Finite Group Databasebeta

← C2×#10607941 #12267713

Subgroups — C2×#10607941 #12267713

Order 1280.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4043C211yesmaximal, characteristic, complemented
2C22×#4043C211yesmaximal, characteristic, complemented
3C22×#4038C211yesmaximal, characteristic, complemented
4C22×C4×D20C211yesmaximal, characteristic
5C2×C5×#389C211yesmaximal, characteristic, complemented
6C2×#65280C212yesmaximal
7C2×#65276C211yesmaximal, characteristic, complemented
8C2×#65274C212yesmaximal, complemented
9C2×#65218C211yesmaximal, characteristic, complemented
10#10607941C2116yesmaximal, direct product factor
11C2×#64963C212yesmaximal, complemented
12C2×#64959C212yesmaximal, complemented
13C23×D20C2211yescharacteristic
14C23×C4×D5C2211yescharacteristic
15C23×C4×D5C2211yescharacteristic
16C23×Dic10C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C4⋊C4C2212yes—
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C42×C5C2211yescharacteristic
21C22×#685C2211yescharacteristic
22C22×#685C2212yes—
23C22×#685C2212yes—
24C22×#685C2212yes—
25C22×#685C2211yescharacteristic
26C22×#685C2212yes—
27C22×C4⋊Dic5C2212yes—
28C22×C4⋊Dic5C2212yes—
29C22×C4⋊Dic5C2211yescharacteristic
30C22×C4⋊Dic5C2211yescharacteristic
31C22×C4×Dic5C2212yes—
32C2×#4043—28no—
33C2×#4043—28no—
34C2×#4043C2218yescomplemented
35C2×#4043C2218yescomplemented
36C2×#4038—28no—
37C2×#4038C2218yescomplemented
38C2×C4×D20—28no—
39C2×C4×D20C2218yes—
40C5×#389C2218yescomplemented
41C2×C5×#95C2212yes—
42C2×#3963C2212yes—
43C2×#3963C2212yes—
44C2×#3963C2212yes—
45#65280C22116yes—
46#65276C2218yescomplemented
47#65274C22116yescomplemented
48#65218C2218yescomplemented
49#64963C22116yescomplemented
50#64959C22116yescomplemented
51C2×#2415—51nomaximal, Sylow
52C24×D5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C23×C4×C5C2311yescharacteristic
55C23×C4×C5C2312yes—
56C23×C4×C5C2312yes—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2311yescharacteristic
59C23×Dic5C2312yes—
60C23×Dic5C2311yescharacteristic
61C23×Dic5C2312yes—
62C23×Dic5C2311yescharacteristic
63C22×D20—24no—
64C22×D20—24no—
65C22×D20—24no—
66C22×D20C2314yes—
67C22×C4×D5—24no—
68C22×C4×D5—24no—
69C22×C4×D5—24no—
70C22×C4×D5—24no—
71C22×C4×D5—24no—
72C22×C4×D5D418yes—
73C22×C4×D5C2314yes—
74C22×C4×D5C2314yes—
75C22×Dic10—24no—
76C22×Dic10—24no—
77C22×Dic10—24no—
78C22×Dic10C2314yes—
79C2×C5×C4⋊C4—24no—
80C2×C5×C4⋊C4—28no—
81C2×C5×C4⋊C4D418yescomplemented
82C2×C5×C4⋊C4C2318yes—
83C2×C5×C4⋊C4C2314yes—
84C2×C5×C4⋊C4C2314yes—
85C2×C42×C5—24no—
86C2×C42×C5C2314yes—
87C2×#685—24no—
88C2×#685—28no—
89C2×#685—28no—
90C2×#685—28no—
91C2×#685—28no—
92C2×#685—24no—
93C2×#685C2318yes—
94C2×#685C2318yes—
95C2×#685C2318yes—
96C2×#685C2318yes—
97C2×#685C2314yes—
98C2×#685C2314yes—
99C2×C4⋊Dic5—24no—
100C2×C4⋊Dic5—28no—
101C2×C4⋊Dic5—24no—
102C2×C4⋊Dic5—28no—
103C2×C4⋊Dic5C2314yes—
104C2×C4⋊Dic5C2314yes—
105C2×C4⋊Dic5C2318yes—
106C2×C4⋊Dic5C2318yes—
107C2×C4×Dic5—28no—
108C2×C4×Dic5C2318yes—
109#4043—232no—
110#4043—232no—
111#4038—232no—
112C4×D20—232no—
113C5×#95C2318yes—
114#3963C2318yes—
115#3963C2318yes—
116#3963C2318yes—
117C22×C22⋊Q8—51no—
118C22×C22⋊Q8—51no—
119C22×C4×D4—51no—
120C22×#117—51no—
121#2415—516no—
122C2×#397—52no—
123C2×#394—52no—
124C2×#391—51no—
125C2×#389—52no—
126C2×#389D511yescharacteristic, complemented
127C2×#388—52no—
128C2×#380—51no—
129C24×C5C2411yescharacteristic
130C23×D5—22no—
131C23×D5—22no—
132C23×D5—22no—
133C23×D5—22no—
134C23×D5—22no—
135C23×D5—22no—
136C23×D5—22no—
137C23×D5—22no—
138C23×D5—22no—
139C23×D5—22no—
140C23×D5—22no—
141C23×D5—22no—
142C23×D5C2×D414yes—
143C23×D5#4114yes—
144C23×D5C2412yes—
145C23×D5C2412yes—
146C22×C4×C5—22no—
147C22×C4×C5—24no—
148C22×C4×C5—22no—
149C22×C4×C5—24no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—24no—
153C22×C4×C5—24no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5C2×D418yes—
157C22×C4×C5C2×D414yes—
158C22×C4×C5C2×D414yes—
159C22×C4×C5#4114yes—
160C22×C4×C5#4114yes—
161C22×C4×C5C2412yes—
162C22×C4×C5C2414yes—
163C22×C4×C5C2414yes—
164C22×C4×C5C2412yes—
165C22×C4×C5C2412yes—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—24no—
173C22×Dic5—24no—
174C22×Dic5—22no—
175C22×Dic5—24no—
176C22×Dic5—24no—
177C22×Dic5C2×D414yes—
178C22×Dic5C2414yes—
179C22×Dic5C2412yes—
180C22×Dic5C2414yes—
181C22×Dic5C2412yes—
182C2×D20—232no—
183C2×D20—28no—
184C2×D20—28no—
185C2×D20—28no—
186C2×C4×D5—232no—
187C2×C4×D5—28no—
188C2×C4×D5—28no—
189C2×C4×D5—28no—
190C2×C4×D5—28no—
191C2×C4×D5—28no—
192C2×C4×D5—416no—
193C2×C4×D5C2×D4116yes—
194C2×Dic10—28no—
195C2×Dic10—28no—
196C2×Dic10—28no—
197C2×Dic10—416no—
198C5×C4⋊C4—28no—
199C5×C4⋊C4—216no—
200C5×C4⋊C4C2×D4116yescomplemented
201C42×C5—28no—
202#685—28no—
203#685—216no—
204#685—216no—
205#685—216no—
206#685—216no—
207#685—28no—
208C4⋊Dic5—28no—
209C4⋊Dic5—216no—
210C4⋊Dic5—28no—
211C4⋊Dic5—216no—
212C4×Dic5—216no—
213C23×Q8—51no—
214C23×D4—51no—
215C24×C4—51no—
216C24×C4—51no—
217C2×C22⋊Q8—58no—
218C2×C22⋊Q8—58no—
219C2×C22⋊Q8—108no—
220C2×C22⋊Q8—108no—
221C2×C4×D4—58no—
222C2×C4×D4—108no—
223C2×#117—58no—
224C2×#117—108no—
225C22×C4⋊C4—52no—
226C22×C4⋊C4—51no—
227C22×C4⋊C4—51no—
228C22×C4⋊C4—52no—
229C22×C4⋊C4C2×D512yes—
230C22×C4⋊C4C2×D511yescharacteristic
231C22×C4⋊C4C2×D511yescharacteristic
232C22×C22⋊C4—52no—
233C22×C22⋊C4—52no—
234C22×C22⋊C4—52no—
235C22×C22⋊C4—52no—
236C22×C22⋊C4—51no—
237C22×C22⋊C4—51no—
238C22×C42—52no—
239C22×C42C2×D511yescharacteristic
240#397—516no—
241#394—516no—
242#391—58no—
243#389—516no—
244#389C2×D518yescomplemented
245#388—516no—
246#380—58no—
247C2×#95—52no—
248C2×#95—52no—
249C2×#95—52no—
250C2×#95C2×D512yes—
251C23×C5C22×D412yes—
252C23×C5C22×D412yes—
253C23×C5C2×#4112yes—
254C23×C5C2×#4112yes—
255C23×C5C2×#4112yes—
256C23×C52−1+412yes—
257C23×C52−1+412yes—
258C23×C5C2511yescharacteristic
259C22×D5—28no—
260C22×D5—28no—
261C22×D5—28no—
262C22×D5—28no—
263C22×D5—28no—
264C22×D5—28no—
265C22×D5—22no—
266C22×D5—22no—
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—22no—
270C22×D5—22no—
271C22×D5—22no—
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—22no—
275C22×D5—22no—
276C22×D5—22no—
277C22×D5—44no—
278C22×D5—44no—
279C22×D5—44no—
280C22×D5—44no—
281C22×D5—44no—
282C22×D5—44no—
283C22×D5—44no—
284C22×D5—44no—
285C22×D5C22×D414yes—
286C22×D5C2×#4114yes—
287C2×C4×C5—28no—
288C2×C4×C5—216no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5—22no—
292C2×C4×C5—24no—
293C2×C4×C5—24no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—24no—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—48no—
300C2×C4×C5—44no—
301C2×C4×C5#121116yes—
302C2×C4×C5C22×D418yes—
303C2×C4×C5C22×D414yes—
304C2×C4×C5C22×D414yes—
305C2×C4×C5C2×#4114yes—
306C2×C4×C5C2×#4114yes—
307C2×Dic5—28no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—24no—
311C2×Dic5—22no—
312C2×Dic5—24no—
313C2×Dic5—24no—
314C2×Dic5—22no—
315C2×Dic5—24no—
316C2×Dic5—24no—
317C2×Dic5—22no—
318C2×Dic5—24no—
319C2×Dic5—48no—
320C2×Dic5—44no—
321C2×Dic5—48no—
322C2×Dic5C22×D414yes—
323D20—232no—
324C4×D5—232no—
325C4×D5—416no—
326Dic10—416no—
327C25—51no—
328C25—51no—
329C22×Q8—54no—
330C22×Q8—104no—
331C22×Q8—104no—
332C22×Q8—104no—
333C22×D4—54no—
334C22×D4—104no—
335C22×D4—104no—
336C22×D4—104no—
337C23×C4—51no—
338C23×C4—58no—
339C23×C4—54no—
340C23×C4—52no—
341C23×C4—52no—
342C23×C4—51no—
343C23×C4—54no—
344C23×C4—104no—
345C23×C4—104no—
346C23×C4—104no—
347C23×C4—104no—
348C23×C4—104no—
349C23×C4C22×D512yes—
350C23×C4C22×D511yescharacteristic
351C23×C4C22×D512yes—
352C23×C4C22×D511yescharacteristic
353C23×C4C22×D511yescharacteristic
354C22⋊Q8—1032no—
355C22⋊Q8—1032no—
356C4×D4—1032no—
357#117—1032no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—28no—
360C2×C4⋊C4—54no—
361C2×C4⋊C4—54no—
362C2×C4⋊C4—58no—
363C2×C4⋊C4—58no—
364C2×C4⋊C4—104no—
365C2×C4⋊C4—108no—
366C2×C4⋊C4—104no—
367C2×C4⋊C4—108no—
368C2×C4⋊C4D2018yescomplemented
369C2×C4⋊C4C22×D514yes—
370C2×C4⋊C4C22×D518yes—
371C2×C4⋊C4C22×D514yes—
372C2×C22⋊C4—58no—
373C2×C22⋊C4—54no—
374C2×C22⋊C4—54no—
375C2×C22⋊C4—58no—
376C2×C22⋊C4—58no—
377C2×C22⋊C4—58no—
378C2×C22⋊C4—104no—
379C2×C22⋊C4—104no—
380C2×C22⋊C4—108no—
381C2×C22⋊C4—108no—
382C2×C22⋊C4—108no—
383C2×C22⋊C4—108no—
384C2×C42—24no—
385C2×C42—58no—
386C2×C42—108no—
387C2×C42C22×D514yes—
388#95—58no—
389#95—58no—
390#95—58no—
391#95C22×D518yes—
392C22×C5C2×#12114yes—
393C22×C5#52514yes—
394C22×C5#53614yes—
395C22×C5#54714yes—
396C22×C5#54814yes—
397C22×C5#55214yes—
398C22×C5#55614yes—
399C22×C5C23×D411yescharacteristic
400C22×C5C23×D411yescharacteristic
401C22×C5C22×#4111yescharacteristic
402C22×C5C22×#4111yescharacteristic
403C22×C5C22×#4111yescharacteristic
404C22×C5C2×2−1+411yescharacteristic
405C22×C5C2×2−1+411yescharacteristic
406C2×D5—24no—
407C2×D5—24no—
408C2×D5—24no—
409C2×D5—24no—
410C2×D5—24no—
411C2×D5—24no—
412C2×D5—42no—
413C2×D5—416no—
414C2×D5—416no—
415C2×D5—42no—
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—42no—
420C2×D5—42no—
421C2×D5—42no—
422C4×C5—24no—
423C4×C5—28no—
424C4×C5—44no—
425C4×C5—42no—
426C4×C5C2×#12118yes—
427Dic5—24no—
428Dic5—44no—
429Dic5—42no—
430Dic5—44no—
431C24—54no—
432C24—54no—
433C24—52no—
434C24—52no—
435C24—102no—
436C24—102no—
437C24—102no—
438C24—102no—
439C24—102no—
440C24—102no—
441C24—102no—
442C24—102no—
443C24—102no—
444C24—102no—
445C24—102no—
446C24—102no—
447C24C23×D511yescharacteristic
448C2×Q8—108no—
449C2×Q8—108no—
450C2×Q8—108no—
451C2×Q8—2016no—
452C2×D4—108no—
453C2×D4—108no—
454C2×D4—1032no—
455C2×D4—108no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—54no—
467C22×C4—52no—
468C22×C4—516no—
469C22×C4—54no—
470C22×C4—54no—
471C22×C4—52no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—108no—
475C22×C4—108no—
476C22×C4—1032no—
477C22×C4—104no—
478C22×C4—104no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—104no—
482C22×C4—102no—
483C22×C4—104no—
484C22×C4—104no—
485C22×C4—104no—
486C22×C4—108no—
487C22×C4—108no—
488C22×C4—108no—
489C22×C4—2016no—
490C22×C4C2×D2018yes—
491C22×C4C2×D2014yes—
492C22×C4D4×D514yes—
493C22×C4#71114yes—
494C22×C4#71314yes—
495C22×C4C23×D512yes—
496C22×C4C23×D514yes—
497C22×C4C23×D514yes—
498C22×C4C23×D512yes—
499C22×C4C23×D512yes—
500C4⋊C4—28no—
501C4⋊C4—216no—
502C4⋊C4—108no—
503C4⋊C4—1016no—
504C4⋊C4—108no—
505C4⋊C4—1016no—
506C4⋊C4C2×D20116yescomplemented
507C22⋊C4—108no—
508C22⋊C4—108no—
509C22⋊C4—1016no—
510C22⋊C4—1016no—
511C22⋊C4—1016no—
512C22⋊C4—1016no—
513C42—28no—
514C42—1016no—
515C2×C5#241518yescomplemented
516C2×C5C22×#12111yescharacteristic
517C2×C5C2×#52511yescharacteristic
518C2×C5C2×#53611yescharacteristic
519C2×C5C2×#54711yescharacteristic
520C2×C5C2×#54811yescharacteristic
521C2×C5C2×#55211yescharacteristic
522C2×C5C2×#55611yescharacteristic
523D5—44no—
524D5—44no—
525C23—54no—
526C23—54no—
527C23—108no—
528C23—108no—
529C23—108no—
530C23—108no—
531C23—108no—
532C23—102no—
533C23—102no—
534C23—102no—
535C23—102no—
536C23—102no—
537C23—102no—
538C23—102no—
539C23—102no—
540C23—102no—
541C23—102no—
542C23—102no—
543C23—102no—
544C23—108no—
545C23—204no—
546C23—204no—
547C23—204no—
548C23—204no—
549C23—204no—
550C23—204no—
551C23—204no—
552C23—204no—
553C23C22×D2012yes—
554C23C2×D4×D512yes—
555C23C2×#71112yes—
556C23C2×#71312yes—
557C23#414712yes—
558C23D5×#4112yes—
559C23#415012yes—
560C23C24×D511yescharacteristic
561Q8—2016no—
562D4—1032no—
563C2×C4—216no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—54no—
578C2×C4—108no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—1032no—
582C2×C4—104no—
583C2×C4—104no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—104no—
587C2×C4—102no—
588C2×C4—104no—
589C2×C4—104no—
590C2×C4—104no—
591C2×C4—204no—
592C2×C4—208no—
593C2×C4—208no—
594C2×C4—2016no—
595C2×C4#4041116yes—
596C2×C4C22×D2018yes—
597C2×C4C22×D2014yes—
598C2×C4C2×D4×D514yes—
599C2×C4C2×#71114yes—
600C2×C4C2×#71314yes—
601C5C2×#241511yescharacteristic, complemented, Sylow
602C22—104no—
603C22—104no—
604C22—104no—
605C22—104no—
606C22—104no—
607C22—104no—
608C22—202no—
609C22—202no—
610C22—202no—
611C22—202no—
612C22—2016no—
613C22—2016no—
614C22—202no—
615C22—202no—
616C22—202no—
617C22—202no—
618C22C2×#404114yes—
619C22#6588914yes—
620C22#6591014yes—
621C22#6594014yes—
622C22#6596214yes—
623C22#6602814yes—
624C22#6606614yes—
625C22C23×D2011yescharacteristic
626C22C22×D4×D511yescharacteristic
627C22C22×#71111yescharacteristic
628C22C22×#71311yescharacteristic
629C22C2×#414711yescharacteristic
630C22C2×D5×#4111yescharacteristic
631C22C2×#415011yescharacteristic
632C4—28no—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—104no—
637C4—202no—
638C4—204no—
639C4—204no—
640C4C2×#404118yes—
641C2—204no—
642C2—204no—
643C2#1060794118yesdirect product factor
644C2C22×#404111yescharacteristic
645C2C2×#6588911yescharacteristic
646C2C2×#6591011yescharacteristic
647C2C2×#6594011yescharacteristic
648C2C2×#6596211yescharacteristic
649C2C2×#6602811yescharacteristic
650C2C2×#6606611yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#389C2
socleC24×C5C24
prime radicalC23×D20C22
hypercenterC2×#389D5
hypocenterC5C2×#2415
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#4043C2semidirect
2. C22×#4043C2semidirect
3. C22×#4038C2semidirect
5. C2×C5×#389C2semidirect
7. C2×#65276C2semidirect
8. C2×#65274C2semidirect
9. C2×#65218C2semidirect
10. #10607941C2direct (N × H)
11. C2×#64963C2semidirect
12. C2×#64959C2semidirect
34. C2×#4043C22semidirect
35. C2×#4043C22semidirect
37. C2×#4038C22semidirect
40. C5×#389C22semidirect
46. #65276C22semidirect
47. #65274C22semidirect
48. #65218C22semidirect
49. #64963C22semidirect
50. #64959C22semidirect
81. C2×C5×C4⋊C4D4semidirect
126. C2×#389D5semidirect
200. C5×C4⋊C4C2×D4semidirect
244. #389C2×D5semidirect
368. C2×C4⋊C4D20semidirect
506. C4⋊C4C2×D20semidirect
515. C2×C5#2415semidirect
601. C5C2×#2415semidirect
643. C2#10607941direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40432
C22×#40382
C22×C4×D202
C2×C5×#3892
C2×#652802
C2×#652762
C2×#652742
C2×#652182
#106079412
C2×#649632
C2×#649592
C2×#24155