Finite Group Databasebeta

← C2×#10609761 #12268356

Subgroups — C2×#10609761 #12268356

Order 1280.

4222 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4040C212yesmaximal, complemented
2C22×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#4029C212yesmaximal, complemented
4C22×#4030C211yesmaximal, characteristic
5C2×C5×#399C211yesmaximal, characteristic, complemented
6C2×#65281C211yesmaximal, characteristic, complemented
7C2×#65265C212yesmaximal, complemented
8C2×#65238C211yesmaximal, characteristic, complemented
9C2×#65246C212yesmaximal, complemented
10#10609761C2116yesmaximal, direct product factor
11C2×#64966C211yesmaximal, characteristic, complemented
12C2×#64955C211yesmaximal, characteristic, complemented
13C23×C22⋊D5C2211yescharacteristic
14C23×C22⋊D5C2211yescharacteristic
15C23×D20C2211yescharacteristic
16C23×C4×D5C2212yes—
17C23×C4×D5C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2212yes—
20C22×C5×C22⋊C4C2212yes—
21C22×C5×C22⋊C4C2211yescharacteristic
22C22×C22⋊Dic5C2212yes—
23C22×#685C2212yes—
24C22×#685C2212yes—
25C22×#685C2211yescharacteristic
26C22×#685C2212yes—
27C22×C4⋊Dic5C2211yescharacteristic
28C22×C4⋊Dic5C2212yes—
29C22×#683C2212yes—
30C22×#683C2212yes—
31C22×#683C2211yescharacteristic
32C22×#683C2212yes—
33C22×C4×Dic5C2212yes—
34C2×#4040—216no—
35C2×#4040C22116yescomplemented
36C2×D5×C4⋊C4—28no—
37C2×D5×C4⋊C4C2218yescomplemented
38C2×#4030—28no—
39C2×#4030C2218yes—
40C2×#4029—216no—
41C2×#4029C22116yescomplemented
42C5×#399C2218yescomplemented
43C2×C5×#95C2211yescharacteristic
44C2×#3963C2211yescharacteristic
45#65281C2218yescomplemented
46#65265C22116yescomplemented
47#65246C22116yescomplemented
48#65238C2218yescomplemented
49#64966C2218yescomplemented
50#64955C2218yescomplemented
51C2×#2501—51nomaximal, Sylow
52C25×C5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C24×D5C2311yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2312yes—
57C23×C4×C5C2312yes—
58C23×C4×C5C2311yescharacteristic
59C22×C22⋊D5—28no—
60C22×C22⋊D5—24no—
61C22×C22⋊D5—24no—
62C22×C22⋊D5—28no—
63C22×C22⋊D5C2314yes—
64C22×C22⋊D5C2314yes—
65C23×Dic5C2311yescharacteristic
66C23×Dic5C2312yes—
67C23×Dic5C2312yes—
68C23×Dic5C2311yescharacteristic
69C22×D20—24no—
70C22×D20—28no—
71C22×D20C2314yes—
72C22×C4×D5—28no—
73C22×C4×D5—28no—
74C22×C4×D5—28no—
75C22×C4×D5—28no—
76C22×C4×D5D418yes—
77C22×C4×D5C2318yes—
78C22×C4×D5C2314yes—
79C2×C5×C4⋊C4—24no—
80C2×C5×C4⋊C4—28no—
81C2×C5×C4⋊C4C2314yes—
82C2×C5×C4⋊C4C2318yes—
83C2×C5×C22⋊C4—28no—
84C2×C5×C22⋊C4—24no—
85C2×C5×C22⋊C4C2318yes—
86C2×C5×C22⋊C4C2314yes—
87C2×C22⋊Dic5—28no—
88C2×C22⋊Dic5C2318yes—
89C2×#685—24no—
90C2×#685—28no—
91C2×#685—28no—
92C2×#685D4116yescomplemented
93C2×#685C2314yes—
94C2×#685C2318yes—
95C2×#685C2318yes—
96C2×#685C2318yes—
97C2×C4⋊Dic5—28no—
98C2×C4⋊Dic5—24no—
99C2×C4⋊Dic5C2318yes—
100C2×C4⋊Dic5C2314yes—
101C2×#683—28no—
102C2×#683—24no—
103C2×#683—28no—
104C2×#683—28no—
105C2×#683C2318yes—
106C2×#683C2314yes—
107C2×#683C2318yes—
108C2×#683C2318yes—
109C2×C4×Dic5—28no—
110C2×C4×Dic5C2318yes—
111#4040—264no—
112D5×C4⋊C4—232no—
113#4030—232no—
114#4029—264no—
115C5×#95C2314yes—
116#3963C2314yes—
117C22×#123—52no—
118C22×#123—52no—
119C22×#121—51no—
120C23×C4⋊C4—51no—
121#2501—516no—
122C2×#399—51no—
123C2×#399—51no—
124C2×#399—51no—
125C2×#399D511yescharacteristic, complemented
126C2×#390—52no—
127C2×#384—52no—
128C2×C4×C22⋊C4—51no—
129C24×C5—24no—
130C24×C5—22no—
131C24×C5—24no—
132C24×C5—22no—
133C24×C5—22no—
134C24×C5C2412yes—
135C24×C5C2411yescharacteristic
136C23×D5—24no—
137C23×D5—22no—
138C23×D5—22no—
139C23×D5—22no—
140C23×D5—24no—
141C23×D5—24no—
142C23×D5—22no—
143C23×D5—22no—
144C23×D5C2×D414yes—
145C23×D5C2×D418yes—
146C23×D5C2412yes—
147C23×D5C2412yes—
148C22×C4×C5—24no—
149C22×C4×C5—24no—
150C22×C4×C5—24no—
151C22×C4×C5—24no—
152C22×C4×C5—22no—
153C22×C4×C5—24no—
154C22×C4×C5—24no—
155C22×C4×C5—24no—
156C22×C4×C5C2×D414yes—
157C22×C4×C5C2×D418yes—
158C22×C4×C5C2414yes—
159C22×C4×C5C2412yes—
160C22×C4×C5C2412yes—
161C22×C4×C5C2414yes—
162C2×C22⋊D5—216no—
163C2×C22⋊D5—28no—
164C2×C22⋊D5—28no—
165C2×C22⋊D5—216no—
166C2×C22⋊D5—416no—
167C2×C22⋊D5—416no—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—24no—
171C22×Dic5—24no—
172C22×Dic5—24no—
173C22×Dic5—24no—
174C22×Dic5—22no—
175C22×Dic5C2×D418yes—
176C22×Dic5C2×D414yes—
177C22×Dic5#4118yes—
178C22×Dic5C2414yes—
179C22×Dic5C2412yes—
180C22×Dic5C2414yes—
181C22×Dic5C2412yes—
182C2×D20—28no—
183C2×D20—216no—
184C2×D20—416no—
185C2×C4×D5—232no—
186C2×C4×D5—216no—
187C2×C4×D5—216no—
188C2×C4×D5—216no—
189C2×C4×D5—216no—
190C2×C4×D5—432no—
191C2×C4×D5C2×D4116yes—
192C5×C4⋊C4—28no—
193C5×C4⋊C4—216no—
194C5×C22⋊C4—216no—
195C5×C22⋊C4—28no—
196C22⋊Dic5—216no—
197#685—28no—
198#685—216no—
199#685—216no—
200#685C2×D4132yescomplemented
201C4⋊Dic5—216no—
202C4⋊Dic5—28no—
203#683—216no—
204#683—28no—
205#683—216no—
206#683—216no—
207C4×Dic5—216no—
208C23×D4—51no—
209C23×D4—51no—
210C23×D4—51no—
211C24×C4—51no—
212C24×C4—52no—
213C2×#123—516no—
214C2×#123—516no—
215C2×#123—1016no—
216C2×#123—1016no—
217C2×#121—58no—
218C2×#121—108no—
219C22×C4⋊C4—52no—
220C22×C4⋊C4—52no—
221C22×C4⋊C4—52no—
222C22×C4⋊C4—51no—
223C22×C4⋊C4—51no—
224C22×C4⋊C4—52no—
225C22×C4⋊C4—58no—
226C22×C4⋊C4—108no—
227C22×C4⋊C4C2×D511yescharacteristic
228C22×C4⋊C4C2×D512yes—
229C22×C22⋊C4—52no—
230C22×C22⋊C4—52no—
231C22×C22⋊C4—51no—
232C22×C22⋊C4—52no—
233C22×C22⋊C4—52no—
234C22×C22⋊C4C2×D511yescharacteristic
235C22×C22⋊C4C2×D512yes—
236C22×C42—52no—
237#399—58no—
238#399—58no—
239#399—58no—
240#399C2×D518yescomplemented
241#390—516no—
242#384—516no—
243C4×C22⋊C4—58no—
244C2×#95—51no—
245C2×#95C2×D511yescharacteristic
246C23×C5—24no—
247C23×C5—22no—
248C23×C5—24no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—44no—
252C23×C5—44no—
253C23×C5—44no—
254C23×C5—48no—
255C23×C5—48no—
256C23×C5C22×D412yes—
257C23×C5C22×D414yes—
258C23×C5C2×#4114yes—
259C23×C52−1+412yes—
260C23×C52−1+412yes—
261C23×C5C2511yescharacteristic
262C22×D5—28no—
263C22×D5—28no—
264C22×D5—216no—
265C22×D5—216no—
266C22×D5—24no—
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—24no—
270C22×D5—22no—
271C22×D5—24no—
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—44no—
275C22×D5—44no—
276C22×D5—44no—
277C22×D5—48no—
278C22×D5—48no—
279C22×D5#127116yes—
280C22×D5C22×D418yes—
281C22×D5C22×D414yes—
282C2×C4×C5—28no—
283C2×C4×C5—216no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—24no—
287C2×C4×C5—24no—
288C2×C4×C5—22no—
289C2×C4×C5—24no—
290C2×C4×C5—24no—
291C2×C4×C5—24no—
292C2×C4×C5—48no—
293C2×C4×C5—44no—
294C2×C4×C5C22×D418yes—
295C2×C4×C5C22×D414yes—
296C22⋊D5—416no—
297C22⋊D5—416no—
298C2×Dic5—216no—
299C2×Dic5—24no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5—22no—
305C2×Dic5—24no—
306C2×Dic5—48no—
307C2×Dic5—44no—
308C2×Dic5#123116yescomplemented
309C2×Dic5C22×D418yes—
310C2×Dic5C22×D414yes—
311C2×Dic5C2×#4118yes—
312D20—416no—
313C4×D5—232no—
314C4×D5—432no—
315C25—51no—
316C25—51no—
317C25C22×D511yescharacteristic
318C22×D4—54no—
319C22×D4—54no—
320C22×D4—54no—
321C22×D4—104no—
322C22×D4—108no—
323C22×D4—104no—
324C22×D4—108no—
325C22×D4—104no—
326C22×D4—108no—
327C23×C4—51no—
328C23×C4—52no—
329C23×C4—58no—
330C23×C4—54no—
331C23×C4—52no—
332C23×C4—51no—
333C23×C4—58no—
334C23×C4—108no—
335C23×C4—108no—
336C23×C4—108no—
337C23×C4—108no—
338C23×C4C22×D512yes—
339C23×C4C22×D511yescharacteristic
340C23×C4C22×D512yes—
341C23×C4C22×D511yescharacteristic
342#123—1064no—
343#123—1064no—
344#121—1032no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—58no—
348C2×C4⋊C4—54no—
349C2×C4⋊C4—54no—
350C2×C4⋊C4—58no—
351C2×C4⋊C4—58no—
352C2×C4⋊C4—58no—
353C2×C4⋊C4—1032no—
354C2×C4⋊C4—108no—
355C2×C4⋊C4—104no—
356C2×C4⋊C4—108no—
357C2×C4⋊C4—108no—
358C2×C4⋊C4—104no—
359C2×C4⋊C4—108no—
360C2×C4⋊C4C22×D514yes—
361C2×C4⋊C4C22×D518yes—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—516no—
365C2×C22⋊C4—58no—
366C2×C22⋊C4—58no—
367C2×C22⋊C4—54no—
368C2×C22⋊C4—58no—
369C2×C22⋊C4—58no—
370C2×C22⋊C4—108no—
371C2×C22⋊C4—108no—
372C2×C22⋊C4—104no—
373C2×C22⋊C4—108no—
374C2×C22⋊C4C22×D514yes—
375C2×C22⋊C4C22×D518yes—
376C2×C42—58no—
377C2×C42—108no—
378#95—54no—
379#95C22×D514yes—
380C22×C5—42no—
381C22×C5—42no—
382C22×C5—42no—
383C22×C5—44no—
384C22×C5—44no—
385C22×C5—88no—
386C22×C5C2×#12314yes—
387C22×C5C2×#12714yes—
388C22×C5#53614yes—
389C22×C5#54718yes—
390C22×C5#54718yes—
391C22×C5C23×D411yescharacteristic
392C22×C5C23×D412yes—
393C22×C5C22×#4112yes—
394C22×C5C2×2−1+411yescharacteristic
395C22×C5C2×2−1+411yescharacteristic
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—28no—
399C2×D5—28no—
400C2×D5—232no—
401C2×D5—42no—
402C2×D5—42no—
403C2×D5—42no—
404C2×D5—44no—
405C2×D5—44no—
406C2×D5—88no—
407C2×D5C2×#12718yes—
408C4×C5—24no—
409C4×C5—28no—
410C4×C5—44no—
411C4×C5—42no—
412Dic5—28no—
413Dic5—44no—
414Dic5—42no—
415Dic5C2×#12318yescomplemented
416C24—24no—
417C24—22no—
418C24—22no—
419C24—24no—
420C24—22no—
421C24—58no—
422C24—54no—
423C24—52no—
424C24—52no—
425C24—102no—
426C24—104no—
427C24—102no—
428C24—104no—
429C24—102no—
430C24—102no—
431C24—104no—
432C24—102no—
433C24C23×D512yes—
434C24C23×D511yescharacteristic
435C2×D4—108no—
436C2×D4—1016no—
437C2×D4—108no—
438C2×D4—1016no—
439C2×D4—108no—
440C2×D4—1016no—
441C2×D4—2016no—
442C2×D4—2016no—
443C2×D4—2016no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—54no—
453C22×C4—58no—
454C22×C4—52no—
455C22×C4—58no—
456C22×C4—54no—
457C22×C4—516no—
458C22×C4—54no—
459C22×C4—52no—
460C22×C4—1032no—
461C22×C4—1016no—
462C22×C4—1016no—
463C22×C4—1016no—
464C22×C4—1016no—
465C22×C4—104no—
466C22×C4—104no—
467C22×C4—104no—
468C22×C4—104no—
469C22×C4—102no—
470C22×C4—104no—
471C22×C4—104no—
472C22×C4—2032no—
473C22×C4D4×D518yes—
474C22×C4D4×D514yes—
475C22×C4C23×D514yes—
476C22×C4C23×D512yes—
477C22×C4C23×D512yes—
478C22×C4C23×D514yes—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—1016no—
482C4⋊C4—108no—
483C4⋊C4—1016no—
484C4⋊C4—1016no—
485C4⋊C4—108no—
486C4⋊C4—1016no—
487C22⋊C4—28no—
488C22⋊C4—216no—
489C22⋊C4—532no—
490C22⋊C4—1016no—
491C22⋊C4—1016no—
492C22⋊C4—108no—
493C22⋊C4—1016no—
494C42—1016no—
495C2×C5—82no—
496C2×C5#250118yescomplemented
497C2×C5C22×#12311yescharacteristic
498C2×C5C22×#12711yescharacteristic
499C2×C5C2×#53611yescharacteristic
500C2×C5C2×#54712yes—
501C2×C5C2×#54712yes—
502D5—28no—
503D5—82no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—44no—
510C23—44no—
511C23—48no—
512C23—44no—
513C23—48no—
514C23—516no—
515C23—58no—
516C23—54no—
517C23—108no—
518C23—1016no—
519C23—108no—
520C23—1016no—
521C23—102no—
522C23—104no—
523C23—102no—
524C23—104no—
525C23—102no—
526C23—102no—
527C23—104no—
528C23—102no—
529C23—204no—
530C23—204no—
531C23—208no—
532C23—204no—
533C23—208no—
534C23C2×D4×D512yes—
535C23C2×D4×D514yes—
536C23#414712yes—
537C23D5×#4114yes—
538C23#415012yes—
539C23C24×D511yescharacteristic
540D4—2016no—
541D4—2016no—
542D4—2016no—
543C2×C4—28no—
544C2×C4—216no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—516no—
556C2×C4—54no—
557C2×C4—58no—
558C2×C4—58no—
559C2×C4—1032no—
560C2×C4—1016no—
561C2×C4—104no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—104no—
565C2×C4—102no—
566C2×C4—104no—
567C2×C4—104no—
568C2×C4—2032no—
569C2×C4—208no—
570C2×C4—204no—
571C2×C4C2×D4×D518yes—
572C2×C4C2×D4×D514yes—
573C5C2×#250111yescharacteristic, complemented, Sylow
574C22—42no—
575C22—44no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—58no—
580C22—88no—
581C22—104no—
582C22—108no—
583C22—104no—
584C22—108no—
585C22—1032no—
586C22—202no—
587C22—204no—
588C22—202no—
589C22—204no—
590C22—202no—
591C22—408no—
592C22#6596214yes—
593C22#6596518yes—
594C22D5×#12314yes—
595C22#6598918yes—
596C22D5×#12714yes—
597C22C22×D4×D511yescharacteristic
598C22C22×D4×D512yes—
599C22C2×#414711yescharacteristic
600C22C2×D5×#4112yes—
601C22C2×#415011yescharacteristic
602C4—24no—
603C4—28no—
604C4—44no—
605C4—42no—
606C4—58no—
607C4—108no—
608C4—204no—
609C4—202no—
610C2—82no—
611C2—108no—
612C2—402no—
613C2#1060976118yesdirect product factor
614C2C2×#6596211yescharacteristic
615C2C2×#6596512yes—
616C2C2×D5×#12311yescharacteristic
617C2C2×#6598912yes—
618C2C2×D5×#12711yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#399C2
socleC24×C5C24
prime radicalC22×#4030C2
hypercenterC2×#399D5
hypocenterC5C2×#2501
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#4040C2semidirect
2. C22×D5×C4⋊C4C2semidirect
3. C22×#4029C2semidirect
5. C2×C5×#399C2semidirect
6. C2×#65281C2semidirect
7. C2×#65265C2semidirect
8. C2×#65238C2semidirect
9. C2×#65246C2semidirect
10. #10609761C2direct (N × H)
11. C2×#64966C2semidirect
12. C2×#64955C2semidirect
35. C2×#4040C22semidirect
37. C2×D5×C4⋊C4C22semidirect
41. C2×#4029C22semidirect
42. C5×#399C22semidirect
45. #65281C22semidirect
46. #65265C22semidirect
47. #65246C22semidirect
48. #65238C22semidirect
49. #64966C22semidirect
50. #64955C22semidirect
92. C2×#685D4semidirect
125. C2×#399D5semidirect
200. #685C2×D4semidirect
240. #399C2×D5semidirect
308. C2×Dic5#123semidirect
415. Dic5C2×#123semidirect
496. C2×C5#2501semidirect
573. C5C2×#2501semidirect
613. C2#10609761direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40402
C22×D5×C4⋊C42
C22×#40292
C22×#40302
C2×C5×#3992
C2×#652812
C2×#652652
C2×#652382
C2×#652462
#106097612
C2×#649662
C2×#649552
C2×#25015