Finite Group Databasebeta

← C2×#10607963 #12268473

Subgroups — C2×#10607963 #12268473

Order 1280.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×C4×D20C211yesmaximal, characteristic
3C2×C5×#389C211yesmaximal, characteristic, complemented
4C2×#65281C212yesmaximal, complemented
5C2×#65278C212yesmaximal, complemented
6C2×#65276C211yesmaximal, characteristic, complemented
7C2×#65276C211yesmaximal, characteristic, complemented
8C2×#65276C211yesmaximal, characteristic, complemented
9C2×#65217C211yesmaximal, characteristic, complemented
10#10607963C2116yesmaximal, direct product factor
11C2×#64954C212yesmaximal
12C2×#64949C212yesmaximal, complemented
13C23×D20C2211yescharacteristic
14C23×C4×D5C2211yescharacteristic
15C23×C4×D5C2211yescharacteristic
16C23×C4×D5C2212yes—
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2212yes—
20C22×C42×C5C2211yescharacteristic
21C22×#685C2212yes—
22C22×#685C2211yescharacteristic
23C22×#685C2211yescharacteristic
24C22×#685C2212yes—
25C22×#685C2212yes—
26C22×C4⋊Dic5C2211yescharacteristic
27C22×C4⋊Dic5C2211yescharacteristic
28C22×C4⋊Dic5C2211yescharacteristic
29C22×#683C2212yes—
30C22×#683C2212yes—
31C2×D5×C4⋊C4—28no—
32C2×D5×C4⋊C4C2218yescomplemented
33C2×C4×D20—28no—
34C2×C4×D20C2218yes—
35C5×#389C2218yescomplemented
36C2×C5×#95C2212yes—
37C2×#3963C2212yes—
38C2×#3963C2212yes—
39C2×#3963C2212yes—
40C2×#3963C2212yes—
41#65281C22116yescomplemented
42#65278C22116yescomplemented
43#65276C2218yescomplemented
44#65276C2218yescomplemented
45#65276C2218yescomplemented
46#65217C2218yescomplemented
47#64954C22116yes—
48#64949C22116yescomplemented
49C2×#2487—51nomaximal, Sylow
50C24×D5C2311yescharacteristic
51C24×D5C2311yescharacteristic
52C23×C4×C5C2312yes—
53C23×C4×C5C2312yes—
54C23×C4×C5C2311yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C23×Dic5C2312yes—
58C23×Dic5C2312yes—
59C23×Dic5C2311yescharacteristic
60C23×Dic5C2311yescharacteristic
61C22×D20—24no—
62C22×D20—24no—
63C22×D20Q818yes—
64C22×D20C2314yes—
65C22×C4×D5—24no—
66C22×C4×D5—24no—
67C22×C4×D5—28no—
68C22×C4×D5—24no—
69C22×C4×D5—24no—
70C22×C4×D5—24no—
71C22×C4×D5—44no—
72C22×C4×D5—44no—
73C22×C4×D5D418yes—
74C22×C4×D5C2314yes—
75C22×C4×D5C2314yes—
76C22×C4×D5C2318yes—
77C2×C5×C4⋊C4—24no—
78C2×C5×C4⋊C4—24no—
79C2×C5×C4⋊C4—44no—
80C2×C5×C4⋊C4C2314yes—
81C2×C5×C4⋊C4C2314yes—
82C2×C5×C4⋊C4C2318yes—
83C2×C42×C5—24no—
84C2×C42×C5C2314yes—
85C2×#685—24no—
86C2×#685—28no—
87C2×#685—24no—
88C2×#685—44no—
89C2×#685—44no—
90C2×#685C2314yes—
91C2×#685C2314yes—
92C2×#685C2318yes—
93C2×#685C2318yes—
94C2×#685C2318yes—
95C2×C4⋊Dic5—24no—
96C2×C4⋊Dic5—24no—
97C2×C4⋊Dic5—24no—
98C2×C4⋊Dic5C2314yes—
99C2×C4⋊Dic5C2314yes—
100C2×C4⋊Dic5C2314yes—
101C2×#683—28no—
102C2×#683—44no—
103C2×#683C2318yes—
104C2×#683C2318yes—
105D5×C4⋊C4—232no—
106C4×D20—232no—
107C5×#95C2318yes—
108#3963C2318yes—
109#3963C2318yes—
110#3963C2318yes—
111#3963C2318yes—
112C22×C4×D4—51no—
113C23×C4⋊C4—51no—
114#2487—516no—
115C2×#400—52no—
116C2×#399—52no—
117C2×#389—51no—
118C2×#389D511yescharacteristic, complemented
119C2×#386—52no—
120C2×#385—52no—
121C2×#380—51no—
122C2×#380—51no—
123C2×#380—51no—
124C24×C5C2411yescharacteristic
125C23×D5—22no—
126C23×D5—22no—
127C23×D5—22no—
128C23×D5—22no—
129C23×D5—22no—
130C23×D5—22no—
131C23×D5—22no—
132C23×D5—22no—
133C23×D5—22no—
134C23×D5—22no—
135C23×D5C2×D414yes—
136C23×D5C2×Q814yes—
137C23×D5C2×Q814yes—
138C23×D5#4114yes—
139C23×D5C2412yes—
140C23×D5C2412yes—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—24no—
145C22×C4×C5—24no—
146C22×C4×C5—24no—
147C22×C4×C5—22no—
148C22×C4×C5—24no—
149C22×C4×C5—24no—
150C22×C4×C5—24no—
151C22×C4×C5—22no—
152C22×C4×C5C2×D414yes—
153C22×C4×C5C2×Q814yes—
154C22×C4×C5#4114yes—
155C22×C4×C5#4114yes—
156C22×C4×C5C2412yes—
157C22×C4×C5C2414yes—
158C22×C4×C5C2414yes—
159C22×C4×C5C2412yes—
160C22×C4×C5C2412yes—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—24no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—24no—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—24no—
171C22×Dic5—24no—
172C22×Dic5C2×D414yes—
173C22×Dic5C2412yes—
174C22×Dic5C2414yes—
175C22×Dic5C2414yes—
176C22×Dic5C2412yes—
177C2×D20—232no—
178C2×D20—28no—
179C2×D20—28no—
180C2×D20C2×Q8116yes—
181C2×C4×D5—232no—
182C2×C4×D5—28no—
183C2×C4×D5—28no—
184C2×C4×D5—216no—
185C2×C4×D5—28no—
186C2×C4×D5—28no—
187C2×C4×D5—28no—
188C2×C4×D5—48no—
189C2×C4×D5—48no—
190C2×C4×D5—432no—
191C2×C4×D5—416no—
192C2×C4×D5C2×D4116yes—
193C5×C4⋊C4—28no—
194C5×C4⋊C4—28no—
195C5×C4⋊C4—48no—
196C42×C5—28no—
197#685—28no—
198#685—216no—
199#685—28no—
200#685—48no—
201#685—48no—
202C4⋊Dic5—28no—
203C4⋊Dic5—28no—
204C4⋊Dic5—28no—
205#683—216no—
206#683—48no—
207C23×D4—51no—
208C24×C4—51no—
209C24×C4—51no—
210C24×C4—52no—
211C2×C4×D4—58no—
212C2×C4×D4—108no—
213C22×C4⋊C4—52no—
214C22×C4⋊C4—51no—
215C22×C4⋊C4—51no—
216C22×C4⋊C4—51no—
217C22×C4⋊C4—52no—
218C22×C4⋊C4—58no—
219C22×C4⋊C4—108no—
220C22×C4⋊C4C2×D511yescharacteristic
221C22×C4⋊C4C2×D511yescharacteristic
222C22×C4⋊C4C2×D512yes—
223C22×C22⋊C4—51no—
224C22×C22⋊C4—51no—
225C22×C22⋊C4—52no—
226C22×C22⋊C4—52no—
227C22×C22⋊C4—52no—
228C22×C42C2×D511yescharacteristic
229#400—516no—
230#399—516no—
231#389—58no—
232#389C2×D518yescomplemented
233#386—516no—
234#385—516no—
235#380—58no—
236#380—58no—
237#380—58no—
238C2×#95—52no—
239C2×#95—52no—
240C2×#95—52no—
241C2×#95—52no—
242C2×#95C2×D512yes—
243C23×C5C22×D412yes—
244C23×C5C22×Q812yes—
245C23×C5C2×#4112yes—
246C23×C5C2×#4112yes—
247C23×C5C2×#4112yes—
248C23×C5H(5,𝔽2)12yes—
249C23×C5H(5,𝔽2)12yes—
250C23×C5C2511yescharacteristic
251C22×D5—28no—
252C22×D5—28no—
253C22×D5—28no—
254C22×D5—28no—
255C22×D5—28no—
256C22×D5—28no—
257C22×D5—22no—
258C22×D5—22no—
259C22×D5—22no—
260C22×D5—22no—
261C22×D5—28no—
262C22×D5—28no—
263C22×D5—28no—
264C22×D5—22no—
265C22×D5—22no—
266C22×D5—22no—
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—22no—
270C22×D5—44no—
271C22×D5—44no—
272C22×D5—44no—
273C22×D5—44no—
274C22×D5C22⋊Q8116yes—
275C22×D5C22×D414yes—
276C22×D5C22×Q814yes—
277C22×D5C22×Q814yes—
278C22×D5C2×#4114yes—
279C2×C4×C5—28no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—24no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—22no—
287C2×C4×C5—24no—
288C2×C4×C5—24no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5—48no—
292C2×C4×C5—44no—
293C2×C4×C5—48no—
294C2×C4×C5#123116yes—
295C2×C4×C5C22×D414yes—
296C2×C4×C5C22×Q814yes—
297C2×C4×C5C2×#4114yes—
298C2×C4×C5C2×#4114yes—
299C2×Dic5—28no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—24no—
303C2×Dic5—22no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—24no—
309C2×Dic5—24no—
310C2×Dic5—24no—
311C2×Dic5—44no—
312C2×Dic5—48no—
313C2×Dic5—48no—
314C2×Dic5C22×D414yes—
315D20—232no—
316C4×D5—232no—
317C4×D5—432no—
318C4×D5—416no—
319C25—51no—
320C25—51no—
321C22×D4—58no—
322C22×D4—54no—
323C22×D4—104no—
324C22×D4—104no—
325C23×C4—51no—
326C23×C4—58no—
327C23×C4—54no—
328C23×C4—52no—
329C23×C4—52no—
330C23×C4—51no—
331C23×C4—54no—
332C23×C4—58no—
333C23×C4—108no—
334C23×C4—104no—
335C23×C4—104no—
336C23×C4—104no—
337C23×C4—104no—
338C23×C4—104no—
339C23×C4—204no—
340C23×C4—204no—
341C23×C4C22×D511yescharacteristic
342C23×C4C22×D512yes—
343C23×C4C22×D512yes—
344C23×C4C22×D511yescharacteristic
345C23×C4C22×D511yescharacteristic
346C4×D4—1032no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—54no—
351C2×C4⋊C4—58no—
352C2×C4⋊C4—54no—
353C2×C4⋊C4—54no—
354C2×C4⋊C4—58no—
355C2×C4⋊C4—1032no—
356C2×C4⋊C4—108no—
357C2×C4⋊C4—104no—
358C2×C4⋊C4—104no—
359C2×C4⋊C4—104no—
360C2×C4⋊C4—204no—
361C2×C4⋊C4C22×D514yes—
362C2×C4⋊C4C22×D514yes—
363C2×C4⋊C4C22×D518yes—
364C2×C22⋊C4—54no—
365C2×C22⋊C4—54no—
366C2×C22⋊C4—58no—
367C2×C22⋊C4—58no—
368C2×C22⋊C4—58no—
369C2×C22⋊C4—104no—
370C2×C22⋊C4—104no—
371C2×C22⋊C4—108no—
372C2×C22⋊C4—204no—
373C2×C22⋊C4—204no—
374C2×C42—24no—
375C2×C42C22×D514yes—
376#95—58no—
377#95—58no—
378#95—58no—
379#95—58no—
380#95C22×D518yes—
381C22×C5C2×C22⋊Q814yes—
382C22×C5C2×#12314yes—
383C22×C5#53514yes—
384C22×C5#55314yes—
385C22×C5#55314yes—
386C22×C5#55414yes—
387C22×C5#55414yes—
388C22×C5C23×D411yescharacteristic
389C22×C5C23×Q811yescharacteristic
390C22×C5C22×#4111yescharacteristic
391C22×C5C22×#4111yescharacteristic
392C22×C5C22×#4111yescharacteristic
393C22×C5C2×H(5,𝔽2)11yescharacteristic
394C22×C5C2×H(5,𝔽2)11yescharacteristic
395C2×D5—24no—
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—24no—
399C2×D5—24no—
400C2×D5—24no—
401C2×D5—232no—
402C2×D5—24no—
403C2×D5—24no—
404C2×D5—24no—
405C2×D5—42no—
406C2×D5—42no—
407C2×D5—42no—
408C2×D5—42no—
409C2×D5—416no—
410C2×D5C2×C22⋊Q818yes—
411C4×C5—24no—
412C4×C5—44no—
413C4×C5—42no—
414C4×C5—44no—
415C4×C5C2×#12318yes—
416Dic5—24no—
417Dic5—42no—
418Dic5—44no—
419Dic5—44no—
420C24—54no—
421C24—54no—
422C24—54no—
423C24—52no—
424C24—54no—
425C24—52no—
426C24—102no—
427C24—102no—
428C24—102no—
429C24—102no—
430C24—102no—
431C24—102no—
432C24—102no—
433C24—102no—
434C24—102no—
435C24—102no—
436C24C23×D511yescharacteristic
437C2×D4—516no—
438C2×D4—108no—
439C2×D4—1032no—
440C2×D4—108no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—54no—
453C22×C4—52no—
454C22×C4—516no—
455C22×C4—52no—
456C22×C4—54no—
457C22×C4—54no—
458C22×C4—1016no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—108no—
462C22×C4—108no—
463C22×C4—1032no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—104no—
467C22×C4—104no—
468C22×C4—104no—
469C22×C4—102no—
470C22×C4—104no—
471C22×C4—104no—
472C22×C4—104no—
473C22×C4—108no—
474C22×C4—108no—
475C22×C4—108no—
476C22×C4—208no—
477C22×C4—208no—
478C22×C4—2032no—
479C22×C4—2016no—
480C22×C4#70914yes—
481C22×C4D4×D514yes—
482C22×C4Q8×D514yes—
483C22×C4#71314yes—
484C22×C4C23×D514yes—
485C22×C4C23×D514yes—
486C22×C4C23×D512yes—
487C22×C4C23×D512yes—
488C22×C4C23×D512yes—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—48no—
492C4⋊C4—1016no—
493C4⋊C4—108no—
494C4⋊C4—108no—
495C4⋊C4—108no—
496C4⋊C4—208no—
497C22⋊C4—108no—
498C22⋊C4—108no—
499C22⋊C4—1016no—
500C22⋊C4—208no—
501C22⋊C4—208no—
502C42—28no—
503C2×C5#248718yescomplemented
504C2×C5C22×C22⋊Q811yescharacteristic
505C2×C5C22×#12311yescharacteristic
506C2×C5C2×#53511yescharacteristic
507C2×C5C2×#55311yescharacteristic
508C2×C5C2×#55311yescharacteristic
509C2×C5C2×#55411yescharacteristic
510C2×C5C2×#55411yescharacteristic
511D5—28no—
512D5—44no—
513C23—516no—
514C23—54no—
515C23—54no—
516C23—54no—
517C23—54no—
518C23—108no—
519C23—108no—
520C23—108no—
521C23—108no—
522C23—108no—
523C23—102no—
524C23—102no—
525C23—102no—
526C23—102no—
527C23—108no—
528C23—108no—
529C23—108no—
530C23—108no—
531C23—102no—
532C23—102no—
533C23—102no—
534C23—102no—
535C23—102no—
536C23—102no—
537C23—204no—
538C23—204no—
539C23—204no—
540C23—204no—
541C23C2×#70912yes—
542C23C2×D4×D512yes—
543C23#414412yes—
544C23C2×Q8×D512yes—
545C23C2×#71312yes—
546C23D5×#4112yes—
547C23#414912yes—
548C23C24×D511yescharacteristic
549D4—1032no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—54no—
566C2×C4—108no—
567C2×C4—102no—
568C2×C4—102no—
569C2×C4—1032no—
570C2×C4—102no—
571C2×C4—102no—
572C2×C4—104no—
573C2×C4—104no—
574C2×C4—104no—
575C2×C4—102no—
576C2×C4—104no—
577C2×C4—104no—
578C2×C4—104no—
579C2×C4—2032no—
580C2×C4—204no—
581C2×C4—208no—
582C2×C4—208no—
583C2×C4—2016no—
584C2×C4#4040116yes—
585C2×C4C2×#70914yes—
586C2×C4C2×D4×D514yes—
587C2×C4C2×Q8×D514yes—
588C2×C4C2×#71314yes—
589C5C2×#248711yescharacteristic, complemented, Sylow
590C22—58no—
591C22—104no—
592C22—104no—
593C22—104no—
594C22—1032no—
595C22—104no—
596C22—104no—
597C22—104no—
598C22—104no—
599C22—104no—
600C22—104no—
601C22—202no—
602C22—202no—
603C22—2016no—
604C22—202no—
605C22—202no—
606C22C2×#404014yes—
607C22#6589214yes—
608C22#6591314yes—
609C22#6594114yes—
610C22D5×C22⋊Q814yes—
611C22#6602914yes—
612C22#6606414yes—
613C22C22×#70911yescharacteristic
614C22C22×D4×D511yescharacteristic
615C22C2×#414411yescharacteristic
616C22C22×Q8×D511yescharacteristic
617C22C22×#71311yescharacteristic
618C22C2×D5×#4111yescharacteristic
619C22C2×#414911yescharacteristic
620C4—24no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—104no—
625C4—202no—
626C4—204no—
627C4—204no—
628C4C2×#404018yes—
629C2—108no—
630C2—204no—
631C2#1060796318yesdirect product factor
632C2C22×#404011yescharacteristic
633C2C2×#6589211yescharacteristic
634C2C2×#6591311yescharacteristic
635C2C2×#6594111yescharacteristic
636C2C2×D5×C22⋊Q811yescharacteristic
637C2C2×#6602911yescharacteristic
638C2C2×#6606411yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#389C2
socleC24×C5C24
prime radicalC23×D20C22
hypercenterC2×#389D5
hypocenterC5C2×#2487
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×D5×C4⋊C4C2semidirect
3. C2×C5×#389C2semidirect
4. C2×#65281C2semidirect
5. C2×#65278C2semidirect
6. C2×#65276C2semidirect
7. C2×#65276C2semidirect
8. C2×#65276C2semidirect
9. C2×#65217C2semidirect
10. #10607963C2direct (N × H)
12. C2×#64949C2semidirect
32. C2×D5×C4⋊C4C22semidirect
35. C5×#389C22semidirect
41. #65281C22semidirect
42. #65278C22semidirect
43. #65276C22semidirect
44. #65276C22semidirect
45. #65276C22semidirect
46. #65217C22semidirect
48. #64949C22semidirect
118. C2×#389D5semidirect
232. #389C2×D5semidirect
503. C2×C5#2487semidirect
589. C5C2×#2487semidirect
631. C2#10607963direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D5×C4⋊C42
C22×C4×D202
C2×C5×#3892
C2×#652812
C2×#652782
C2×#652762
C2×#652172
#106079632
C2×#649542
C2×#649492
C2×#24875