Finite Group Databasebeta

← C2×#10618933 #13007482

Subgroups — C2×#10618933 #13007482

Order 1280.

6826 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#557C211yesmaximal, characteristic, complemented
2#10618933C2132yesmaximal, direct product factor
3C2×#66063C213yesmaximal
4C2×#66056C213yesmaximal, complemented
5C2×#66023C213yesmaximal
6C2×#66019C213yesmaximal, complemented
7C2×#66019C213yesmaximal, complemented
8C2×#66022C213yesmaximal
9C2×#65912C213yesmaximal
10C2×#65906C211yesmaximal, characteristic
11C2×#65906C211yesmaximal, characteristic
12C2×C4×Q8×D5C211yesmaximal, characteristic
13C2×Q8×Dic10C211yesmaximal, characteristic, complemented
14C2×#65903C211yesmaximal, characteristic, complemented
15C2×#65903C211yesmaximal, characteristic, complemented
16C2×#65901C213yesmaximal, complemented
17C22×Q8×D5C2211yescharacteristic
18C5×#557C22116yescomplemented
19C2×C5×C4⋊Q8C2213yes—
20C2×C5×#125C2213yes—
21C2×C5×#125C2213yes—
22C2×C4×C5×Q8C2213yes—
23C2×C4×C5×Q8C2211yescharacteristic
24C2×C4×C5×Q8C2211yescharacteristic
25C2×C4×C5×Q8C2211yescharacteristic
26C2×#4092C2213yes—
27C2×Q8×Dic5C2211yescharacteristic
28C2×Q8×Dic5C2211yescharacteristic
29C2×Q8×Dic5C2211yescharacteristic
30C2×#4090C2216yes—
31C2×#4090C2213yes—
32#66063C22148yes—
33#66056C22148yescomplemented
34#66023C22148yes—
35#66022C22148yes—
36#66019C22148yescomplemented
37#66019C22148yescomplemented
38#65912C22148yes—
39#65906C22116yes—
40#65906C22116yes—
41C4×Q8×D5C22116yes—
42#65903C22116yescomplemented
43#65903C22116yescomplemented
44#65901C22148yescomplemented
45Q8×Dic10C22116yescomplemented
46C2×#4043C2213yes—
47C2×#4042C2211yescharacteristic
48C2×#4042C2213yes—
49C2×#4042C2216yes—
50C2×#4038C2213yes—
51C2×#4038C2213yes—
52C2×D5×C4⋊C4C2213yes—
53C2×#4036C2213yes—
54C2×#4036C2213yes—
55C2×#4035C2213yes—
56C2×#4035C2216yes—
57C2×#4035C2213yes—
58C2×#4035C2216yes—
59C2×#4034C2213yes—
60C2×#4034C2213yes—
61C2×#4034C2213yes—
62C2×#4034C2213yes—
63C2×#4034C2213yes—
64C2×#4034C2216yes—
65C2×#4033C2216yes—
66C2×#4033C2213yes—
67C2×#4033C2213yes—
68C2×#4033C2213yes—
69C2×#4033C2216yes—
70C2×#4018C2213yes—
71C2×#4018C2213yes—
72C2×C42×D5C2213yes—
73C2×#4016C2213yes—
74C2×#4016C2213yes—
75C2×#4015C2213yes—
76C2×C4×Dic10C2213yes—
77C2×C4×Dic10C2213yes—
78C2×C4×Dic10C2213yes—
79C2×C4×Dic10C2213yes—
80C2×C4×Dic10C2213yes—
81C2×C4×Dic10C2213yes—
82C2×#3502—51nomaximal, Sylow
83C22×C5×Q8C2313yes—
84C22×C5×Q8C2311yescharacteristic
85C2×Q8×D5Q8116yescomplemented
86C2×Q8×D5C2318yes—
87C22×C4×D5C2311yescharacteristic
88C22×C4×D5C2313yes—
89C22×C4×D5C2313yes—
90C22×Dic10C2311yescharacteristic
91C22×Dic10C2313yes—
92C22×Dic10C2313yes—
93C22×Dic10C2316yes—
94C22×Dic10C2313yes—
95C5×C4⋊Q8C23124yes—
96C5×#125C23124yes—
97C5×#125C23124yes—
98C4×C5×Q8C2318yes—
99C4×C5×Q8C23124yes—
100C4×C5×Q8C2318yes—
101C4×C5×Q8C2318yes—
102C2×C5×C4⋊C4C2313yes—
103C2×C5×C4⋊C4C2313yes—
104C2×C5×C4⋊C4C2316yes—
105C2×C5×C4⋊C4C2313yes—
106C2×C5×C4⋊C4C2313yes—
107C2×C5×C4⋊C4C2313yes—
108C2×C5×C4⋊C4C2311yescharacteristic
109C2×C42×C5C2313yes—
110C2×C42×C5C2313yes—
111C2×C42×C5C2313yes—
112#4092C23124yes—
113Q8×Dic5C2318yes—
114Q8×Dic5C2318yes—
115Q8×Dic5C2318yes—
116#4090C23124yes—
117#4090C23148yes—
118C2×#685C2311yescharacteristic
119C2×#685C2311yescharacteristic
120C2×#685C2313yes—
121C2×#685C2313yes—
122C2×C4⋊Dic5C2313yes—
123C2×C4⋊Dic5C2313yes—
124C2×C4⋊Dic5C2313yes—
125C2×C4⋊Dic5C2313yes—
126C2×C4⋊Dic5C2313yes—
127C2×C4⋊Dic5C2313yes—
128C2×#683C2311yescharacteristic
129C2×#683C2311yescharacteristic
130C2×#683C2313yes—
131C2×#683C2313yes—
132C2×#683C2313yes—
133C2×#683C2313yes—
134C2×#683C2313yes—
135C2×#683C2316yes—
136C2×#683C2313yes—
137C2×#683C2316yes—
138C2×#683C2316yes—
139C2×#683C2313yes—
140C2×#683C2313yes—
141C2×C4×Dic5C2313yes—
142C2×C4×Dic5C2313yes—
143C2×C4×Dic5C2313yes—
144C2×C4×Dic5C2313yes—
145C2×C4×Dic5C2316yes—
146C2×C4×Dic5C2316yes—
147C2×C4×Dic5C2313yes—
148#4043C23124yes—
149#4042C23148yes—
150#4042C2318yes—
151#4042C23124yes—
152#4038C23124yes—
153#4038C23124yes—
154D5×C4⋊C4C23124yes—
155#4036C23124yes—
156#4036C23124yes—
157#4035C23148yes—
158#4035C23124yes—
159#4035C23148yes—
160#4035C23124yes—
161#4034C23124yes—
162#4034C23124yes—
163#4034C23148yes—
164#4034C23124yes—
165#4034C23124yes—
166#4034C23124yes—
167#4033C23124yes—
168#4033C23124yes—
169#4033C23148yes—
170#4033C23148yes—
171#4033C23124yes—
172#4018C23124yes—
173#4018C23124yes—
174C42×D5C23124yes—
175#4016C23124yes—
176#4016C23124yes—
177#4015C23124yes—
178C4×Dic10C23124yes—
179C4×Dic10C23124yes—
180C4×Dic10C23124yes—
181C4×Dic10C23124yes—
182C4×Dic10C23124yes—
183C4×Dic10C23124yes—
184#3502—532no—
185C2×Q82—53no—
186C2×Q82—51no—
187C2×#557—53no—
188C2×#557—53no—
189C2×#557—53no—
190C2×#557—51no—
191C2×#557—51no—
192C2×#557D511yescharacteristic, complemented
193C2×#544—53no—
194C2×#544—53no—
195C2×#533—53no—
196C2×#533—53no—
197C2×#519—51no—
198C2×#519—51no—
199C22×C4×Q8—51no—
200C23×D5C2411yescharacteristic
201C2×C5×Q8—46no—
202C2×C5×Q8C2×Q818yes—
203C2×C5×Q8C24112yes—
204C2×C5×Q8C2414yes—
205C22×C4×C5C2411yescharacteristic
206C22×C4×C5C2411yescharacteristic
207C22×C4×C5C2411yescharacteristic
208C22×C4×C5C2413yes—
209C22×C4×C5C2413yes—
210C22×C4×C5C2413yes—
211C22×C4×C5C2413yes—
212C22×Dic5C2411yescharacteristic
213C22×Dic5C2413yes—
214C22×Dic5C2413yes—
215C22×Dic5C2411yescharacteristic
216C22×Dic5C2413yes—
217C22×Dic5C2413yes—
218C22×Dic5C2411yescharacteristic
219Q8×D5C2×Q8164yescomplemented
220C2×C4×D5—24no—
221C2×C4×D5—46no—
222C2×C4×D5C2×Q8124yes—
223C2×C4×D5C2414yes—
224C2×C4×D5C24112yes—
225C2×C4×D5C24112yes—
226C2×Dic10—24no—
227C2×Dic10—212no—
228C2×Dic10—46no—
229C2×Dic10—412no—
230C2×Dic10C2×Q8124yes—
231C2×Dic10C2414yes—
232C2×Dic10C24112yes—
233C2×Dic10C24112yes—
234C2×Dic10C24124yes—
235C2×Dic10C24112yes—
236C5×C4⋊C4C2414yes—
237C5×C4⋊C4C24112yes—
238C5×C4⋊C4C24112yes—
239C5×C4⋊C4C24112yes—
240C5×C4⋊C4C24112yes—
241C5×C4⋊C4C24112yes—
242C5×C4⋊C4C24124yes—
243C42×C5C24112yes—
244C42×C5C24112yes—
245C42×C5C24112yes—
246#685C2414yes—
247#685C2414yes—
248#685C24112yes—
249#685C24112yes—
250C4⋊Dic5C24112yes—
251C4⋊Dic5C24112yes—
252C4⋊Dic5C24112yes—
253C4⋊Dic5C24112yes—
254C4⋊Dic5C24112yes—
255C4⋊Dic5C24112yes—
256#683C2414yes—
257#683C2414yes—
258#683C24112yes—
259#683C24112yes—
260#683C24112yes—
261#683C24112yes—
262#683C24112yes—
263#683C24124yes—
264#683C24112yes—
265#683C24124yes—
266#683C24124yes—
267#683C24112yes—
268#683C24112yes—
269C4×Dic5C24112yes—
270C4×Dic5C24112yes—
271C4×Dic5C24112yes—
272C4×Dic5C24112yes—
273C4×Dic5C24124yes—
274C4×Dic5C24124yes—
275C4×Dic5C24112yes—
276C23×Q8—51no—
277Q82—548no—
278Q82—516no—
279#557—548no—
280#557—548no—
281#557—548no—
282#557—516no—
283#557—516no—
284#557C2×D5116yescomplemented
285#544—548no—
286#544—548no—
287#533—548no—
288#533—548no—
289C2×C4⋊Q8—53no—
290C2×C4⋊Q8—56no—
291C2×C4⋊Q8—53no—
292C2×C4⋊Q8—53no—
293C2×C4⋊Q8—53no—
294C2×C4⋊Q8—56no—
295C2×C4⋊Q8—53no—
296C2×C4⋊Q8—53no—
297C2×C4⋊Q8—53no—
298C2×C4⋊Q8C2×D513yes—
299C2×#125—53no—
300C2×#125—53no—
301C2×#125—56no—
302C2×#125—56no—
303C2×#125—53no—
304C2×#125—53no—
305C2×#125—53no—
306C2×#125—53no—
307C2×#125C2×D513yes—
308C2×#125C2×D513yes—
309C2×C22⋊Q8—53no—
310C2×C22⋊Q8—53no—
311C2×C22⋊Q8—51no—
312C2×C22⋊Q8—56no—
313C2×C22⋊Q8—53no—
314#519—516no—
315#519—516no—
316C2×C4×Q8—51no—
317C2×C4×Q8—51no—
318C2×C4×Q8—51no—
319C2×C4×Q8—516no—
320C2×C4×Q8—53no—
321C2×C4×Q8—53no—
322C2×C4×Q8—53no—
323C2×C4×Q8—56no—
324C2×C4×Q8—56no—
325C2×C4×Q8—53no—
326C2×C4×Q8—53no—
327C2×C4×Q8—53no—
328C2×C4×Q8—53no—
329C2×C4×Q8—53no—
330C2×C4×Q8—53no—
331C2×C4×Q8C2×D511yescharacteristic
332C2×C4×Q8C2×D511yescharacteristic
333C2×C4×Q8C2×D511yescharacteristic
334C2×C4×Q8C2×D513yes—
335C2×#117—53no—
336C2×#117—53no—
337C2×#117—53no—
338C2×#117—53no—
339C22×C4⋊C4—53no—
340C22×C42—53no—
341C23×C5C2511yescharacteristic
342C22×D5—22no—
343C22×D5—22no—
344C22×D5C22×Q814yes—
345C22×D5C2512yes—
346C5×Q8—412no—
347C5×Q8C22×Q8116yes—
348C2×C4×C5—22no—
349C2×C4×C5—26no—
350C2×C4×C5—22no—
351C2×C4×C5—26no—
352C2×C4×C5—26no—
353C2×C4×C5—22no—
354C2×C4×C5C22×Q8112yes—
355C2×C4×C5C2512yes—
356C2×C4×C5C2516yes—
357C2×C4×C5C2516yes—
358C2×C4×C5C2516yes—
359C2×C4×C5C2512yes—
360C2×C4×C5C2512yes—
361C2×C4×C5C2516yes—
362C2×Dic5—22no—
363C2×Dic5—26no—
364C2×Dic5—22no—
365C2×Dic5—26no—
366C2×Dic5—26no—
367C2×Dic5C22×Q8112yes—
368C2×Dic52−1+414yes—
369C2×Dic5C2512yes—
370C2×Dic5C2516yes—
371C2×Dic5C2512yes—
372C2×Dic5C2516yes—
373C2×Dic5C2516yes—
374C2×Dic5C2516yes—
375C2×Dic5C2512yes—
376C4×D5—28no—
377C4×D5—412no—
378C4×D5C22×Q8148yes—
379Dic10—28no—
380Dic10—224no—
381Dic10—412no—
382Dic10—424no—
383Dic10C22×Q8148yes—
384C22×Q8—516no—
385C22×Q8—58no—
386C22×Q8—53no—
387C22×Q8—53no—
388C22×Q8—51no—
389C22×Q8—53no—
390C22×Q8—56no—
391C22×Q8C22×D513yes—
392C22×Q8C22×D511yescharacteristic
393C23×C4—53no—
394C23×C4—51no—
395C23×C4—53no—
396C4⋊Q8—524no—
397C4⋊Q8—548no—
398C4⋊Q8—524no—
399C4⋊Q8—524no—
400C4⋊Q8—524no—
401C4⋊Q8—548no—
402C4⋊Q8—524no—
403C4⋊Q8—524no—
404C4⋊Q8—524no—
405C4⋊Q8C22×D5124yes—
406#125—524no—
407#125—524no—
408#125—548no—
409#125—548no—
410#125—524no—
411#125—524no—
412#125—524no—
413#125—524no—
414#125C22×D5124yes—
415#125C22×D5124yes—
416C22⋊Q8—524no—
417C22⋊Q8—524no—
418C22⋊Q8—58no—
419C22⋊Q8—548no—
420C22⋊Q8—524no—
421C4×Q8—58no—
422C4×Q8—58no—
423C4×Q8—58no—
424C4×Q8—524no—
425C4×Q8—524no—
426C4×Q8—524no—
427C4×Q8—548no—
428C4×Q8—548no—
429C4×Q8—524no—
430C4×Q8—524no—
431C4×Q8—524no—
432C4×Q8—524no—
433C4×Q8—524no—
434C4×Q8—524no—
435C4×Q8C22×D518yes—
436C4×Q8C22×D518yes—
437C4×Q8C22×D518yes—
438C4×Q8C22×D5124yes—
439#117—524no—
440#117—524no—
441#117—524no—
442#117—524no—
443C2×C4⋊C4—56no—
444C2×C4⋊C4—53no—
445C2×C4⋊C4—53no—
446C2×C4⋊C4—53no—
447C2×C4⋊C4—53no—
448C2×C4⋊C4—53no—
449C2×C4⋊C4—53no—
450C2×C4⋊C4—53no—
451C2×C4⋊C4—53no—
452C2×C4⋊C4—51no—
453C2×C4⋊C4—53no—
454C2×C4⋊C4—51no—
455C2×C4⋊C4—56no—
456C2×C4⋊C4—53no—
457C2×C4⋊C4—56no—
458C2×C4⋊C4—53no—
459C2×C4⋊C4—53no—
460C2×C4⋊C4—53no—
461C2×C4⋊C4—53no—
462C2×C4⋊C4—524no—
463C2×C4⋊C4C22×D513yes—
464C2×C4⋊C4C22×D513yes—
465C2×C4⋊C4C22×D513yes—
466C2×C4⋊C4C22×D513yes—
467C2×C4⋊C4C22×D516yes—
468C2×C4⋊C4C22×D511yescharacteristic
469C2×C4⋊C4C22×D513yes—
470C2×C22⋊C4—53no—
471C2×C22⋊C4—51no—
472C2×C22⋊C4—53no—
473C2×C22⋊C4—51no—
474C2×C42—56no—
475C2×C42—56no—
476C2×C42—53no—
477C2×C42—53no—
478C2×C42—53no—
479C2×C42—53no—
480C2×C42—53no—
481C2×C42—524no—
482C2×C42C22×D513yes—
483C2×C42C22×D513yes—
484C2×C42C22×D513yes—
485C22×C5C23×Q812yes—
486C22×C5C2×2−1+412yes—
487C22×C5#58512yes—
488C22×C5C2611yescharacteristic
489C2×D5—22no—
490C2×D5—22no—
491C2×D5—28no—
492C2×D5C23×Q814yes—
493C4×C5—26no—
494C4×C5—22no—
495C4×C5—22no—
496C4×C5—26no—
497C4×C5—26no—
498C4×C5—22no—
499C4×C5C23×Q8112yes—
500Dic5—22no—
501Dic5—26no—
502Dic5—22no—
503Dic5—26no—
504Dic5—26no—
505Dic5C23×Q8112yes—
506Dic5C2×2−1+414yes—
507C24—51no—
508C2×Q8—46no—
509C2×Q8—564no—
510C2×Q8—524no—
511C2×Q8—512no—
512C2×Q8—512no—
513C2×Q8—54no—
514C2×Q8—512no—
515C2×Q8—524no—
516C2×Q8—1012no—
517C2×Q8—104no—
518C2×Q8—206no—
519C2×Q8—2012no—
520C2×Q8Q8×D518yes—
521C2×Q8C23×D5112yes—
522C2×Q8C23×D514yes—
523C22×C4—53no—
524C22×C4—51no—
525C22×C4—524no—
526C22×C4—512no—
527C22×C4—51no—
528C22×C4—53no—
529C22×C4—51no—
530C22×C4—53no—
531C22×C4—53no—
532C22×C4—54no—
533C22×C4—512no—
534C22×C4—104no—
535C22×C4—206no—
536C22×C4C23×D511yescharacteristic
537C22×C4C23×D513yes—
538C22×C4C23×D513yes—
539C22×C4C23×D513yes—
540C22×C4C23×D511yescharacteristic
541C22×C4C23×D511yescharacteristic
542C22×C4C23×D513yes—
543C4⋊C4—524no—
544C4⋊C4—512no—
545C4⋊C4—512no—
546C4⋊C4—512no—
547C4⋊C4—512no—
548C4⋊C4—512no—
549C4⋊C4—512no—
550C4⋊C4—512no—
551C4⋊C4—512no—
552C4⋊C4—54no—
553C4⋊C4—512no—
554C4⋊C4—54no—
555C4⋊C4—524no—
556C4⋊C4—512no—
557C4⋊C4—524no—
558C4⋊C4—512no—
559C4⋊C4—512no—
560C4⋊C4—512no—
561C4⋊C4—512no—
562C4⋊C4C23×D5112yes—
563C4⋊C4C23×D5112yes—
564C4⋊C4C23×D5112yes—
565C4⋊C4C23×D5112yes—
566C4⋊C4C23×D5124yes—
567C4⋊C4C23×D514yes—
568C4⋊C4C23×D5112yes—
569C22⋊C4—512no—
570C22⋊C4—54no—
571C22⋊C4—512no—
572C22⋊C4—54no—
573C42—512no—
574C42—512no—
575C42—512no—
576C42—524no—
577C42—524no—
578C42—512no—
579C42—512no—
580C42C23×D5112yes—
581C42C23×D5112yes—
582C42C23×D5112yes—
583C2×C5#350214yescomplemented
584C2×C5C24×Q811yescharacteristic
585C2×C5C22×2−1+411yescharacteristic
586C2×C5C2×#58511yescharacteristic
587D5—24no—
588C23—54no—
589C23—52no—
590C23—102no—
591C23—102no—
592C23C24×D511yescharacteristic
593Q8—412no—
594Q8—548no—
595Q8—1024no—
596Q8—108no—
597Q8—2012no—
598Q8—2024no—
599Q8C2×Q8×D5116yes—
600C2×C4—26no—
601C2×C4—22no—
602C2×C4—26no—
603C2×C4—22no—
604C2×C4—26no—
605C2×C4—22no—
606C2×C4—512no—
607C2×C4—54no—
608C2×C4—56no—
609C2×C4—52no—
610C2×C4—548no—
611C2×C4—52no—
612C2×C4—56no—
613C2×C4—52no—
614C2×C4—56no—
615C2×C4—56no—
616C2×C4—108no—
617C2×C4—106no—
618C2×C4—102no—
619C2×C4—106no—
620C2×C4—102no—
621C2×C4—106no—
622C2×C4—2012no—
623C2×C4C2×Q8×D5112yes—
624C2×C4C24×D512yes—
625C2×C4C24×D516yes—
626C2×C4C24×D516yes—
627C2×C4C24×D516yes—
628C2×C4C24×D512yes—
629C2×C4C24×D512yes—
630C2×C4C24×D516yes—
631C5C2×#350211yescharacteristic, complemented, Sylow
632C22—54no—
633C22—102no—
634C22—102no—
635C22—108no—
636C22C22×Q8×D512yes—
637C22#6628312yes—
638C22D5×2−1+412yes—
639C22C25×D511yescharacteristic
640C4—26no—
641C4—22no—
642C4—26no—
643C4—22no—
644C4—26no—
645C4—22no—
646C4—512no—
647C4—54no—
648C4—106no—
649C4—102no—
650C4—106no—
651C4—102no—
652C4—106no—
653C4C22×Q8×D5112yes—
654C2—104no—
655C2#1061893314yesdirect product factor
656C2C23×Q8×D511yescharacteristic
657C2C2×#6628311yescharacteristic
658C2C2×D5×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×#557C2
socleC23×C5C25
prime radicalC23×D5C24
hypercenterC2×#557D5
hypocenterC5C2×#3502
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C5×#557C2semidirect
2. #10618933C2direct (N × H)
4. C2×#66056C2semidirect
6. C2×#66019C2semidirect
7. C2×#66019C2semidirect
13. C2×Q8×Dic10C2semidirect
14. C2×#65903C2semidirect
15. C2×#65903C2semidirect
16. C2×#65901C2semidirect
18. C5×#557C22semidirect
33. #66056C22semidirect
36. #66019C22semidirect
37. #66019C22semidirect
42. #65903C22semidirect
43. #65903C22semidirect
44. #65901C22semidirect
45. Q8×Dic10C22semidirect
85. C2×Q8×D5Q8semidirect
192. C2×#557D5semidirect
219. Q8×D5C2×Q8semidirect
284. #557C2×D5semidirect
583. C2×C5#3502semidirect
631. C5C2×#3502semidirect
655. C2#10618933direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5572
#106189332
C2×#660632
C2×#660562
C2×#660232
C2×#660192
C2×#660222
C2×#659122
C2×#659062
C2×C4×Q8×D52
C2×Q8×Dic102
C2×#659032
C2×#659012
C2×#35025