611yescharacteristic
636C22C2×#409611yescharacteristic
637C22#6615611yescharacteristic
638C22C2×#409711yescharacteristic
639C22#6616211yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—104no—
649C4—202no—
650C4—202no—
651C4—202no—
652C4#6534314yesdirect product factor
653C4C4×#71111yescharacteristic
654C4C4×D4×D511yescharacteristic
655C4#6588211yescharacteristic
656C4#6589111yescharacteristic
657C2—82no—
658C2C4×#408411yescharacteristic
659C2C4×#409611yescharacteristic
660C2C4×#409711yescharacteristic
661C2C2×#6534311yescharacteristic
662C2#1061487111yescharacteristic
663C2#1061488311yescharacteristic
664C2#1061502711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×#124C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC4×#124D5
hypocenterC5C4×#464
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
3. C2×C4×#689C2semidirect
5. C42×Dic10C2semidirect
7. C4×#3967C2semidirect
9. #10600483C2semidirect
10. #10600459C2semidirect
13. #10600352C2semidirect
14. #10600333C2semidirect
15. C4×C4⋊#168C2semidirect
38. #65343C4direct (N × H)
91. C2×#689C2×C4semidirect
115. C4×Dic10C2×C4semidirect
123. #3967C2×C4semidirect
135. C4⋊#168C2×C4semidirect
452. C4×C5#464semidirect
625. C5C4×#464semidirect
652. C4#65343direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C5×#1242
C2×#653432
C2×C4×#6892
C2×C4×#6872
C42×Dic102
#106025142
C4×#39672
C4×#39642
#106004832
#106004592
#106004532
#106004292
#106003522
#106003332
C4×C4⋊#1682
C4×#4645