Finite Group Databasebeta

← C2×#10613654 #12971508

Subgroups — C2×#10613654 #12971508

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×#66023C211yesmaximal, characteristic
2C2×#66031C211yesmaximal, characteristic
3C2×C5×#499C211yesmaximal, characteristic, complemented
4C2×#65354C211yesmaximal, characteristic, complemented
5C2×#65358C211yesmaximal, characteristic, complemented
6#10613654C2116yesmaximal, direct product factor
7C2×#65166C211yesmaximal, characteristic, complemented
8C2×#65164C211yesmaximal, characteristic, complemented
9C2×#65173C211yesmaximal, characteristic
10C2×#65177C211yesmaximal, characteristic
11C2×#65146C211yesmaximal, characteristic, complemented
12C2×#65143C211yesmaximal, characteristic, complemented
13C2×#65151C211yesmaximal, characteristic
14C2×#65156C211yesmaximal, characteristic
15C2×#64970C211yesmaximal, characteristic, complemented
16C2×#64979C211yesmaximal, characteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×#125C2211yescharacteristic
19#66031—24no—
20#66031C2218yes—
21#66023—24no—
22#66023C2218yes—
23C2×#4044—21no—
24C2×#4044—21no—
25C2×#4043—21no—
26C2×#4043C2211yescharacteristic
27C2×#4043C2211yescharacteristic
28C2×#4042—21no—
29C2×#4038C2211yescharacteristic
30C2×D5×C4⋊C4C2211yescharacteristic
31C2×#4036—21no—
32C2×#4036—21no—
33C2×#4036C2211yescharacteristic
34C2×#4035—21no—
35C2×#4034—21no—
36C2×#4034C2211yescharacteristic
37C2×#4033C2211yescharacteristic
38C2×#4033C2211yescharacteristic
39C2×#4018C2211yescharacteristic
40C2×#4015C2211yescharacteristic
41C2×#4015C2211yescharacteristic
42C5×#499C2218yescomplemented
43C2×C5×#107C2211yescharacteristic
44C2×C5×#107C2211yescharacteristic
45C2×C5×#106C2211yescharacteristic
46C2×C5×#106C2211yescharacteristic
47C2×C4×C5×C8C2211yescharacteristic
48C2×#3952C2211yescharacteristic
49C2×#3952C2211yescharacteristic
50C2×#3948C2211yescharacteristic
51C2×#3948C2211yescharacteristic
52C2×#3948C2211yescharacteristic
53C2×#3948C2211yescharacteristic
54C2×#3946C2211yescharacteristic
55C2×#3946C2211yescharacteristic
56#65358C2218yescomplemented
57#65354C2218yescomplemented
58C2×#3942C2211yescharacteristic
59C2×#3942C2211yescharacteristic
60C2×#3942C2211yescharacteristic
61C2×#3942C2211yescharacteristic
62C2×#3940C2211yescharacteristic
63C2×#3940C2211yescharacteristic
64C2×#3939C2211yescharacteristic
65C2×#3939C2211yescharacteristic
66C2×#3935C2211yescharacteristic
67#65177C2218yes—
68#65173C2218yes—
69#65166C2218yescomplemented
70#65164C2218yescomplemented
71#65156C2218yes—
72#65151C2218yes—
73#65146C2218yescomplemented
74#65143C2218yescomplemented
75#64979C2218yes—
76#64970C2218yescomplemented
77C2×#3242—51nomaximal, Sylow
78C22×C4×D5—21no—
79C22×C4×D5C2311yescharacteristic
80C22×Dic10—21no—
81C22×Dic10—21no—
82C22×Dic10C2311yescharacteristic
83C22×Dic10C2311yescharacteristic
84C22×C5×C8C2311yescharacteristic
85C22×C5×C8C2311yescharacteristic
86C5×#125—22no—
87C5×#125—22no—
88C5×#125C2314yes—
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×C5×C4⋊C4C2311yescharacteristic
97C2×C5×C4⋊C4C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C2×#685—21no—
100C2×#685—21no—
101C2×#685—21no—
102C2×#685D411yescharacteristic
103C2×#685D411yescharacteristic
104C2×C4⋊Dic5—21no—
105C2×C4⋊Dic5—21no—
106C2×C4⋊Dic5—21no—
107C2×C4⋊Dic5—21no—
108C2×C4⋊Dic5—21no—
109C2×C4⋊Dic5—21no—
110C2×C4⋊Dic5C2311yescharacteristic
111C2×C4⋊Dic5C2311yescharacteristic
112C2×#683—21no—
113C2×#683—21no—
114C2×#683—21no—
115C2×#683—21no—
116C2×#683—21no—
117C2×#683D411yescharacteristic
118C2×#683D411yescharacteristic
119C2×C4×Dic5—21no—
120C2×C4×Dic5—21no—
121C2×C4×Dic5—21no—
122C2×C4×Dic5C2311yescharacteristic
123C22×#168C2311yescharacteristic
124C22×#168C2311yescharacteristic
125#4044—28no—
126#4044—28no—
127#4043—22no—
128#4043—22no—
129#4043—28no—
130#4043C2314yes—
131#4043C2314yes—
132#4042—28no—
133#4038—22no—
134#4038C2314yes—
135D5×C4⋊C4—22no—
136D5×C4⋊C4C2314yes—
137#4036—28no—
138#4036—28no—
139#4036—22no—
140#4036C2314yes—
141#4035—28no—
142#4034—28no—
143#4034—22no—
144#4034C2314yes—
145#4033—22no—
146#4033—22no—
147#4033C2314yes—
148#4033C2314yes—
149#4018D414yes—
150#4018C2314yes—
151#4015—22no—
152#4015—22no—
153#4015C2314yes—
154#4015C2314yes—
155C5×#107C2314yes—
156C5×#107C2314yes—
157C5×#106C2314yes—
158C5×#106C2314yes—
159C4×C5×C8C2314yes—
160#3952C2314yes—
161#3952C2314yes—
162#3948C2314yes—
163#3948C2314yes—
164#3948C2314yes—
165#3948C2314yes—
166#3946C2314yes—
167#3946C2314yes—
168#3942C2314yes—
169#3942C2314yes—
170#3942C2314yes—
171#3942C2314yes—
172#3940C2314yes—
173#3940C2314yes—
174#3939C2314yes—
175#3939C2314yes—
176#3935C2314yes—
177C2×#544—51no—
178C2×#541—51no—
179#3242—516no—
180C2×#501—51no—
181C2×#499D511yescharacteristic, complemented
182C2×#491—51no—
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
194C22×C4×C5—21no—
195C22×C4×C5—21no—
196C22×C4×C5—21no—
197C22×C4×C5—21no—
198C22×C4×C5C2×D411yescharacteristic
199C22×C4×C5C2×D411yescharacteristic
200C22×C4×C5C2411yescharacteristic
201C22×Dic5—21no—
202C22×Dic5—21no—
203C22×Dic5—21no—
204C22×Dic5—21no—
205C22×Dic5C2×D411yescharacteristic
206C22×Dic5C2×D411yescharacteristic
207C22×Dic5C2×D411yescharacteristic
208C2×C4×D5—22no—
209C2×C4×D5—22no—
210C2×C4×D5—24no—
211C2×C4×D5—82no—
212C2×C4×D5C2×D412yes—
213C2×C4×D5C2412yes—
214C2×Dic10—22no—
215C2×Dic10—22no—
216C2×Dic10—22no—
217C2×Dic10—22no—
218C2×Dic10—21no—
219C2×Dic10—21no—
220C2×Dic10—24no—
221C2×Dic10—24no—
222C2×Dic10—82no—
223C2×Dic10—82no—
224C2×Dic10C2412yes—
225C2×Dic10C2412yes—
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⋊C4—21no—
237C5×C4⋊C4—21no—
238C5×C4⋊C4C2412yes—
239C5×C4⋊C4C2412yes—
240C5×C4⋊C4C2412yes—
241C5×C4⋊C4C2412yes—
242C42×C5C2×D412yes—
243C42×C5C2412yes—
244#685—24no—
245#685—24no—
246#685—24no—
247#685C2×D414yes—
248#685C2×D414yes—
249C4⋊Dic5—21no—
250C4⋊Dic5—21no—
251C4⋊Dic5—24no—
252C4⋊Dic5—24no—
253C4⋊Dic5—24no—
254C4⋊Dic5—24no—
255C4⋊Dic5—24no—
256C4⋊Dic5—24no—
257C4⋊Dic5C2412yes—
258C4⋊Dic5C2412yes—
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—51no—
282C2×C4⋊Q8—101no—
283C2×#126—101no—
284C2×#126—101no—
285C2×#125—51no—
286C2×#125—101no—
287C2×#125—101no—
288C2×#125—101no—
289C2×#125C2×D511yescharacteristic
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#499C2×D518yescomplemented
302#491—58no—
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×#107C2×D511yescharacteristic
316C2×#107C2×D511yescharacteristic
317C2×#106—51no—
318C2×#106—51no—
319C2×#106C2×D511yescharacteristic
320C2×#106C2×D511yescharacteristic
321C2×C4⋊C8—51no—
322C2×C4⋊C8—51no—
323C2×#103—51no—
324C2×#103—51no—
325C2×#103—51no—
326C2×#103—51no—
327C2×#103—51no—
328C2×#103—51no—
329C2×#103—51no—
330C2×#103—51no—
331C2×C22⋊C8—51no—
332C2×C22⋊C8—51no—
333C2×#97—51no—
334C2×C4×C8C2×D511yescharacteristic
335C23×C5C22×D411yescharacteristic
336C22×D5—22no—
337C22×D5—22no—
338C22×D5—22no—
339C22×D5C22×D412yes—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5—22no—
343C2×C4×C5—22no—
344C2×C4×C5—22no—
345C2×C4×C5—22no—
346C2×C4×C5—42no—
347C2×C4×C5—42no—
348C2×C4×C5—42no—
349C2×C4×C5—42no—
350C2×C4×C5C22×D412yes—
351C2×C4×C5C22×D412yes—
352C2×C4×C5C22×D411yescharacteristic
353C2×C4×C52−1+412yes—
354C2×C4×C52−1+412yes—
355C2×C4×C5C2511yescharacteristic
356C2×Dic5—22no—
357C2×Dic5—22no—
358C2×Dic5—22no—
359C2×Dic5—22no—
360C2×Dic5—22no—
361C2×Dic5—22no—
362C2×Dic5—22no—
363C2×Dic5—42no—
364C2×Dic5—42no—
365C2×Dic5—42no—
366C2×Dic5—42no—
367C2×Dic5C22×D412yes—
368C2×Dic5C22×D412yes—
369C2×Dic5C22×D412yes—
370C4×D5—24no—
371C4×D5—22no—
372C4×D5—22no—
373C4×D5—84no—
374Dic10—22no—
375Dic10—22no—
376Dic10—22no—
377Dic10—22no—
378Dic10—42no—
379Dic10—42no—
380Dic10—84no—
381Dic10—84no—
382C5×C8—22no—
383C5×C8—22no—
384#168—22no—
385#168—22no—
386C22×Q8—51no—
387C22×Q8—51no—
388C22×Q8—101no—
389C22×Q8—101no—
390C23×C4—51no—
391C23×C4—101no—
392C22×C8—51no—
393C22×C8—51no—
394C22×C8C22×D511yescharacteristic
395C22×C8C22×D511yescharacteristic
396C4⋊Q8—54no—
397C4⋊Q8—54no—
398C4⋊Q8—54no—
399C4⋊Q8—102no—
400C4⋊Q8—108no—
401C4⋊Q8—102no—
402C4⋊Q8—102no—
403#126—108no—
404#126—108no—
405#125—22no—
406#125—22no—
407#125—54no—
408#125—102no—
409#125—108no—
410#125—108no—
411#125—108no—
412#125C22×D514yes—
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—54no—
437C2×C4⋊C4—102no—
438C2×C4⋊C4—101no—
439C2×C4⋊C4—101no—
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⋊C4C22×D511yescharacteristic
450C2×C4⋊C4C22×D511yescharacteristic
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#107C22×D514yes—
466#107C22×D514yes—
467#106—54no—
468#106—54no—
469#106C22×D514yes—
470#106C22×D514yes—
471C4⋊C8—54no—
472C4⋊C8—54no—
473#103—54no—
474#103—54no—
475#103—54no—
476#103—54no—
477#103—54no—
478#103—54no—
479#103—54no—
480#103—54no—
481C22⋊C8—54no—
482C22⋊C8—54no—
483#97—54no—
484C4×C8C22×D514yes—
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—52no—
511C2×Q8—52no—
512C2×Q8—104no—
513C2×Q8—101no—
514C2×Q8—101no—
515C2×Q8—104no—
516C2×Q8—102no—
517C2×Q8—102no—
518C2×Q8—102no—
519C2×Q8—102no—
520C2×Q8—402no—
521C2×Q8—402no—
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—21no—
557C4⋊C4—21no—
558C4⋊C4—54no—
559C4⋊C4—54no—
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—104no—
573C4⋊C4—104no—
574C4⋊C4—104no—
575C4⋊C4C23×D512yes—
576C4⋊C4C23×D512yes—
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—102no—
602Q8—102no—
603Q8—102no—
604Q8—102no—
605Q8—202no—
606Q8—202no—
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#414712yes—
641C2×C4#414712yes—
642C2×C4C24×D511yescharacteristic
643C8—22no—
644C8—22no—
645C8—102no—
646C8—102no—
647C5C2×#324211yescharacteristic, complemented, Sylow
648C22—102no—
649C22—102no—
650C22—102no—
651C22—204no—
652C22#6605812yes—
653C22#6610412yes—
654C22#6610512yes—
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#6605812yes—
670C4C2×#414711yescharacteristic
671C4C2×#414711yescharacteristic
672C2—202no—
673C2#1061365414yesdirect product factor
674C2C2×#6605811yescharacteristic
675C2C2×#6610411yescharacteristic
676C2C2×#6610511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#499C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
hypercenterC2×#499D5
hypocenterC5C2×#3242
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
3. C2×C5×#499C2semidirect
4. C2×#65354C2semidirect
5. C2×#65358C2semidirect
6. #10613654C2direct (N × H)
7. C2×#65166C2semidirect
8. C2×#65164C2semidirect
11. C2×#65146C2semidirect
12. C2×#65143C2semidirect
15. C2×#64970C2semidirect
42. C5×#499C22semidirect
56. #65358C22semidirect
57. #65354C22semidirect
69. #65166C22semidirect
70. #65164C22semidirect
73. #65146C22semidirect
74. #65143C22semidirect
76. #64970C22semidirect
181. C2×#499D5semidirect
301. #499C2×D5semidirect
591. C2×C5#3242semidirect
647. C5C2×#3242semidirect
673. C2#10613654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660232
C2×#660312
C2×C5×#4992
C2×#653542
C2×#653582
#106136542
C2×#651662
C2×#651642
C2×#651732
C2×#651772
C2×#651462
C2×#651432
C2×#651512
C2×#651562
C2×#649702
C2×#649792
C2×#32425