Finite Group Databasebeta

← C2×#10600831 #12126726

Subgroups — C2×#10600831 #12126726

Order 1280.

1234 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65851C211yesmaximal, characteristic, complemented
2C2×C5×#410C211yesmaximal, characteristic
3C2×#65525C211yesmaximal, characteristic
4C22×#3969C211yesmaximal, characteristic
5C2×#64753C211yesmaximal, characteristic, complemented
6C2×#64751C211yesmaximal, characteristic, complemented
7C2×#64752C211yesmaximal, characteristic, complemented
8#10600831C218yesmaximal, direct product factor
9C22×C5×#41C2211yescharacteristic
10C2×C5×#117C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C2×#4072C2211yescharacteristic
13C22×C4×Dic5C2211yescharacteristic
14C22×#681C2211yescharacteristic
15C22×#681C2211yescharacteristic
16#65851—22no—
17#65851—22no—
18#65851—22no—
19#65851C2214yescomplemented
20C2×#4036—21no—
21C2×#4034—21no—
22C2×#4024—22no—
23C2×C4×Dic10—22no—
24C5×#410—22no—
25C5×#410C2214yes—
26C2×C5×#99—22no—
27#65525—22no—
28#65525C2214yes—
29C2×#3969—21no—
30C2×#3969—21no—
31C2×#3969—22no—
32C2×#3969—22no—
33C2×#3969C2214yes—
34#64753C2214yescomplemented
35#64752C2214yescomplemented
36#64751C2214yescomplemented
37C2×#2031—51nomaximal, Sylow
38C2×C5×#41—21no—
39C2×C5×#41—21no—
40C2×C5×#41—21no—
41C2×C5×#41—21no—
42C2×C5×#41—21no—
43C2×C5×#41—21no—
44C2×C5×#41—21no—
45C2×C5×#41—21no—
46C2×C5×#41—21no—
47C2×C5×#41—21no—
48C2×C5×#41Q812yescomplemented
49C2×C5×#41C2312yes—
50C22×C5×Q8Q811yescharacteristic, complemented
51C22×C5×D4—21no—
52C22×C5×D4Q811yescharacteristic, complemented
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C23×Dic5—21no—
56C22×Dic10—22no—
57C5×#117—21no—
58C5×#117—21no—
59C5×#117—21no—
60C5×#117—21no—
61C5×#117—21no—
62C5×#117D412yes—
63C5×#117C2312yes—
64C5×#117C2312yes—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4D412yes—
69C2×C42×C5—21no—
70C2×C42×C5—21no—
71C2×C22⋊Dic5—21no—
72#4072—21no—
73#4072—21no—
74#4072D412yes—
75#4072C2312yes—
76C2×C4⋊Dic5—22no—
77C2×C4⋊Dic5—21no—
78C2×#683—22no—
79C2×#683—22no—
80C2×C4×Dic5—21no—
81C2×C4×Dic5—21no—
82C2×C4×Dic5—21no—
83C2×C4×Dic5D412yes—
84C2×C4×Dic5D411yescharacteristic
85C2×C4×Dic5D411yescharacteristic
86C2×C4×Dic5D412yes—
87C2×C4×Dic5C2312yes—
88C2×#681—21no—
89C2×#681—21no—
90C2×#681—21no—
91C2×#681—21no—
92C2×#681—21no—
93C2×#681—21no—
94C2×#681—21no—
95C2×#681—21no—
96C2×#681—21no—
97C2×#681—21no—
98C2×#681C2312yes—
99C2×#681C2312yes—
100C22×#168—21no—
101C22×#168—21no—
102#4036—24no—
103#4036—24no—
104#4034—24no—
105#4034—24no—
106#4024—28no—
107#4024—28no—
108C4×Dic10—28no—
109C4×Dic10—28no—
110C5×#99—28no—
111#3969—24no—
112#3969—24no—
113#3969—28no—
114C2×#533—51no—
115C2×#421—51no—
116C22×#104—51no—
117C2×#410D511yescharacteristic
118#2031—58no—
119C2×#344—51no—
120C2×#339—51no—
121C2×H(3,ℤ/4)—51no—
122C24×C5—21no—
123C24×C5C2×D411yescharacteristic
124C5×#41—22no—
125C5×#41—22no—
126C5×#41—22no—
127C5×#41—22no—
128C5×#41—22no—
129C5×#41—22no—
130C5×#41—22no—
131C5×#41—22no—
132C5×#41—41no—
133C5×#41—41no—
134C5×#41—41no—
135C5×#41—41no—
136C5×#41—41no—
137C5×#41—41no—
138C5×#41—41no—
139C5×#41—41no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8C2×Q812yescomplemented
146C2×C5×D4—22no—
147C2×C5×D4—22no—
148C2×C5×D4—21no—
149C2×C5×D4—21no—
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4—41no—
154C2×C5×D4—41no—
155C2×C5×D4—41no—
156C2×C5×D4—41no—
157C2×C5×D4C2×Q812yescomplemented
158C22×C4×C5—22no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—41no—
163C22×C4×C5—41no—
164C22×C4×C5—41no—
165C22×C4×C5—41no—
166C22×C4×C5—41no—
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×Q811yescharacteristic
173C22×C4×C5C2×Q811yescharacteristic
174C22×C4×C5#4111yescharacteristic
175C22×C4×C5#4111yescharacteristic
176C22×C4×C5#4111yescharacteristic
177C22×C4×C5#4111yescharacteristic
178C22×C4×C5#4111yescharacteristic
179C22×C4×C5#4111yescharacteristic
180C22×C4×C5C2411yescharacteristic
181C22×Dic5—21no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5—21no—
185C22×Dic5—22no—
186C22×Dic5—44no—
187C2×Dic10—28no—
188C2×Dic10—48no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—41no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—22no—
195C5×C22⋊C4—41no—
196C5×C22⋊C4C2×D414yes—
197C42×C5—22no—
198C42×C5—22no—
199C42×C5—22no—
200C42×C5—41no—
201C22⋊Dic5—22no—
202C22⋊Dic5—22no—
203C4⋊Dic5—28no—
204C4⋊Dic5—22no—
205C4⋊Dic5—22no—
206#683—28no—
207#683—28no—
208C4×Dic5—22no—
209C4×Dic5—22no—
210C4×Dic5—21no—
211C4×Dic5—21no—
212C4×Dic5—22no—
213C4×Dic5C2×D414yes—
214C4×Dic5C2×D412yes—
215C4×Dic5C2×D412yes—
216#681—22no—
217#681—22no—
218#681—22no—
219#681—22no—
220#681—41no—
221#681—41no—
222#681—41no—
223#681—41no—
224C2×#168—22no—
225C2×#168—22no—
226C2×#168—41no—
227C2×#168—41no—
228C2×#168—41no—
229C2×#168—41no—
230C22×#41C2×D511yescharacteristic
231C22×M16—51no—
232C22×M16—51no—
233#533—54no—
234#533—102no—
235#533—102no—
236#533—102no—
237C2×C4⋊Q8—101no—
238C2×#125—101no—
239C2×C22⋊Q8—102no—
240C2×C4×Q8—102no—
241C2×#117—51no—
242C2×#117C2×D511yescharacteristic
243C2×#117C2×D511yescharacteristic
244C22×C42—51no—
245#421—54no—
246#421—102no—
247C2×#104—54no—
248C2×#104—101no—
249C2×#104—101no—
250C2×#104—102no—
251C2×#104—102no—
252#410—22no—
253#410C2×D514yes—
254C2×#99—22no—
255#344—54no—
256#339—54no—
257H(3,ℤ/4)—54no—
258C23×C5—22no—
259C23×C5—42no—
260C23×C5—41no—
261C23×C5—41no—
262C23×C5—82no—
263C23×C5#12011yescharacteristic
264C23×C5#12011yescharacteristic
265C23×C5C22⋊Q811yescharacteristic
266C23×C5C22⋊Q811yescharacteristic
267C23×C5#12411yescharacteristic
268C23×C5#12411yescharacteristic
269C23×C5C22×D411yescharacteristic
270C5×Q8—22no—
271C5×Q8—22no—
272C5×Q8—41no—
273C5×Q8—41no—
274C5×D4—22no—
275C5×D4—22no—
276C5×D4—42no—
277C5×D4—42no—
278C5×D4—42no—
279C5×D4—42no—
280C5×D4—41no—
281C5×D4—41no—
282C2×C4×C5—24no—
283C2×C4×C5—22no—
284C2×C4×C5—44no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—82no—
293C2×C4×C5#12011yescharacteristic
294C2×C4×C5#12011yescharacteristic
295C2×C4×C5C22⋊Q811yescharacteristic
296C2×C4×C5C22⋊Q811yescharacteristic
297C2×C4×C5C22⋊Q811yescharacteristic
298C2×C4×C5C22⋊Q811yescharacteristic
299C2×C4×C5C22⋊Q811yescharacteristic
300C2×C4×C5C22⋊Q811yescharacteristic
301C2×C4×C5C22⋊Q811yescharacteristic
302C2×C4×C5C22⋊Q811yescharacteristic
303C2×C4×C5C22⋊Q811yescharacteristic
304C2×C4×C5C22⋊Q811yescharacteristic
305C2×C4×C5#12411yescharacteristic
306C2×C4×C5#12411yescharacteristic
307C2×C4×C5#12411yescharacteristic
308C2×C4×C5#12411yescharacteristic
309C2×C4×C5#12411yescharacteristic
310C2×C4×C5#12411yescharacteristic
311C2×C4×C5#12411yescharacteristic
312C2×C4×C5#12411yescharacteristic
313C2×C4×C5#12411yescharacteristic
314C2×C4×C5#12411yescharacteristic
315C2×C4×C5C22×D411yescharacteristic
316C2×C4×C5C22×D411yescharacteristic
317C2×C4×C5C22×Q811yescharacteristic
318C2×C4×C5C2×#4111yescharacteristic
319C2×C4×C5C2×#4111yescharacteristic
320C2×C4×C5C2×#4111yescharacteristic
321C2×Dic5—24no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—24no—
325C2×Dic5—44no—
326C2×Dic5—44no—
327C2×Dic5—44no—
328C2×Dic5—42no—
329Dic10—416no—
330#168—42no—
331#168—42no—
332C2×#41—21no—
333C2×#41—21no—
334C2×#41—21no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—21no—
342C2×#41Dic1012yescomplemented
343C2×#41C22×D512yes—
344C22×Q8—102no—
345C22×Q8Dic1011yescharacteristic, complemented
346C22×D4—21no—
347C22×D4Dic1011yescharacteristic, complemented
348C23×C4—21no—
349C23×C4—101no—
350C23×C4C22×D511yescharacteristic
351C2×M16—52no—
352C2×M16—52no—
353C2×M16—101no—
354C2×M16—101no—
355C2×M16—101no—
356C2×M16—101no—
357C2×M16—101no—
358C2×M16—101no—
359C2×M16—101no—
360C2×M16—101no—
361C2×M16—101no—
362C2×M16—101no—
363C22×C8—101no—
364C22×C8—101no—
365C4⋊Q8—104no—
366C4⋊Q8—104no—
367#125—104no—
368#125—104no—
369C22⋊Q8—108no—
370C22⋊Q8—108no—
371C4×Q8—108no—
372C4×Q8—108no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—52no—
379#117—52no—
380#117—101no—
381#117—101no—
382#117C22⋊D512yes—
383#117C22×D512yes—
384#117C22×D512yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—102no—
388C2×C4⋊C4—102no—
389C2×C4⋊C4—102no—
390C2×C4⋊C4—101no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—101no—
393C2×C22⋊C4C22⋊D512yes—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—52no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—52no—
400C2×C42—52no—
401C2×C42—101no—
402C2×C42—101no—
403C2×C42—101no—
404#104—104no—
405#104—104no—
406#104—108no—
407#99—28no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—41no—
414C22×C5—41no—
415C22×C5—42no—
416C22×C5—82no—
417C22×C5—82no—
418C22×C5—82no—
419C22×C5#39311yescharacteristic
420C22×C5#39311yescharacteristic
421C22×C5#39311yescharacteristic
422C22×C5#39311yescharacteristic
423C22×C5C2×#12011yescharacteristic
424C22×C5C2×C22⋊Q811yescharacteristic
425C22×C5C2×#12411yescharacteristic
426C4×C5—44no—
427C4×C5—42no—
428C4×C5—82no—
429C4×C5#39311yescharacteristic
430C4×C5#39311yescharacteristic
431C4×C5#39311yescharacteristic
432C4×C5#39311yescharacteristic
433C4×C5C2×C22⋊Q811yescharacteristic
434C4×C5C2×C22⋊Q811yescharacteristic
435C4×C5C2×#12411yescharacteristic
436C4×C5C2×#12411yescharacteristic
437Dic5—24no—
438Dic5—44no—
439Dic5—42no—
440C24—21no—
441C24C2×C22⋊D511yescharacteristic
442#41—22no—
443#41—22no—
444#41—22no—
445#41—22no—
446#41—22no—
447#41—22no—
448#41—22no—
449#41—22no—
450#41—41no—
451#41—41no—
452#41—41no—
453#41—41no—
454#41—41no—
455#41—41no—
456#41—41no—
457#41—41no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—108no—
464C2×Q8—208no—
465C2×Q8C2×Dic1012yescomplemented
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4C2×Dic1012yescomplemented
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—101no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—204no—
493C22×C4C2×Dic1011yescharacteristic
494C22×C4C2×Dic1011yescharacteristic
495C22×C4#70911yescharacteristic
496C22×C4#70911yescharacteristic
497C22×C4D4×D511yescharacteristic
498C22×C4D4×D511yescharacteristic
499C22×C4D4×D511yescharacteristic
500C22×C4D4×D511yescharacteristic
501C22×C4#71111yescharacteristic
502C22×C4#71111yescharacteristic
503C22×C4#71111yescharacteristic
504C22×C4#71111yescharacteristic
505C22×C4C2×C22⋊D511yescharacteristic
506C22×C4C23×D511yescharacteristic
507M16—102no—
508M16—102no—
509M16—102no—
510M16—102no—
511M16—201no—
512M16—201no—
513M16—201no—
514M16—201no—
515C2×C8—102no—
516C2×C8—102no—
517C2×C8—201no—
518C2×C8—201no—
519C2×C8—201no—
520C2×C8—201no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—108no—
526C4⋊C4—108no—
527C4⋊C4—108no—
528C4⋊C4—102no—
529C4⋊C4—102no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—102no—
534C22⋊C4—102no—
535C22⋊C4C2×C22⋊D514yes—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—52no—
541C42—52no—
542C42—54no—
543C42—102no—
544C42—102no—
545C42—101no—
546C42—101no—
547C42—102no—
548C2×C5—21no—
549C2×C5—21no—
550C2×C5—21no—
551C2×C5—21no—
552C2×C5—21no—
553C2×C5—21no—
554C2×C5—82no—
555C2×C5#203112yescomplemented
556C2×C5C2×#39311yescharacteristic
557C23—22no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—82no—
562C23#407011yescharacteristic
563C23#407011yescharacteristic
564C23#408211yescharacteristic
565C23#408211yescharacteristic
566C23#408311yescharacteristic
567C23#408311yescharacteristic
568C23C22×C22⋊D511yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—2016no—
574D4—22no—
575D4—22no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—41no—
581D4—41no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—104no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—104no—
597C2×C4—204no—
598C2×C4—204no—
599C2×C4—202no—
600C2×C4—204no—
601C2×C4#402411yescharacteristic
602C2×C4#402411yescharacteristic
603C2×C4#402411yescharacteristic
604C2×C4#402411yescharacteristic
605C2×C4#402411yescharacteristic
606C2×C4#402411yescharacteristic
607C2×C4#402411yescharacteristic
608C2×C4#402411yescharacteristic
609C2×C4#403111yescharacteristic
610C2×C4#403111yescharacteristic
611C2×C4#403111yescharacteristic
612C2×C4#403111yescharacteristic
613C2×C4#403111yescharacteristic
614C2×C4#403111yescharacteristic
615C2×C4#403111yescharacteristic
616C2×C4#403111yescharacteristic
617C2×C4#407011yescharacteristic
618C2×C4#407011yescharacteristic
619C2×C4#408211yescharacteristic
620C2×C4#408211yescharacteristic
621C2×C4#408311yescharacteristic
622C2×C4#408311yescharacteristic
623C2×C4C22×Dic1011yescharacteristic
624C2×C4C2×#70911yescharacteristic
625C2×C4C2×D4×D511yescharacteristic
626C2×C4C2×D4×D511yescharacteristic
627C2×C4C2×#71111yescharacteristic
628C2×C4C2×#71111yescharacteristic
629C8—202no—
630C8—202no—
631C5C2×#203111yescharacteristic, complemented, Sylow
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22#6523611yescharacteristic
644C22#6523611yescharacteristic
645C22#6523611yescharacteristic
646C22#6523611yescharacteristic
647C22C2×#407011yescharacteristic
648C22C2×#408211yescharacteristic
649C22C2×#408311yescharacteristic
650C4—44no—
651C4—42no—
652C4—82no—
653C4—104no—
654C4—202no—
655C4—204no—
656C4#6523611yescharacteristic
657C4#6523611yescharacteristic
658C4#6523611yescharacteristic
659C4#6523611yescharacteristic
660C4C2×#402411yescharacteristic
661C4C2×#402411yescharacteristic
662C4C2×#403111yescharacteristic
663C4C2×#403111yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2#1060083112yesdirect product factor
672C2C2×#6523611yescharacteristic
subgroupG/H
centerC2×C4#4031
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#410C2
socleC22×C5#393
prime radicalC22×C5×#41C22
hypercenterC2×#410D5
hypocenterC5C2×#2031
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. C2×#65851C2semidirect
5. C2×#64753C2semidirect
6. C2×#64751C2semidirect
7. C2×#64752C2semidirect
8. #10600831C2direct (N × H)
19. #65851C22semidirect
34. #64753C22semidirect
35. #64752C22semidirect
36. #64751C22semidirect
48. C2×C5×#41Q8semidirect
50. C22×C5×Q8Q8semidirect
52. C22×C5×D4Q8semidirect
145. C2×C5×Q8C2×Q8semidirect
157. C2×C5×D4C2×Q8semidirect
342. C2×#41Dic10semidirect
345. C22×Q8Dic10semidirect
347. C22×D4Dic10semidirect
465. C2×Q8C2×Dic10semidirect
477. C2×D4C2×Dic10semidirect
555. C2×C5#2031semidirect
631. C5C2×#2031semidirect
671. C2#10600831direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658512
C2×C5×#4102
C2×#655252
C22×#39692
C2×#647532
C2×#647512
C2×#647522
#106008312
C2×#20315