Finite Group Databasebeta

← C2×#10601158 #12100788

Subgroups — C2×#10601158 #12100788

Order 1280.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66161C211yesmaximal, characteristic
2C22×#4097C211yesmaximal, characteristic
3C22×C5×#104C211yesmaximal, characteristic
4C2×#65409C211yesmaximal, characteristic, complemented
5C2×#65218C211yesmaximal, characteristic, complemented
6C2×#64774C211yesmaximal, characteristic, complemented
7C2×#64743C211yesmaximal, characteristic, complemented
8#10601158C218yesmaximal, direct product factor
9C22×C5×#41C2211yescharacteristic
10C23×Dic10C2211yescharacteristic
11C22×C5×M16C2211yescharacteristic
12C22×C42×C5C2211yescharacteristic
13#66161—22no—
14#66161—22no—
15#66161—22no—
16#66161C2214yes—
17C2×#4097—22no—
18C2×#4097—24no—
19C2×#4097C418yescomplemented
20C2×#4097C2214yes—
21C2×Q8×Dic5—21no—
22C22×#689—21no—
23C2×D4×Dic5—21no—
24C2×D4×Dic5—22no—
25C22×#687—21no—
26C2×#4072—21no—
27C2×#4072C2211yescharacteristic
28C22×C4⋊Dic5—21no—
29C22×C4⋊Dic5C2211yescharacteristic
30C22×#681C2211yescharacteristic
31C2×C5×#104—24no—
32C2×C5×#104C2214yes—
33C2×#3963—22no—
34#65409—22no—
35#65409C2214yescomplemented
36C2×#3948—21no—
37C2×#3948—21no—
38#65218—22no—
39#65218C2214yescomplemented
40#64774C2214yescomplemented
41#64743C2214yescomplemented
42C2×#1882—51nomaximal, Sylow
43C2×C5×#41—21no—
44C2×C5×#41—21no—
45C2×C5×#41—22no—
46C2×C5×#41—22no—
47C2×C5×#41C2×C414yes—
48C2×C5×#41D412yes—
49C2×C5×#41D414yes—
50C2×C5×#41C2312yes—
51C22×C5×Q8D411yescharacteristic
52C22×C5×D4—21no—
53C22×C5×D4D411yescharacteristic
54C23×C4×C5—21no—
55C23×C4×C5—21no—
56C23×C4×C5C2311yescharacteristic
57C23×Dic5—21no—
58C23×Dic5—22no—
59C23×Dic5D412yes—
60C22×Dic10—21no—
61C22×Dic10—21no—
62C22×Dic10—22no—
63C22×Dic10—21no—
64C22×Dic10—21no—
65C22×Dic10—22no—
66C22×Dic10—21no—
67C22×Dic10—22no—
68C22×Dic10C2×C414yescomplemented
69C22×Dic10C2312yes—
70C2×C5×M16—21no—
71C2×C5×M16—22no—
72C2×C5×M16—22no—
73C2×C5×M16C2312yes—
74C22×C5×C8—21no—
75C2×C42×C5—22no—
76C2×C42×C5—22no—
77C2×C42×C5—21no—
78C2×C42×C5C2312yes—
79C2×C22⋊Dic5—22no—
80C2×C22⋊Dic5—21no—
81#4097—216no—
82#4097—28no—
83#4097C2×C4116yescomplemented
84Q8×Dic5—24no—
85Q8×Dic5—24no—
86C2×#689—28no—
87C2×#689—24no—
88D4×Dic5—28no—
89D4×Dic5—24no—
90D4×Dic5—24no—
91D4×Dic5—28no—
92C2×#687—28no—
93C2×#687—24no—
94#4072—24no—
95#4072—21no—
96#4072—21no—
97#4072—24no—
98#4072D412yes—
99#4072C2312yes—
100C2×C4⋊Dic5—22no—
101C2×C4⋊Dic5—24no—
102C2×C4⋊Dic5—21no—
103C2×C4⋊Dic5—21no—
104C2×C4⋊Dic5—21no—
105C2×C4⋊Dic5—21no—
106C2×C4⋊Dic5—24no—
107C2×C4⋊Dic5D412yes—
108C2×C4⋊Dic5D411yescharacteristic
109C2×C4⋊Dic5D411yescharacteristic
110C2×C4⋊Dic5C2312yes—
111C2×C4×Dic5—22no—
112C2×C4×Dic5—21no—
113C2×#681—21no—
114C2×#681—22no—
115C2×#681C2×C414yescomplemented
116C2×#681C2312yes—
117C22×#168—21no—
118C5×#104—28no—
119#3963—28no—
120#3948—24no—
121#3948—24no—
122C22×#137—51no—
123C2×#520—51no—
124C22×#104D511yescharacteristic
125C2×#419—51no—
126C2×#391—51no—
127#1882—58no—
128C2×#343—51no—
129C2×#339—51no—
130C24×C5—21no—
131C24×C5C2×D411yescharacteristic
132C5×#41—24no—
133C5×#41—24no—
134C5×#41—24no—
135C5×#41—24no—
136C5×#41—24no—
137C5×#41—22no—
138C5×#41—22no—
139C5×#41C22⋊C418yes—
140C5×#41C22×C414yes—
141C5×#41C2×D414yes—
142C2×C5×Q8—21no—
143C2×C5×Q8—22no—
144C2×C5×Q8C22⋊C414yes—
145C2×C5×Q8C2×D412yes—
146C2×C5×D4—24no—
147C2×C5×D4—22no—
148C2×C5×D4—24no—
149C2×C5×D4—22no—
150C2×C5×D4—21no—
151C2×C5×D4—22no—
152C2×C5×D4C22⋊C414yes—
153C2×C5×D4C2×D412yes—
154C22×C4×C5—21no—
155C22×C4×C5—24no—
156C22×C4×C5—22no—
157C22×C4×C5—24no—
158C22×C4×C5—22no—
159C22×C4×C5—24no—
160C22×C4×C5—22no—
161C22×C4×C5—42no—
162C22×C4×C5—42no—
163C22×C4×C5C22×C412yes—
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D412yes—
170C22×C4×C5#4112yes—
171C22×C4×C5#4112yes—
172C22×C4×C5C2411yescharacteristic
173C22×Dic5—24no—
174C22×Dic5—21no—
175C22×Dic5—21no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—28no—
179C22×Dic5—22no—
180C22×Dic5—41no—
181C22×Dic5—41no—
182C22×Dic5—42no—
183C22×Dic5—44no—
184C22×Dic5—84no—
185C22×Dic5C2×D414yes—
186C2×Dic10—24no—
187C2×Dic10—24no—
188C2×Dic10—24no—
189C2×Dic10—24no—
190C2×Dic10—22no—
191C2×Dic10—22no—
192C2×Dic10—22no—
193C2×Dic10—22no—
194C2×Dic10—24no—
195C2×Dic10—22no—
196C2×Dic10—41no—
197C2×Dic10—41no—
198C2×Dic10—42no—
199C2×Dic10—42no—
200C2×Dic10—42no—
201C2×Dic10—42no—
202C2×Dic10—42no—
203C2×Dic10C22×C414yescomplemented
204C5×M16—22no—
205C5×M16—22no—
206C5×M16—42no—
207C2×C5×C8—22no—
208C2×C5×C8—42no—
209C42×C5—22no—
210C42×C5—22no—
211C42×C5—42no—
212C22⋊Dic5—28no—
213C22⋊Dic5—22no—
214C22⋊Dic5—22no—
215#689—216no—
216#687—216no—
217C4⋊Dic5—28no—
218C4⋊Dic5—22no—
219C4⋊Dic5—21no—
220C4⋊Dic5—22no—
221C4⋊Dic5—21no—
222C4⋊Dic5—22no—
223C4⋊Dic5—22no—
224C4⋊Dic5C2×D412yes—
225C4⋊Dic5C2×D412yes—
226C4×Dic5—28no—
227C4×Dic5—22no—
228C4×Dic5—22no—
229#681—24no—
230#681—22no—
231#681C22×C414yescomplemented
232C2×#168—24no—
233C2×#168—22no—
234C22×#41C2×D511yescharacteristic
235C23×Q8—51no—
236C2×#137—58no—
237C2×#137—54no—
238C2×#137—102no—
239C2×#137—104no—
240C22×Q16—101no—
241C22×SD16—101no—
242C22×M16—51no—
243C22×M16C2×D511yescharacteristic
244#520—54no—
245#520—102no—
246#520—102no—
247#520—102no—
248C2×C4×Q8—101no—
249C2×C4×D4—102no—
250C2×C4×D4—101no—
251C2×#117—51no—
252C2×#117—101no—
253C22×C4⋊C4—51no—
254C22×C4⋊C4—101no—
255C22×C42C2×D511yescharacteristic
256C2×#104—24no—
257C2×#104C2×D514yes—
258#419—54no—
259#419—102no—
260C2×#103—101no—
261C2×#103—101no—
262#391—54no—
263#391—102no—
264C2×#95—102no—
265#343—54no—
266#339—54no—
267C23×C5—22no—
268C23×C5—22no—
269C23×C5—22no—
270C23×C5—24no—
271C23×C5—42no—
272C23×C5—42no—
273C23×C5C4×D412yes—
274C23×C5#12011yescharacteristic
275C23×C5#12011yescharacteristic
276C23×C5#12112yes—
277C23×C5C22×D411yescharacteristic
278C5×Q8—22no—
279C5×Q8—24no—
280C5×Q8C2×C22⋊C414yes—
281C5×D4—24no—
282C5×D4—24no—
283C5×D4—28no—
284C5×D4—22no—
285C5×D4—24no—
286C5×D4C2×C22⋊C414yes—
287C2×C4×C5—22no—
288C2×C4×C5—24no—
289C2×C4×C5—24no—
290C2×C4×C5—24no—
291C2×C4×C5—28no—
292C2×C4×C5—24no—
293C2×C4×C5—48no—
294C2×C4×C5—42no—
295C2×C4×C5—42no—
296C2×C4×C5C2×C22⋊C412yes—
297C2×C4×C5C2×C22⋊C412yes—
298C2×C4×C5C4×D412yes—
299C2×C4×C5C4×D412yes—
300C2×C4×C5C4×D412yes—
301C2×C4×C5#12011yescharacteristic
302C2×C4×C5#12011yescharacteristic
303C2×C4×C5#12112yes—
304C2×C4×C5#12112yes—
305C2×C4×C5#12112yes—
306C2×C4×C5#12312yes—
307C2×C4×C5#12312yes—
308C2×C4×C5C23×C411yescharacteristic
309C2×C4×C5C22×D411yescharacteristic
310C2×C4×C5C22×D411yescharacteristic
311C2×C4×C5C22×D411yescharacteristic
312C2×C4×C5C2×#4111yescharacteristic
313C2×C4×C5C2×#4111yescharacteristic
314C2×Dic5—24no—
315C2×Dic5—22no—
316C2×Dic5—24no—
317C2×Dic5—22no—
318C2×Dic5—42no—
319C2×Dic5—42no—
320C2×Dic5—44no—
321C2×Dic5—44no—
322C2×Dic5—44no—
323C2×Dic5—42no—
324C2×Dic5—42no—
325C2×Dic5—44no—
326C2×Dic5—82no—
327C2×Dic5—88no—
328Dic10—24no—
329Dic10—24no—
330Dic10—42no—
331Dic10—42no—
332Dic10—44no—
333Dic10—44no—
334C5×C8—42no—
335#168—24no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41C4×D514yes—
341C2×#41D2012yes—
342C2×#41C22⋊D514yes—
343C2×#41C22×D512yes—
344C22×Q8—54no—
345C22×Q8—52no—
346C22×Q8—101no—
347C22×Q8—102no—
348C22×Q8—101no—
349C22×Q8—101no—
350C22×Q8—101no—
351C22×Q8—101no—
352C22×Q8—102no—
353C22×Q8—102no—
354C22×Q8D2011yescharacteristic
355C22×D4—21no—
356C22×D4D2011yescharacteristic
357C23×C4—21no—
358C23×C4—21no—
359C23×C4—52no—
360C23×C4—101no—
361C23×C4—102no—
362C23×C4C22×D511yescharacteristic
363#137—516no—
364#137—1016no—
365#137—108no—
366C2×Q16—108no—
367C2×Q16—104no—
368C2×SD16—108no—
369C2×SD16—104no—
370C2×M16—21no—
371C2×M16—22no—
372C2×M16—22no—
373C2×M16—54no—
374C2×M16—52no—
375C2×M16—101no—
376C2×M16—102no—
377C2×M16C22×D512yes—
378C22×C8—21no—
379C22×C8—101no—
380C4×Q8—104no—
381C4×Q8—104no—
382C4×D4—108no—
383C4×D4—108no—
384C4×D4—104no—
385C4×D4—104no—
386#117—52no—
387#117—52no—
388#117—104no—
389#117—104no—
390#117—101no—
391#117—101no—
392C2×C4⋊C4—51no—
393C2×C4⋊C4—51no—
394C2×C4⋊C4—52no—
395C2×C4⋊C4—52no—
396C2×C4⋊C4—104no—
397C2×C4⋊C4—102no—
398C2×C4⋊C4—104no—
399C2×C4⋊C4—101no—
400C2×C4⋊C4—101no—
401C2×C4⋊C4—101no—
402C2×C4⋊C4—101no—
403C2×C22⋊C4—102no—
404C2×C22⋊C4—101no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—21no—
408C2×C42—102no—
409C2×C42—101no—
410C2×C42C22×D512yes—
411#104—28no—
412#103—104no—
413#103—104no—
414#95—108no—
415C22×C5—24no—
416C22×C5—24no—
417C22×C5—21no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5—24no—
422C22×C5—48no—
423C22×C5—42no—
424C22×C5—42no—
425C22×C5#38612yes—
426C22×C5#38612yes—
427C22×C5#45512yes—
428C22×C5#45612yes—
429C22×C5C2×C4×D411yescharacteristic
430C22×C5C2×#12011yescharacteristic
431C22×C5C2×#12111yescharacteristic
432C4×C5—24no—
433C4×C5—44no—
434C4×C5#38612yes—
435C4×C5#38612yes—
436C4×C5C22×C22⋊C411yescharacteristic
437C4×C5C2×C4×D411yescharacteristic
438C4×C5C2×#12111yescharacteristic
439C4×C5C2×#12311yescharacteristic
440Dic5—44no—
441Dic5—42no—
442Dic5—42no—
443Dic5—44no—
444C24—21no—
445C24D4×D511yescharacteristic
446#41—22no—
447#41—22no—
448#41—24no—
449#41—24no—
450#41—24no—
451#41—24no—
452#41—24no—
453#41#68518yes—
454#41C2×C4×D514yes—
455#41C2×C22⋊D514yes—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—54no—
459C2×Q8—102no—
460C2×Q8—104no—
461C2×Q8—102no—
462C2×Q8—102no—
463C2×Q8—102no—
464C2×Q8—102no—
465C2×Q8—104no—
466C2×Q8—104no—
467C2×Q8—104no—
468C2×Q8—104no—
469C2×Q8—201no—
470C2×Q8—201no—
471C2×Q8—202no—
472C2×Q8—202no—
473C2×Q8—202no—
474C2×Q8—202no—
475C2×Q8—202no—
476C2×Q8#68514yes—
477C2×Q8C2×D2012yes—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—22no—
481C2×D4—24no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4#68514yes—
485C2×D4C2×D2012yes—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—54no—
496C22×C4—102no—
497C22×C4—104no—
498C22×C4—101no—
499C22×C4—101no—
500C22×C4—102no—
501C22×C4—108no—
502C22×C4—102no—
503C22×C4—201no—
504C22×C4—201no—
505C22×C4—202no—
506C22×C4—204no—
507C22×C4—404no—
508C22×C4C2×C4×D512yes—
509C22×C4C2×D2011yescharacteristic
510C22×C4C2×D2011yescharacteristic
511C22×C4D4×D511yescharacteristic
512C22×C4D4×D511yescharacteristic
513C22×C4D4×D511yescharacteristic
514C22×C4#71112yes—
515C22×C4#71112yes—
516C22×C4C2×C22⋊D512yes—
517C22×C4C23×D511yescharacteristic
518Q16—1016no—
519SD16—1016no—
520M16—22no—
521M16—22no—
522M16—42no—
523M16—54no—
524M16—102no—
525M16—104no—
526C2×C8—22no—
527C2×C8—42no—
528C2×C8—102no—
529C2×C8—104no—
530C4⋊C4—52no—
531C4⋊C4—52no—
532C4⋊C4—108no—
533C4⋊C4—102no—
534C4⋊C4—102no—
535C4⋊C4—101no—
536C4⋊C4—102no—
537C4⋊C4—102no—
538C4⋊C4—101no—
539C22⋊C4—108no—
540C22⋊C4—102no—
541C22⋊C4—102no—
542C42—22no—
543C42—22no—
544C42—42no—
545C42—108no—
546C42—102no—
547C42—102no—
548C2×C5—21no—
549C2×C5—21no—
550C2×C5—22no—
551C2×C5—44no—
552C2×C5#188214yescomplemented
553C2×C5C2×#38611yescharacteristic
554C2×C5C2×#45511yescharacteristic
555C2×C5C2×#45611yescharacteristic
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—42no—
561C23—42no—
562C23#402712yes—
563C23#402811yescharacteristic
564C23#402811yescharacteristic
565C23#408512yes—
566C23C2×D4×D511yescharacteristic
567Q8—22no—
568Q8—24no—
569Q8—104no—
570Q8—104no—
571Q8—202no—
572Q8—202no—
573Q8—204no—
574Q8—204no—
575Q8C2×#68514yes—
576D4—24no—
577D4—24no—
578D4—28no—
579D4—22no—
580D4—24no—
581D4C2×#68514yes—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—104no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—104no—
595C2×C4—202no—
596C2×C4—202no—
597C2×C4—204no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—204no—
601C2×C4—204no—
602C2×C4—204no—
603C2×C4—402no—
604C2×C4—408no—
605C2×C4#402712yes—
606C2×C4#402712yes—
607C2×C4#402712yes—
608C2×C4#402811yescharacteristic
609C2×C4#402811yescharacteristic
610C2×C4#403212yes—
611C2×C4#403212yes—
612C2×C4C2×#68512yes—
613C2×C4C2×#68512yes—
614C2×C4#408512yes—
615C2×C4#408512yes—
616C2×C4#408512yes—
617C2×C4C22×C4×D511yescharacteristic
618C2×C4C22×D2011yescharacteristic
619C2×C4C2×D4×D511yescharacteristic
620C2×C4C2×#71111yescharacteristic
621C2×C4C2×#71111yescharacteristic
622C2×C4C22×C22⋊D511yescharacteristic
623C8—42no—
624C8—104no—
625C5C2×#188211yescharacteristic, complemented, Sylow
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—48no—
636C22#6511412yes—
637C22#6511512yes—
638C22#6524712yes—
639C22#6524712yes—
640C22C2×#402711yescharacteristic
641C22C2×#402811yescharacteristic
642C22C2×#408511yescharacteristic
643C4—24no—
644C4—44no—
645C4—204no—
646C4—202no—
647C4—202no—
648C4—204no—
649C4#6524712yes—
650C4#6524712yes—
651C4C2×#402711yescharacteristic
652C4C2×#403211yescharacteristic
653C4C22×#68511yescharacteristic
654C4C2×#408511yescharacteristic
655C2—21no—
656C2—21no—
657C2—22no—
658C2—44no—
659C2#1060115814yesdirect product factor
660C2C2×#6511411yescharacteristic
661C2C2×#6511511yescharacteristic
662C2C2×#6524711yescharacteristic
subgroupG/H
centerC23#4028
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC22×C5×#104C2
socleC23×C5#120
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenterC22×#104D5
hypocenterC5C2×#1882
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
4. C2×#65409C2semidirect
5. C2×#65218C2semidirect
6. C2×#64774C2semidirect
7. C2×#64743C2semidirect
8. #10601158C2direct (N × H)
19. C2×#4097C4semidirect
35. #65409C22semidirect
39. #65218C22semidirect
40. #64774C22semidirect
41. #64743C22semidirect
68. C22×Dic10C2×C4semidirect
83. #4097C2×C4semidirect
115. C2×#681C2×C4semidirect
203. C2×Dic10C22×C4semidirect
231. #681C22×C4semidirect
552. C2×C5#1882semidirect
625. C5C2×#1882semidirect
659. C2#10601158direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#661612
C22×#40972
C22×C5×#1042
C2×#654092
C2×#652182
C2×#647742
C2×#647432
#106011582
C2×#18825