Finite Group Databasebeta

← C2×#10613478 #12971500

Subgroups — C2×#10613478 #12971500

Order 1280.

1678 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66065C211yesmaximal, characteristic
2C2×#66023C211yesmaximal, characteristic
3C2×C5×#491C211yesmaximal, characteristic, complemented
4C2×#65370C211yesmaximal, characteristic, complemented
5C2×#65353C211yesmaximal, characteristic, complemented
6#10613478C2116yesmaximal, direct product factor
7C2×#65098C211yesmaximal, characteristic
8C2×#65104C211yesmaximal, characteristic
9C2×#65094C211yesmaximal, characteristic
10C2×#65103C211yesmaximal, characteristic
11C2×#65085C211yesmaximal, characteristic, complemented
12C2×#65088C211yesmaximal, characteristic, complemented
13C2×#65083C211yesmaximal, characteristic, complemented
14C2×#65080C211yesmaximal, characteristic, complemented
15C2×#64998C211yesmaximal, characteristic
16C2×#64991C211yesmaximal, characteristic, complemented
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×#125C2211yescharacteristic
19C2×#4092C2211yescharacteristic
20C2×#4092C2211yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23#66065—24no—
24#66065C2218yes—
25#66023—24no—
26#66023C2218yes—
27C2×#4044—21no—
28C2×#4044—21no—
29C2×#4043—21no—
30C2×#4042—21no—
31C2×#4038C2211yescharacteristic
32C2×D5×C4⋊C4C2211yescharacteristic
33C2×#4036C2211yescharacteristic
34C2×#4035—21no—
35C2×#4035—21no—
36C2×#4035—21no—
37C2×#4034—21no—
38C2×#4034C2211yescharacteristic
39C2×#4018C2211yescharacteristic
40C2×#4016C2211yescharacteristic
41C2×#4015C2211yescharacteristic
42C5×#491C2218yescomplemented
43C2×C5×#103C2211yescharacteristic
44C2×C5×#103C2211yescharacteristic
45C2×C5×#103C2211yescharacteristic
46C2×C5×#103C2211yescharacteristic
47C2×C5×#97C2211yescharacteristic
48C2×#3967C2211yescharacteristic
49C2×#3967C2211yescharacteristic
50C2×#3967C2211yescharacteristic
51C2×#3967C2211yescharacteristic
52C2×#3952C2211yescharacteristic
53C2×#3952C2211yescharacteristic
54C2×#3950C2211yescharacteristic
55C2×#3950C2211yescharacteristic
56C2×#3949C2211yescharacteristic
57C2×#3949C2211yescharacteristic
58C2×#3946C2211yescharacteristic
59C2×#3946C2211yescharacteristic
60#65370C2218yescomplemented
61#65353C2218yescomplemented
62C2×#3940C2211yescharacteristic
63C2×#3940C2211yescharacteristic
64C2×#3939C2211yescharacteristic
65C2×#3939C2211yescharacteristic
66C2×C4×#168C2211yescharacteristic
67#65104C2218yes—
68#65103C2218yes—
69#65098C2218yes—
70#65094C2218yes—
71#65088C2218yescomplemented
72#65085C2218yescomplemented
73#65083C2218yescomplemented
74#65080C2218yescomplemented
75#64998C2218yes—
76#64991C2218yescomplemented
77C2×#3242—51nomaximal, Sylow
78C22×C5×Q8C2311yescharacteristic
79C22×C5×Q8C2311yescharacteristic
80C22×C4×D5—21no—
81C22×C4×D5C2311yescharacteristic
82C22×Dic10—21no—
83C22×Dic10—21no—
84C22×C5×C8C2311yescharacteristic
85C22×C5×C8C2311yescharacteristic
86C5×C4⋊Q8—22no—
87C5×C4⋊Q8C2314yes—
88C5×#125—22no—
89C5×#125C2314yes—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4—21no—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C4⋊C4C2311yescharacteristic
96C2×C42×C5C2311yescharacteristic
97#4092—22no—
98#4092—22no—
99#4092C2314yes—
100#4092C2314yes—
101Q8×Dic5—22no—
102Q8×Dic5—22no—
103Q8×Dic5C2314yes—
104Q8×Dic5C2314yes—
105C2×#685—21no—
106C2×#685—21no—
107C2×#685—21no—
108C2×#685D411yescharacteristic
109C2×#685D411yescharacteristic
110C2×C4⋊Dic5—21no—
111C2×C4⋊Dic5—21no—
112C2×C4⋊Dic5—21no—
113C2×C4⋊Dic5—21no—
114C2×C4⋊Dic5C2311yescharacteristic
115C2×C4⋊Dic5C2311yescharacteristic
116C2×C4⋊Dic5C2311yescharacteristic
117C2×C4⋊Dic5C2311yescharacteristic
118C2×#683—21no—
119C2×#683—21no—
120C2×#683—21no—
121C2×#683—21no—
122C2×#683—21no—
123C2×#683—21no—
124C2×#683—21no—
125C2×#683D411yescharacteristic
126C2×#683D411yescharacteristic
127C2×C4×Dic5—21no—
128C2×C4×Dic5—21no—
129C2×C4×Dic5—21no—
130C2×C4×Dic5C2311yescharacteristic
131C22×#168C2311yescharacteristic
132C22×#168C2311yescharacteristic
133#4044—28no—
134#4044—28no—
135#4043—28no—
136#4042—28no—
137#4038—22no—
138#4038C2314yes—
139D5×C4⋊C4—22no—
140D5×C4⋊C4C2314yes—
141#4036—22no—
142#4036C2314yes—
143#4035—28no—
144#4035—28no—
145#4035—28no—
146#4034—28no—
147#4034—22no—
148#4034C2314yes—
149#4018D414yes—
150#4018C2314yes—
151#4016—22no—
152#4016C2314yes—
153#4015—22no—
154#4015C2314yes—
155C5×#103C2314yes—
156C5×#103C2314yes—
157C5×#103C2314yes—
158C5×#103C2314yes—
159C5×#97C2314yes—
160#3967C2314yes—
161#3967C2314yes—
162#3967C2314yes—
163#3967C2314yes—
164#3952C2314yes—
165#3952C2314yes—
166#3950C2314yes—
167#3950C2314yes—
168#3949C2314yes—
169#3949C2314yes—
170#3946C2314yes—
171#3946C2314yes—
172#3940C2314yes—
173#3940C2314yes—
174#3939C2314yes—
175#3939C2314yes—
176C4×#168C2314yes—
177C2×#544—51no—
178C2×#541—51no—
179#3242—516no—
180C2×#501—51no—
181C2×#499—51no—
182C2×#491D511yescharacteristic, complemented
183C2×#487—51no—
184C2×#484—51no—
185C2×#485—51no—
186C2×#485—51no—
187C2×#483—51no—
188C2×#479—51no—
189C2×#479—51no—
190C2×#477—51no—
191C2×#475—51no—
192C2×#433—51no—
193C23×D5C2×D411yescharacteristic
194C2×C5×Q8—21no—
195C2×C5×Q8—21no—
196C2×C5×Q8—22no—
197C2×C5×Q8—22no—
198C2×C5×Q8—22no—
199C2×C5×Q8—22no—
200C2×C5×Q8C2412yes—
201C2×C5×Q8C2412yes—
202C22×C4×C5—21no—
203C22×C4×C5—21no—
204C22×C4×C5—21no—
205C22×C4×C5—21no—
206C22×C4×C5C2×D411yescharacteristic
207C22×C4×C5C2×D411yescharacteristic
208C22×C4×C5C2411yescharacteristic
209C22×Dic5—21no—
210C22×Dic5—21no—
211C22×Dic5—21no—
212C22×Dic5—21no—
213C22×Dic5C2×D411yescharacteristic
214C22×Dic5C2×D411yescharacteristic
215C22×Dic5C2×D411yescharacteristic
216C2×C4×D5—22no—
217C2×C4×D5—22no—
218C2×C4×D5—24no—
219C2×C4×D5—82no—
220C2×C4×D5C2×D412yes—
221C2×C4×D5C2412yes—
222C2×Dic10—24no—
223C2×Dic10—24no—
224C2×Dic10—82no—
225C2×Dic10—82no—
226C2×C5×C8—22no—
227C2×C5×C8—22no—
228C2×C5×C8C2412yes—
229C2×C5×C8C2412yes—
230C5×C4⋊C4—24no—
231C5×C4⋊C4—24no—
232C5×C4⋊C4—24no—
233C5×C4⋊C4—24no—
234C5×C4⋊C4—21no—
235C5×C4⋊C4—21no—
236C5×C4⋊C4C2412yes—
237C5×C4⋊C4C2412yes—
238C42×C5C2×D412yes—
239C42×C5C2412yes—
240#685—24no—
241#685—24no—
242#685—24no—
243#685C2×D414yes—
244#685C2×D414yes—
245C4⋊Dic5—21no—
246C4⋊Dic5—21no—
247C4⋊Dic5—24no—
248C4⋊Dic5—21no—
249C4⋊Dic5—21no—
250C4⋊Dic5—24no—
251C4⋊Dic5—24no—
252C4⋊Dic5—24no—
253C4⋊Dic5C2412yes—
254C4⋊Dic5C2412yes—
255C4⋊Dic5C2412yes—
256C4⋊Dic5C2412yes—
257#683—24no—
258#683—24no—
259#683—24no—
260#683—24no—
261#683—24no—
262#683—24no—
263#683—24no—
264#683C2×D414yes—
265#683C2×D414yes—
266C4×Dic5—24no—
267C4×Dic5—24no—
268C4×Dic5—24no—
269C4×Dic5C2×D412yes—
270C4×Dic5C2412yes—
271C2×#168—22no—
272C2×#168—22no—
273C2×#168C2412yes—
274C2×#168C2412yes—
275#544—58no—
276#544—104no—
277#541—58no—
278#541—104no—
279C2×C4⋊Q8—51no—
280C2×C4⋊Q8—51no—
281C2×C4⋊Q8—101no—
282C2×C4⋊Q8C2×D511yescharacteristic
283C2×#126—101no—
284C2×#126—101no—
285C2×#125—51no—
286C2×#125—51no—
287C2×#125—101no—
288C2×#125—101no—
289C2×#125—101no—
290C2×#125C2×D511yescharacteristic
291C2×C22⋊Q8—51no—
292C2×C22⋊Q8—51no—
293C2×C22⋊Q8—101no—
294C2×C22⋊Q8—101no—
295C2×C4×Q8—51no—
296C2×C4×Q8—51no—
297C2×#117—51no—
298C2×#117—51no—
299C22×C4⋊C4—51no—
300#501—58no—
301#499—58no—
302#491C2×D518yescomplemented
303#487—58no—
304#485—58no—
305#485—58no—
306#484—58no—
307#483—58no—
308#479—58no—
309#479—58no—
310#477—58no—
311#475—58no—
312#433—58no—
313C2×#107—51no—
314C2×#107—51no—
315C2×#107—51no—
316C2×#107—51no—
317C2×#106—51no—
318C2×#106—51no—
319C2×#106—51no—
320C2×#106—51no—
321C2×C4⋊C8—51no—
322C2×C4⋊C8—51no—
323C2×#103—51no—
324C2×#103—51no—
325C2×#103—51no—
326C2×#103—51no—
327C2×#103C2×D511yescharacteristic
328C2×#103C2×D511yescharacteristic
329C2×#103C2×D511yescharacteristic
330C2×#103C2×D511yescharacteristic
331C2×C22⋊C8—51no—
332C2×C22⋊C8—51no—
333C2×#97C2×D511yescharacteristic
334C2×C4×C8—51no—
335C23×C5C22×D411yescharacteristic
336C22×D5—22no—
337C22×D5—22no—
338C22×D5—22no—
339C22×D5C22×D412yes—
340C5×Q8—22no—
341C5×Q8—22no—
342C5×Q8—22no—
343C5×Q8—22no—
344C5×Q8—42no—
345C5×Q8—42no—
346C2×C4×C5—22no—
347C2×C4×C5—22no—
348C2×C4×C5—22no—
349C2×C4×C5—22no—
350C2×C4×C5—22no—
351C2×C4×C5—22no—
352C2×C4×C5—42no—
353C2×C4×C5—42no—
354C2×C4×C5—42no—
355C2×C4×C5—42no—
356C2×C4×C5C22×D412yes—
357C2×C4×C5C22×D412yes—
358C2×C4×C5C22×D411yescharacteristic
359C2×C4×C52−1+412yes—
360C2×C4×C52−1+412yes—
361C2×C4×C5C2511yescharacteristic
362C2×Dic5—22no—
363C2×Dic5—22no—
364C2×Dic5—22no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—42no—
370C2×Dic5—42no—
371C2×Dic5—42no—
372C2×Dic5—42no—
373C2×Dic5C22×D412yes—
374C2×Dic5C22×D412yes—
375C2×Dic5C22×D412yes—
376C4×D5—24no—
377C4×D5—22no—
378C4×D5—22no—
379C4×D5—84no—
380Dic10—84no—
381Dic10—84no—
382C5×C8—22no—
383C5×C8—22no—
384#168—22no—
385#168—22no—
386C22×Q8—101no—
387C22×Q8—101no—
388C22×Q8C22×D511yescharacteristic
389C22×Q8C22×D511yescharacteristic
390C23×C4—51no—
391C23×C4—101no—
392C22×C8—51no—
393C22×C8—51no—
394C22×C8C22×D511yescharacteristic
395C22×C8C22×D511yescharacteristic
396C4⋊Q8—22no—
397C4⋊Q8—54no—
398C4⋊Q8—54no—
399C4⋊Q8—102no—
400C4⋊Q8—108no—
401C4⋊Q8—102no—
402C4⋊Q8C22×D514yes—
403#126—108no—
404#126—108no—
405#125—22no—
406#125—54no—
407#125—54no—
408#125—102no—
409#125—108no—
410#125—108no—
411#125—108no—
412#125—102no—
413#125C22×D514yes—
414C22⋊Q8—54no—
415C22⋊Q8—54no—
416C22⋊Q8—108no—
417C22⋊Q8—108no—
418C22⋊Q8—102no—
419C22⋊Q8—102no—
420C4×Q8—54no—
421C4×Q8—54no—
422C4×Q8—102no—
423C4×Q8—102no—
424#117—54no—
425#117—54no—
426#117—54no—
427#117—102no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—51no—
433C2×C4⋊C4—51no—
434C2×C4⋊C4—51no—
435C2×C4⋊C4—51no—
436C2×C4⋊C4—51no—
437C2×C4⋊C4—51no—
438C2×C4⋊C4—54no—
439C2×C4⋊C4—102no—
440C2×C4⋊C4—101no—
441C2×C4⋊C4—101no—
442C2×C4⋊C4—101no—
443C2×C4⋊C4—101no—
444C2×C4⋊C4—101no—
445C2×C4⋊C4—101no—
446C2×C4⋊C4—101no—
447C2×C4⋊C4—101no—
448C2×C4⋊C4—101no—
449C2×C4⋊C4—101no—
450C2×C4⋊C4—101no—
451C2×C4⋊C4C22×D511yescharacteristic
452C2×C4⋊C4C22×D511yescharacteristic
453C2×C22⋊C4—51no—
454C2×C22⋊C4—51no—
455C2×C22⋊C4—101no—
456C2×C22⋊C4—101no—
457C2×C22⋊C4—101no—
458C2×C42—51no—
459C2×C42—101no—
460C2×C42—101no—
461C2×C42—101no—
462C2×C42C22×D511yescharacteristic
463#107—54no—
464#107—54no—
465#107—54no—
466#107—54no—
467#106—54no—
468#106—54no—
469#106—54no—
470#106—54no—
471C4⋊C8—54no—
472C4⋊C8—54no—
473#103—54no—
474#103—54no—
475#103—54no—
476#103—54no—
477#103C22×D514yes—
478#103C22×D514yes—
479#103C22×D514yes—
480#103C22×D514yes—
481C22⋊C8—54no—
482C22⋊C8—54no—
483#97C22×D514yes—
484C4×C8—54no—
485C22×C5#53612yes—
486C22×C5#57712yes—
487C22×C5#57812yes—
488C22×C5C23×D411yescharacteristic
489C2×D5—22no—
490C2×D5—22no—
491C2×D5—22no—
492C2×D5—44no—
493C4×C5—22no—
494C4×C5—22no—
495C4×C5—42no—
496C4×C5—42no—
497C4×C5—42no—
498C4×C5—42no—
499C4×C5#53612yes—
500C4×C5C2×2−1+411yescharacteristic
501C4×C5C2×2−1+411yescharacteristic
502Dic5—22no—
503Dic5—22no—
504Dic5—22no—
505Dic5—42no—
506Dic5—42no—
507Dic5—42no—
508Dic5—42no—
509C24—51no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—22no—
513C2×Q8—22no—
514C2×Q8—22no—
515C2×Q8—22no—
516C2×Q8—104no—
517C2×Q8—104no—
518C2×Q8—402no—
519C2×Q8—402no—
520C2×Q8C23×D512yes—
521C2×Q8C23×D512yes—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—51no—
527C22×C4—51no—
528C22×C4—51no—
529C22×C4—52no—
530C22×C4—52no—
531C22×C4—104no—
532C22×C4—102no—
533C22×C4—102no—
534C22×C4—101no—
535C22×C4—101no—
536C22×C4—101no—
537C22×C4—101no—
538C22×C4—402no—
539C22×C4D4×D511yescharacteristic
540C22×C4D4×D511yescharacteristic
541C22×C4C23×D511yescharacteristic
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—52no—
545C2×C8—52no—
546C2×C8—102no—
547C2×C8—102no—
548C2×C8C23×D512yes—
549C2×C8C23×D512yes—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—54no—
557C4⋊C4—54no—
558C4⋊C4—52no—
559C4⋊C4—52no—
560C4⋊C4—52no—
561C4⋊C4—52no—
562C4⋊C4—104no—
563C4⋊C4—104no—
564C4⋊C4—104no—
565C4⋊C4—104no—
566C4⋊C4—104no—
567C4⋊C4—104no—
568C4⋊C4—104no—
569C4⋊C4—104no—
570C4⋊C4—101no—
571C4⋊C4—101no—
572C4⋊C4—101no—
573C4⋊C4—101no—
574C4⋊C4—104no—
575C4⋊C4—104no—
576C4⋊C4—104no—
577C4⋊C4C23×D512yes—
578C4⋊C4C23×D512yes—
579C22⋊C4—54no—
580C22⋊C4—54no—
581C22⋊C4—104no—
582C22⋊C4—104no—
583C22⋊C4—104no—
584C42—52no—
585C42—52no—
586C42—104no—
587C42—104no—
588C42—104no—
589C42D4×D512yes—
590C42C23×D512yes—
591C2×C5#324214yescomplemented
592C2×C5C2×#53611yescharacteristic
593C2×C5C2×#57711yescharacteristic
594C2×C5C2×#57811yescharacteristic
595D5—42no—
596C23—52no—
597C23—102no—
598C23—102no—
599C23—102no—
600C23C2×D4×D511yescharacteristic
601Q8—22no—
602Q8—22no—
603Q8—22no—
604Q8—22no—
605Q8—42no—
606Q8—42no—
607Q8—404no—
608Q8—404no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—52no—
620C2×C4—52no—
621C2×C4—52no—
622C2×C4—102no—
623C2×C4—102no—
624C2×C4—102no—
625C2×C4—104no—
626C2×C4—102no—
627C2×C4—102no—
628C2×C4—102no—
629C2×C4—102no—
630C2×C4—102no—
631C2×C4—102no—
632C2×C4—202no—
633C2×C4—202no—
634C2×C4—202no—
635C2×C4—202no—
636C2×C4—404no—
637C2×C4C2×D4×D512yes—
638C2×C4C2×D4×D512yes—
639C2×C4C2×D4×D511yescharacteristic
640C2×C4#415012yes—
641C2×C4#415012yes—
642C2×C4C24×D511yescharacteristic
643C8—22no—
644C8—22no—
645C8—102no—
646C8—102no—
647C5C2×#324211yescharacteristic, complemented, Sylow
648C22—102no—
649C22—102no—
650C22—102no—
651C22—204no—
652C22#6600912yes—
653C22#6611212yes—
654C22#6611312yes—
655C22C22×D4×D511yescharacteristic
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—102no—
663C4—102no—
664C4—102no—
665C4—202no—
666C4—202no—
667C4—202no—
668C4—202no—
669C4#6600912yes—
670C4C2×#415011yescharacteristic
671C4C2×#415011yescharacteristic
672C2—202no—
673C2#1061347814yesdirect product factor
674C2C2×#6600911yescharacteristic
675C2C2×#6611211yescharacteristic
676C2C2×#6611311yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#491C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
hypercenterC2×#491D5
hypocenterC5C2×#3242
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
3. C2×C5×#491C2semidirect
4. C2×#65370C2semidirect
5. C2×#65353C2semidirect
6. #10613478C2direct (N × H)
11. C2×#65085C2semidirect
12. C2×#65088C2semidirect
13. C2×#65083C2semidirect
14. C2×#65080C2semidirect
16. C2×#64991C2semidirect
42. C5×#491C22semidirect
60. #65370C22semidirect
61. #65353C22semidirect
71. #65088C22semidirect
72. #65085C22semidirect
73. #65083C22semidirect
74. #65080C22semidirect
76. #64991C22semidirect
182. C2×#491D5semidirect
302. #491C2×D5semidirect
591. C2×C5#3242semidirect
647. C5C2×#3242semidirect
673. C2#10613478direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660652
C2×#660232
C2×C5×#4912
C2×#653702
C2×#653532
#106134782
C2×#650982
C2×#651042
C2×#650942
C2×#651032
C2×#650852
C2×#650882
C2×#650832
C2×#650802
C2×#649982
C2×#649912
C2×#32425