Finite Group Databasebeta

← C2×#10605642 #12268348

Subgroups — C2×#10605642 #12268348

Order 1280.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D5C211yesmaximal, characteristic
2C22×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#4029C212yesmaximal
4C2×C5×#380C211yesmaximal, characteristic, complemented
5C2×#65501C211yesmaximal, characteristic, complemented
6C2×#65276C211yesmaximal, characteristic, complemented
7C2×#65274C211yesmaximal, characteristic, complemented
8C2×#65262C211yesmaximal, characteristic, complemented
9C2×#65241C212yesmaximal, complemented
10#10605642C2116yesmaximal, direct product factor
11C2×#64953C212yesmaximal, complemented
12C2×#64946C212yesmaximal, complemented
13C24×C4×C5C2211yescharacteristic
14C23×C22⋊D5C2211yescharacteristic
15C23×C4×D5C2212yes—
16C23×C4×D5C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C22⋊C4C2212yes—
20C22×C22⋊Dic5C2211yescharacteristic
21C22×C22⋊Dic5C2212yes—
22C2×C4×C22⋊D5—28no—
23C2×C4×C22⋊D5C2218yes—
24C22×#685C2211yescharacteristic
25C22×#685C2212yes—
26C22×C4⋊Dic5C2211yescharacteristic
27C22×C4⋊Dic5C2212yes—
28C22×C4⋊Dic5C2211yescharacteristic
29C22×#683C2212yes—
30C22×#683C2211yescharacteristic
31C22×#683C2212yes—
32C22×C4×Dic5C2211yescharacteristic
33C2×D5×C4⋊C4—28no—
34C2×D5×C4⋊C4C2218yescomplemented
35C2×#4029—216no—
36C2×#4029C22116yes—
37C5×#380C2218yescomplemented
38C2×C5×#95C2212yes—
39#65501C2218yescomplemented
40C2×#3963C2212yes—
41C2×#3963C2212yes—
42C2×#3963C2212yes—
43C2×#3963C2212yes—
44#65276C2218yescomplemented
45#65274C2218yescomplemented
46#65262C2218yescomplemented
47#65241C22116yescomplemented
48#64953C22116yescomplemented
49#64946C22116yescomplemented
50C2×#2486—51nomaximal, Sylow
51C25×C5C2311yescharacteristic
52C24×D5C2311yescharacteristic
53C23×C4×C5—24no—
54C23×C4×C5—24no—
55C23×C4×C5—24no—
56C23×C4×C5C2311yescharacteristic
57C23×C4×C5C2312yes—
58C23×C4×C5C2312yes—
59C23×C4×C5C2314yes—
60C23×C4×C5C2311yescharacteristic
61C22×C22⋊D5—24no—
62C22×C22⋊D5—24no—
63C22×C22⋊D5D418yes—
64C22×C22⋊D5C2314yes—
65C23×Dic5C2311yescharacteristic
66C23×Dic5C2312yes—
67C23×Dic5C2312yes—
68C23×Dic5C2311yescharacteristic
69C23×Dic5C2311yescharacteristic
70C22×C4×D5—24no—
71C22×C4×D5—24no—
72C22×C4×D5—28no—
73C22×C4×D5—28no—
74C22×C4×D5—28no—
75C22×C4×D5D418yes—
76C22×C4×D5C2318yes—
77C22×C4×D5C2314yes—
78C2×C5×C4⋊C4—24no—
79C2×C5×C4⋊C4—24no—
80C2×C5×C4⋊C4C2314yes—
81C2×C5×C4⋊C4C2314yes—
82C2×C5×C22⋊C4—28no—
83C2×C5×C22⋊C4C2318yes—
84C2×C22⋊Dic5—28no—
85C2×C22⋊Dic5—24no—
86C2×C22⋊Dic5C2318yes—
87C2×C22⋊Dic5C2314yes—
88C4×C22⋊D5—232no—
89C2×#685—28no—
90C2×#685—24no—
91C2×#685C2318yes—
92C2×#685C2314yes—
93C2×C4⋊Dic5—24no—
94C2×C4⋊Dic5—24no—
95C2×C4⋊Dic5—28no—
96C2×C4⋊Dic5C2318yes—
97C2×C4⋊Dic5C2314yes—
98C2×C4⋊Dic5C2314yes—
99C2×#683—28no—
100C2×#683—28no—
101C2×#683—24no—
102C2×#683C2318yes—
103C2×#683C2318yes—
104C2×#683C2314yes—
105C2×C4×Dic5—24no—
106C2×C4×Dic5C2314yes—
107D5×C4⋊C4—232no—
108#4029—264no—
109C5×#95C2318yes—
110#3963C2318yes—
111#3963C2318yes—
112#3963C2318yes—
113#3963C2318yes—
114C22×#123—52no—
115C22×C4×D4—51no—
116C23×C4⋊C4—51no—
117#2486—516no—
118C2×#398—52no—
119C2×#397—52no—
120C2×#389—51no—
121C2×#389—51no—
122C2×#381—52no—
123C2×#380—51no—
124C2×#380—51no—
125C2×#380D511yescharacteristic, complemented
126C24×C5—22no—
127C24×C5—22no—
128C24×C5—22no—
129C24×C5—22no—
130C24×C5—22no—
131C24×C5—22no—
132C24×C5C2×D414yes—
133C24×C5C2412yes—
134C24×C5C2411yescharacteristic
135C23×D5—22no—
136C23×D5—22no—
137C23×D5—22no—
138C23×D5—22no—
139C23×D5C2×D414yes—
140C23×D5C2×D414yes—
141C23×D5#4114yes—
142C23×D5C2412yes—
143C22×C4×C5—24no—
144C22×C4×C5—24no—
145C22×C4×C5—24no—
146C22×C4×C5—22no—
147C22×C4×C5—24no—
148C22×C4×C5—24no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—28no—
153C22×C4×C5—28no—
154C22×C4×C5—28no—
155C22×C4×C5—416no—
156C22×C4×C5C2×D414yes—
157C22×C4×C5#4114yes—
158C22×C4×C5#4114yes—
159C22×C4×C5C2414yes—
160C22×C4×C5C2414yes—
161C22×C4×C5C2412yes—
162C22×C4×C5C2412yes—
163C2×C22⋊D5—232no—
164C2×C22⋊D5—28no—
165C2×C22⋊D5—28no—
166C2×C22⋊D5C2×D4116yes—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—24no—
172C22×Dic5—22no—
173C22×Dic5—24no—
174C22×Dic5—24no—
175C22×Dic5—24no—
176C22×Dic5—24no—
177C22×Dic5—22no—
178C22×Dic5—24no—
179C22×Dic5—22no—
180C22×Dic5C2×D414yes—
181C22×Dic5C2×D414yes—
182C22×Dic5C2414yes—
183C22×Dic5C2412yes—
184C22×Dic5C2414yes—
185C22×Dic5C2412yes—
186C22×Dic5C2412yes—
187C2×C4×D5—232no—
188C2×C4×D5—28no—
189C2×C4×D5—28no—
190C2×C4×D5—216no—
191C2×C4×D5—216no—
192C2×C4×D5—216no—
193C2×C4×D5—432no—
194C2×C4×D5C2×D4116yes—
195C5×C4⋊C4—28no—
196C5×C4⋊C4—28no—
197C5×C22⋊C4—216no—
198C22⋊Dic5—216no—
199C22⋊Dic5—28no—
200#685—216no—
201#685—28no—
202C4⋊Dic5—28no—
203C4⋊Dic5—28no—
204C4⋊Dic5—216no—
205#683—216no—
206#683—216no—
207#683—28no—
208C4×Dic5—28no—
209C23×D4—51no—
210C24×C4—51no—
211C24×C4—52no—
212C24×C4C2×D511yescharacteristic
213C2×#123—516no—
214C2×#123—1016no—
215C2×C4×D4—58no—
216C2×C4×D4—108no—
217C22×C4⋊C4—52no—
218C22×C4⋊C4—51no—
219C22×C4⋊C4—52no—
220C22×C4⋊C4—52no—
221C22×C4⋊C4—51no—
222C22×C4⋊C4—51no—
223C22×C4⋊C4—58no—
224C22×C4⋊C4—108no—
225C22×C4⋊C4C2×D511yescharacteristic
226C22×C4⋊C4C2×D511yescharacteristic
227C22×C22⋊C4—51no—
228C22×C22⋊C4—52no—
229C22×C22⋊C4—51no—
230C22×C22⋊C4—52no—
231C22×C22⋊C4C2×D512yes—
232C22×C42—51no—
233#398—516no—
234#397—516no—
235#389—58no—
236#389—58no—
237#381—516no—
238#380—58no—
239#380—58no—
240#380C2×D518yescomplemented
241C2×#95—52no—
242C2×#95—52no—
243C2×#95—52no—
244C2×#95—52no—
245C2×#95C2×D512yes—
246C23×C5—28no—
247C23×C5—28no—
248C23×C5—28no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—22no—
252C23×C5—22no—
253C23×C5—22no—
254C23×C5—22no—
255C23×C5—44no—
256C23×C5—44no—
257C23×C5—44no—
258C23×C5—44no—
259C23×C5C22×D414yes—
260C23×C5C22×D412yes—
261C23×C5C22×D412yes—
262C23×C5C2×#4112yes—
263C23×C5C2×#4112yes—
264C23×C5C2×#4112yes—
265C23×C52−1+412yes—
266C23×C52−1+412yes—
267C23×C5C2511yescharacteristic
268C22×D5—28no—
269C22×D5—28no—
270C22×D5—28no—
271C22×D5—28no—
272C22×D5—28no—
273C22×D5—28no—
274C22×D5—22no—
275C22×D5—22no—
276C22×D5—22no—
277C22×D5—22no—
278C22×D5#121116yes—
279C22×D5C22×D414yes—
280C22×D5C22×D414yes—
281C22×D5C2×#4114yes—
282C2×C4×C5—22no—
283C2×C4×C5—24no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—24no—
287C2×C4×C5—24no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5—48no—
292C2×C4×C5—44no—
293C2×C4×C5—48no—
294C2×C4×C5—416no—
295C2×C4×C5#123116yescomplemented
296C2×C4×C5C22×D414yes—
297C2×C4×C5C2×#4114yes—
298C2×C4×C5C2×#4114yes—
299C22⋊D5—232no—
300C2×Dic5—28no—
301C2×Dic5—28no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—24no—
309C2×Dic5—24no—
310C2×Dic5—24no—
311C2×Dic5—24no—
312C2×Dic5—22no—
313C2×Dic5—24no—
314C2×Dic5—22no—
315C2×Dic5—48no—
316C2×Dic5—44no—
317C2×Dic5—48no—
318C2×Dic5C22×D414yes—
319C2×Dic5C22×D414yes—
320C4×D5—232no—
321C4×D5—432no—
322C25—51no—
323C25C22×D511yescharacteristic
324C22×D4—58no—
325C22×D4—54no—
326C22×D4—104no—
327C22×D4—104no—
328C23×C4—24no—
329C23×C4—24no—
330C23×C4—24no—
331C23×C4—51no—
332C23×C4—51no—
333C23×C4—58no—
334C23×C4—54no—
335C23×C4—52no—
336C23×C4—51no—
337C23×C4—52no—
338C23×C4—58no—
339C23×C4—108no—
340C23×C4—108no—
341C23×C4—108no—
342C23×C4—104no—
343C23×C4—104no—
344C23×C4C22×D514yes—
345C23×C4C22×D512yes—
346C23×C4C22×D511yescharacteristic
347C23×C4C22×D512yes—
348C23×C4C22×D511yescharacteristic
349#123—1064no—
350C4×D4—1032no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—54no—
354C2×C4⋊C4—58no—
355C2×C4⋊C4—54no—
356C2×C4⋊C4—58no—
357C2×C4⋊C4—58no—
358C2×C4⋊C4—54no—
359C2×C4⋊C4—1032no—
360C2×C4⋊C4—108no—
361C2×C4⋊C4—104no—
362C2×C4⋊C4—108no—
363C2×C4⋊C4—108no—
364C2×C4⋊C4—104no—
365C2×C4⋊C4—104no—
366C2×C4⋊C4C22×D514yes—
367C2×C4⋊C4C22×D514yes—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—54no—
370C2×C22⋊C4—58no—
371C2×C22⋊C4—54no—
372C2×C22⋊C4—58no—
373C2×C22⋊C4—104no—
374C2×C22⋊C4—108no—
375C2×C22⋊C4—104no—
376C2×C22⋊C4—108no—
377C2×C22⋊C4C22×D518yes—
378C2×C42—54no—
379C2×C42—104no—
380#95—58no—
381#95—58no—
382#95—58no—
383#95—58no—
384#95C22×D518yes—
385C22×C5—24no—
386C22×C5—24no—
387C22×C5—24no—
388C22×C5—42no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—416no—
393C22×C5C2×#12114yes—
394C22×C5C2×#12314yes—
395C22×C5#53614yes—
396C22×C5#54714yes—
397C22×C5#54714yes—
398C22×C5#55214yes—
399C22×C5#55214yes—
400C22×C5C23×D411yescharacteristic
401C22×C5C23×D411yescharacteristic
402C22×C5C22×#4111yescharacteristic
403C22×C5C22×#4111yescharacteristic
404C22×C5C22×#4111yescharacteristic
405C22×C5C2×2−1+411yescharacteristic
406C22×C5C2×2−1+411yescharacteristic
407C2×D5—24no—
408C2×D5—24no—
409C2×D5—24no—
410C2×D5—24no—
411C2×D5—24no—
412C2×D5—24no—
413C2×D5—232no—
414C2×D5C2×#12118yes—
415C4×C5—44no—
416C4×C5—42no—
417C4×C5—44no—
418C4×C5C2×#12318yescomplemented
419Dic5—24no—
420Dic5—24no—
421Dic5—44no—
422Dic5—42no—
423Dic5—44no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—54no—
431C24—54no—
432C24—54no—
433C24—52no—
434C24—102no—
435C24—102no—
436C24—102no—
437C24—102no—
438C24D4×D514yes—
439C24C23×D512yes—
440C24C23×D511yescharacteristic
441C2×D4—516no—
442C2×D4—108no—
443C2×D4—1032no—
444C2×D4—108no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—416no—
458C22×C4—54no—
459C22×C4—54no—
460C22×C4—52no—
461C22×C4—52no—
462C22×C4—516no—
463C22×C4—52no—
464C22×C4—54no—
465C22×C4—54no—
466C22×C4—1016no—
467C22×C4—1016no—
468C22×C4—1016no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—108no—
474C22×C4—108no—
475C22×C4—1032no—
476C22×C4—104no—
477C22×C4—102no—
478C22×C4—104no—
479C22×C4—104no—
480C22×C4—104no—
481C22×C4—104no—
482C22×C4—102no—
483C22×C4—104no—
484C22×C4—102no—
485C22×C4—2032no—
486C22×C4#70914yes—
487C22×C4D4×D514yes—
488C22×C4#71114yes—
489C22×C4C23×D512yes—
490C22×C4C23×D514yes—
491C22×C4C23×D514yes—
492C22×C4C23×D512yes—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—1016no—
496C4⋊C4—108no—
497C4⋊C4—1016no—
498C4⋊C4—1016no—
499C4⋊C4—108no—
500C4⋊C4—108no—
501C22⋊C4—216no—
502C22⋊C4—108no—
503C22⋊C4—1016no—
504C22⋊C4—108no—
505C22⋊C4—1016no—
506C42—108no—
507C2×C5—44no—
508C2×C5#248618yescomplemented
509C2×C5C22×#12111yescharacteristic
510C2×C5C22×#12311yescharacteristic
511C2×C5C2×#53611yescharacteristic
512C2×C5C2×#54711yescharacteristic
513C2×C5C2×#54711yescharacteristic
514C2×C5C2×#55211yescharacteristic
515C2×C5C2×#55211yescharacteristic
516D5—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—516no—
531C23—54no—
532C23—54no—
533C23—54no—
534C23—108no—
535C23—108no—
536C23—108no—
537C23—108no—
538C23—108no—
539C23—108no—
540C23—102no—
541C23—102no—
542C23—102no—
543C23—102no—
544C23C2×#70912yes—
545C23C2×D4×D514yes—
546C23C2×D4×D512yes—
547C23C2×D4×D512yes—
548C23C2×#71112yes—
549C23#414712yes—
550C23D5×#4112yes—
551C23#415012yes—
552C23C24×D511yescharacteristic
553D4—1032no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—416no—
567C2×C4—54no—
568C2×C4—54no—
569C2×C4—108no—
570C2×C4—108no—
571C2×C4—102no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—1032no—
576C2×C4—104no—
577C2×C4—102no—
578C2×C4—104no—
579C2×C4—104no—
580C2×C4—104no—
581C2×C4—104no—
582C2×C4—102no—
583C2×C4—104no—
584C2×C4—102no—
585C2×C4—2032no—
586C2×C4—208no—
587C2×C4—208no—
588C2×C4—204no—
589C2×C4#4029116yescomplemented
590C2×C4C2×#70914yes—
591C2×C4C2×D4×D514yes—
592C2×C4C2×#71114yes—
593C5C2×#248611yescharacteristic, complemented, Sylow
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—416no—
601C22—42no—
602C22—58no—
603C22—104no—
604C22—104no—
605C22—104no—
606C22—104no—
607C22—1032no—
608C22—104no—
609C22—104no—
610C22C2×#402914yes—
611C22#6584114yes—
612C22#6586314yes—
613C22D5×#12114yes—
614C22#6594514yes—
615C22#6596214yes—
616C22#6597314yes—
617C22C22×#70911yescharacteristic
618C22C22×D4×D511yescharacteristic
619C22C22×D4×D511yescharacteristic
620C22C22×#71111yescharacteristic
621C22C2×#414711yescharacteristic
622C22C2×D5×#4111yescharacteristic
623C22C2×#415011yescharacteristic
624C4—44no—
625C4—44no—
626C4—42no—
627C4—104no—
628C4—104no—
629C4—204no—
630C4—202no—
631C4—204no—
632C4C2×#402918yescomplemented
633C2—44no—
634C2—108no—
635C2#1060564218yesdirect product factor
636C2C22×#402911yescharacteristic
637C2C2×#6584111yescharacteristic
638C2C2×#6586311yescharacteristic
639C2C2×#6594511yescharacteristic
640C2C2×D5×#12111yescharacteristic
641C2C2×#6596211yescharacteristic
642C2C2×#6597311yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#380C2
socleC24×C5C24
prime radicalC23×C22⋊D5C22
hypercenterC2×#380D5
hypocenterC5C2×#2486
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×D5×C4⋊C4C2semidirect
4. C2×C5×#380C2semidirect
5. C2×#65501C2semidirect
6. C2×#65276C2semidirect
7. C2×#65274C2semidirect
8. C2×#65262C2semidirect
9. C2×#65241C2semidirect
10. #10605642C2direct (N × H)
11. C2×#64953C2semidirect
12. C2×#64946C2semidirect
34. C2×D5×C4⋊C4C22semidirect
37. C5×#380C22semidirect
39. #65501C22semidirect
44. #65276C22semidirect
45. #65274C22semidirect
46. #65262C22semidirect
47. #65241C22semidirect
48. #64953C22semidirect
49. #64946C22semidirect
125. C2×#380D5semidirect
240. #380C2×D5semidirect
295. C2×C4×C5#123semidirect
418. C4×C5C2×#123semidirect
508. C2×C5#2486semidirect
589. C2×C4#4029semidirect
593. C5C2×#2486semidirect
632. C4C2×#4029semidirect
635. C2#10605642direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D52
C22×D5×C4⋊C42
C22×#40292
C2×C5×#3802
C2×#655012
C2×#652762
C2×#652742
C2×#652622
C2×#652412
#106056422
C2×#649532
C2×#649462
C2×#24865