Finite Group Databasebeta

← C2×#10612970 #12971466

Subgroups — C2×#10612970 #12971466

Order 1280.

1790 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66022C211yesmaximal, characteristic, complemented
2C2×#65914C211yesmaximal, characteristic
3C2×C5×#479C211yesmaximal, characteristic, complemented
4C2×#65355C211yesmaximal, characteristic, complemented
5C2×#65311C211yesmaximal, characteristic, complemented
6#10612970C2116yesmaximal, direct product factor
7C2×#65174C211yesmaximal, characteristic
8C2×#65165C211yesmaximal, characteristic, complemented
9C2×#65155C211yesmaximal, characteristic
10C2×#65145C211yesmaximal, characteristic, complemented
11C2×#65136C211yesmaximal, characteristic, complemented
12C2×#65124C211yesmaximal, characteristic, complemented
13C2×#65104C211yesmaximal, characteristic, complemented
14C2×#65084C211yesmaximal, characteristic, complemented
15C2×#65084C211yesmaximal, characteristic, complemented
16C2×#65103C211yesmaximal, characteristic, complemented
17C2×C5×#125C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C2×#4093C2211yescharacteristic
20C2×#4092C2211yescharacteristic
21#66022—24no—
22#66022C2218yescomplemented
23#65914—24no—
24#65914C2218yes—
25C2×#4044—21no—
26C2×#4042—21no—
27C2×#4042—21no—
28C2×#4041C2211yescharacteristic
29C2×#4040—21no—
30C2×#4038C2211yescharacteristic
31C2×#4038C2211yescharacteristic
32C2×#4036C2211yescharacteristic
33C2×#4034C2211yescharacteristic
34C2×#4034C2211yescharacteristic
35C2×#4033—21no—
36C2×#4033—21no—
37C2×C4×D20—21no—
38C2×C4×D20C2211yescharacteristic
39C2×#4018—21no—
40C2×C42×D5C2211yescharacteristic
41C2×#4016C2211yescharacteristic
42C5×#479C2218yescomplemented
43C2×C5×#107C2211yescharacteristic
44C2×C5×#106C2211yescharacteristic
45C2×C5×C4⋊C8C2211yescharacteristic
46C2×C5×#103C2211yescharacteristic
47C2×C5×#103C2211yescharacteristic
48C2×#3967C2211yescharacteristic
49C2×#3967C2211yescharacteristic
50C2×#3953C2211yescharacteristic
51C2×#3953C2211yescharacteristic
52C2×#3952C2211yescharacteristic
53C2×#3952C2211yescharacteristic
54C2×#3950C2211yescharacteristic
55C2×#3949C2211yescharacteristic
56C2×#3947C2211yescharacteristic
57C2×#3947C2211yescharacteristic
58#65355C2218yescomplemented
59#65311C2218yescomplemented
60C2×#3941C2211yescharacteristic
61C2×#3941C2211yescharacteristic
62C2×#3940C2211yescharacteristic
63C2×#3940C2211yescharacteristic
64C2×#3939C2211yescharacteristic
65C2×#3939C2211yescharacteristic
66C2×C4⋊#168C2211yescharacteristic
67#65174C2218yes—
68#65165C2218yescomplemented
69#65155C2218yes—
70#65145C2218yescomplemented
71#65136C2218yescomplemented
72#65124C2218yescomplemented
73#65104C2218yescomplemented
74#65103C2218yescomplemented
75#65084C2218yescomplemented
76#65084C2218yescomplemented
77C2×#3240—51nomaximal, Sylow
78C22×C5×Q8C2311yescharacteristic
79C22×D20—21no—
80C22×D20C2311yescharacteristic
81C22×C4×D5—21no—
82C22×C4×D5D411yescharacteristic
83C22×C4×D5D411yescharacteristic
84C22×C4×D5C2311yescharacteristic
85C22×Dic10—21no—
86C22×Dic10—21no—
87C22×C5×C8C2311yescharacteristic
88C22×C5×C8C2311yescharacteristic
89C5×#125—22no—
90C5×#125C2314yes—
91C4×C5×Q8—22no—
92C4×C5×Q8C2314yes—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4—21no—
95C2×C5×C4⋊C4—21no—
96C2×C5×C4⋊C4C2311yescharacteristic
97C2×C5×C4⋊C4C2311yescharacteristic
98C2×C5×C4⋊C4C2311yescharacteristic
99C2×C42×C5—21no—
100C2×C42×C5C2311yescharacteristic
101#4093—22no—
102#4093C2314yes—
103#4092—22no—
104#4092C2314yes—
105C2×#685—21no—
106C2×#685—21no—
107C2×#685—21no—
108C2×#685—21no—
109C2×#685—21no—
110C2×#685—21no—
111C2×#685—21no—
112C2×C4⋊Dic5—21no—
113C2×C4⋊Dic5C2311yescharacteristic
114C2×C4⋊Dic5C2311yescharacteristic
115C2×C4⋊Dic5C2311yescharacteristic
116C2×#683—21no—
117C2×#683—21no—
118C2×#683—21no—
119C2×#683—21no—
120C2×#683—21no—
121C2×#683—21no—
122C2×C4×Dic5—21no—
123C2×C4×Dic5—21no—
124C2×C4×Dic5D411yescharacteristic
125C2×C4×Dic5D411yescharacteristic
126C2×C4×Dic5C2311yescharacteristic
127C22×#168C2311yescharacteristic
128C22×#168C2311yescharacteristic
129#4044—28no—
130#4042—28no—
131#4042—28no—
132#4041—22no—
133#4041C2314yes—
134#4040—28no—
135#4038—22no—
136#4038—22no—
137#4038C2314yes—
138#4038C2314yes—
139#4036—22no—
140#4036C2314yes—
141#4034—22no—
142#4034—22no—
143#4034C2314yes—
144#4034C2314yes—
145#4033—28no—
146#4033—28no—
147C4×D20—28no—
148C4×D20—22no—
149C4×D20C2314yes—
150#4018—28no—
151C42×D5D414yes—
152C42×D5C2314yes—
153#4016—22no—
154#4016C2314yes—
155C5×#107C2314yes—
156C5×#106C2314yes—
157C5×C4⋊C8C2314yes—
158C5×#103C2314yes—
159C5×#103C2314yes—
160#3967C2314yes—
161#3967C2314yes—
162#3953C2314yes—
163#3953C2314yes—
164#3952C2314yes—
165#3952C2314yes—
166#3950C2314yes—
167#3949C2314yes—
168#3947C2314yes—
169#3947C2314yes—
170#3941C2314yes—
171#3941C2314yes—
172#3940C2314yes—
173#3940C2314yes—
174#3939C2314yes—
175#3939C2314yes—
176C4⋊#168C2314yes—
177C2×#533—51no—
178C2×#529—51no—
179#3240—516no—
180C2×#501—51no—
181C2×#501—51no—
182C2×#489—51no—
183C2×#489—51no—
184C2×#485—51no—
185C2×#485—51no—
186C2×#482—51no—
187C2×#482—51no—
188C2×#479—51no—
189C2×#479D511yescharacteristic, complemented
190C2×#478—51no—
191C2×#478—51no—
192C2×#432—51no—
193C23×D5—21no—
194C23×D5C2×D411yescharacteristic
195C2×C5×Q8—21no—
196C2×C5×Q8—22no—
197C2×C5×Q8—22no—
198C2×C5×Q8C2412yes—
199C22×C4×C5—21no—
200C22×C4×C5—21no—
201C22×C4×C5—21no—
202C22×C4×C5—21no—
203C22×C4×C5C2×D411yescharacteristic
204C22×C4×C5C2×D411yescharacteristic
205C22×C4×C5C2411yescharacteristic
206C22×Dic5—21no—
207C22×Dic5—21no—
208C22×Dic5—21no—
209C22×Dic5C2×D411yescharacteristic
210C22×Dic5C2×D411yescharacteristic
211C22×Dic5C2×D411yescharacteristic
212C2×D20—22no—
213C2×D20—22no—
214C2×D20—24no—
215C2×D20—21no—
216C2×D20—44no—
217C2×D20C2412yes—
218C2×C4×D5—22no—
219C2×C4×D5—22no—
220C2×C4×D5—24no—
221C2×C4×D5—42no—
222C2×C4×D5—42no—
223C2×C4×D5—44no—
224C2×C4×D5C2×D412yes—
225C2×C4×D5C2×D414yes—
226C2×C4×D5C2×D414yes—
227C2×C4×D5C2412yes—
228C2×Dic10—24no—
229C2×Dic10—24no—
230C2×Dic10—44no—
231C2×Dic10—44no—
232C2×C5×C8—22no—
233C2×C5×C8—22no—
234C2×C5×C8C2412yes—
235C2×C5×C8C2412yes—
236C5×C4⋊C4—24no—
237C5×C4⋊C4—24no—
238C5×C4⋊C4—24no—
239C5×C4⋊C4—21no—
240C5×C4⋊C4—21no—
241C5×C4⋊C4—21no—
242C5×C4⋊C4C2412yes—
243C5×C4⋊C4C2412yes—
244C5×C4⋊C4C2412yes—
245C42×C5—24no—
246C42×C5C2×D412yes—
247C42×C5C2412yes—
248#685—24no—
249#685—24no—
250#685—24no—
251#685—24no—
252#685—24no—
253#685—24no—
254#685—24no—
255C4⋊Dic5—21no—
256C4⋊Dic5—21no—
257C4⋊Dic5—21no—
258C4⋊Dic5—24no—
259C4⋊Dic5C2412yes—
260C4⋊Dic5C2412yes—
261C4⋊Dic5C2412yes—
262#683—24no—
263#683—24no—
264#683—24no—
265#683—24no—
266#683—24no—
267#683—24no—
268C4×Dic5—24no—
269C4×Dic5—24no—
270C4×Dic5C2×D412yes—
271C4×Dic5C2×D414yes—
272C4×Dic5C2×D414yes—
273C4×Dic5C2412yes—
274C2×#168—22no—
275C2×#168—22no—
276C2×#168C2412yes—
277C2×#168C2412yes—
278#533—58no—
279#533—104no—
280C2×C4⋊Q8—51no—
281C2×C4⋊Q8—51no—
282#529—58no—
283#529—104no—
284C2×#126—101no—
285C2×#125—51no—
286C2×#125—51no—
287C2×#125C2×D511yescharacteristic
288C2×#124—51no—
289C2×#123—101no—
290C2×C22⋊Q8—51no—
291C2×C22⋊Q8—101no—
292C2×C22⋊Q8—101no—
293C2×#121—51no—
294C2×C4×Q8—101no—
295C2×C4×Q8—101no—
296C2×C4×Q8C2×D511yescharacteristic
297C2×C4×D4—51no—
298C2×C4×D4—101no—
299C2×#117—51no—
300C2×#117—51no—
301C2×#117—101no—
302C22×C42—51no—
303#501—58no—
304#501—58no—
305#489—58no—
306#489—58no—
307#485—58no—
308#485—58no—
309#482—58no—
310#482—58no—
311#479—58no—
312#479C2×D518yescomplemented
313#478—58no—
314#478—58no—
315#432—58no—
316C2×#107—51no—
317C2×#107—51no—
318C2×#107—51no—
319C2×#107C2×D511yescharacteristic
320C2×#106—51no—
321C2×#106—51no—
322C2×#106—51no—
323C2×#106C2×D511yescharacteristic
324C2×C4⋊C8—51no—
325C2×C4⋊C8C2×D511yescharacteristic
326C2×#103—51no—
327C2×#103—51no—
328C2×#103C2×D511yescharacteristic
329C2×#103C2×D511yescharacteristic
330C2×#102—51no—
331C2×#102—51no—
332C2×#102—51no—
333C2×#102—51no—
334C2×C22⋊C8—51no—
335C2×C22⋊C8—51no—
336C2×#97—51no—
337C2×#97—51no—
338C23×C5C22×D411yescharacteristic
339C22×D5—22no—
340C22×D5—22no—
341C22×D5—22no—
342C22×D5—22no—
343C22×D5—42no—
344C22×D5—42no—
345C22×D5—42no—
346C22×D5C22×D412yes—
347C5×Q8—22no—
348C5×Q8—22no—
349C5×Q8—42no—
350C2×C4×C5—22no—
351C2×C4×C5—22no—
352C2×C4×C5—22no—
353C2×C4×C5—22no—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—42no—
357C2×C4×C5—42no—
358C2×C4×C5—42no—
359C2×C4×C5—42no—
360C2×C4×C5C22×D412yes—
361C2×C4×C5C22×D412yes—
362C2×C4×C5C22×D411yescharacteristic
363C2×C4×C52−1+412yes—
364C2×C4×C52−1+412yes—
365C2×C4×C5C2511yescharacteristic
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—42no—
373C2×Dic5—42no—
374C2×Dic5—42no—
375C2×Dic5C22×D412yes—
376C2×Dic5C22×D412yes—
377C2×Dic5C22×D412yes—
378D20—22no—
379D20—22no—
380D20—48no—
381D20—42no—
382C4×D5—24no—
383C4×D5—22no—
384C4×D5—22no—
385C4×D5—44no—
386C4×D5—44no—
387C4×D5—48no—
388Dic10—48no—
389Dic10—48no—
390C5×C8—22no—
391C5×C8—22no—
392#168—22no—
393#168—22no—
394C22×Q8—101no—
395C22×Q8—101no—
396C22×Q8C22×D511yescharacteristic
397C22×D4—51no—
398C22×D4—101no—
399C23×C4—51no—
400C23×C4—51no—
401C23×C4—51no—
402C23×C4—101no—
403C22×C8—51no—
404C22×C8—51no—
405C22×C8C22×D511yescharacteristic
406C22×C8C22×D511yescharacteristic
407C4⋊Q8—54no—
408C4⋊Q8—54no—
409C4⋊Q8—102no—
410C4⋊Q8—102no—
411#126—108no—
412#125—22no—
413#125—54no—
414#125—54no—
415#125—102no—
416#125—102no—
417#125C22×D514yes—
418#124—54no—
419#124—102no—
420#123—108no—
421C22⋊Q8—54no—
422C22⋊Q8—108no—
423C22⋊Q8—108no—
424C22⋊Q8—102no—
425#121—54no—
426#121—102no—
427C4×Q8—22no—
428C4×Q8—108no—
429C4×Q8—108no—
430C4×Q8C22×D514yes—
431C4×D4—54no—
432C4×D4—108no—
433C4×D4—102no—
434#117—54no—
435#117—54no—
436#117—108no—
437#117—102no—
438#117—102no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—51no—
443C2×C4⋊C4—51no—
444C2×C4⋊C4—51no—
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⋊C4—101no—
452C2×C4⋊C4C22×D511yescharacteristic
453C2×C4⋊C4C22×D511yescharacteristic
454C2×C4⋊C4C22×D511yescharacteristic
455C2×C22⋊C4—101no—
456C2×C22⋊C4—101no—
457C2×C22⋊C4—101no—
458C2×C22⋊C4—101no—
459C2×C22⋊C4—101no—
460C2×C22⋊C4—101no—
461C2×C22⋊C4—101no—
462C2×C42—21no—
463C2×C42—51no—
464C2×C42—51no—
465C2×C42—51no—
466C2×C42—54no—
467C2×C42—54no—
468C2×C42—101no—
469C2×C42—101no—
470C2×C42C22×D511yescharacteristic
471#107—54no—
472#107—54no—
473#107—54no—
474#107C22×D514yes—
475#106—54no—
476#106—54no—
477#106—54no—
478#106C22×D514yes—
479C4⋊C8—54no—
480C4⋊C8C22×D514yes—
481#103—54no—
482#103—54no—
483#103C22×D514yes—
484#103C22×D514yes—
485#102—54no—
486#102—54no—
487#102—54no—
488#102—54no—
489C22⋊C8—54no—
490C22⋊C8—54no—
491#97—54no—
492#97—54no—
493C22×C5#53612yes—
494C22×C5#57512yes—
495C22×C5#57512yes—
496C22×C5C23×D411yescharacteristic
497C2×D5—22no—
498C2×D5—22no—
499C2×D5—22no—
500C2×D5—44no—
501C2×D5—42no—
502C2×D5—42no—
503C2×D5—42no—
504C2×D5—84no—
505C4×C5—22no—
506C4×C5—22no—
507C4×C5—42no—
508C4×C5—42no—
509C4×C5—42no—
510C4×C5—42no—
511C4×C5#53612yes—
512C4×C5C2×2−1+411yescharacteristic
513C4×C5C2×2−1+411yescharacteristic
514Dic5—22no—
515Dic5—22no—
516Dic5—22no—
517Dic5—42no—
518Dic5—42no—
519Dic5—42no—
520C24—51no—
521C24—101no—
522C2×Q8—21no—
523C2×Q8—22no—
524C2×Q8—22no—
525C2×Q8—104no—
526C2×Q8—104no—
527C2×Q8—204no—
528C2×Q8—204no—
529C2×Q8C23×D512yes—
530C2×D4—52no—
531C2×D4—104no—
532C2×D4—101no—
533C2×D4—102no—
534C2×D4—102no—
535C2×D4—204no—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—51no—
541C22×C4—51no—
542C22×C4—51no—
543C22×C4—52no—
544C22×C4—52no—
545C22×C4—54no—
546C22×C4—54no—
547C22×C4—102no—
548C22×C4—102no—
549C22×C4—101no—
550C22×C4—101no—
551C22×C4—101no—
552C22×C4—104no—
553C22×C4—202no—
554C22×C4—202no—
555C22×C4—204no—
556C22×C4D4×D511yescharacteristic
557C22×C4D4×D511yescharacteristic
558C22×C4C23×D511yescharacteristic
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—52no—
562C2×C8—52no—
563C2×C8—102no—
564C2×C8—102no—
565C2×C8C23×D512yes—
566C2×C8C23×D512yes—
567C4⋊C4—24no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—52no—
574C4⋊C4—52no—
575C4⋊C4—52no—
576C4⋊C4—104no—
577C4⋊C4—104no—
578C4⋊C4—104no—
579C4⋊C4—104no—
580C4⋊C4—104no—
581C4⋊C4—101no—
582C4⋊C4—101no—
583C4⋊C4—101no—
584C4⋊C4—104no—
585C4⋊C4—104no—
586C4⋊C4C23×D512yes—
587C4⋊C4C23×D512yes—
588C4⋊C4C23×D512yes—
589C22⋊C4—104no—
590C22⋊C4—104no—
591C22⋊C4—104no—
592C22⋊C4—104no—
593C22⋊C4—104no—
594C22⋊C4—104no—
595C22⋊C4—104no—
596C42—24no—
597C42—54no—
598C42—54no—
599C42—52no—
600C42—52no—
601C42—104no—
602C42—104no—
603C42D4×D512yes—
604C42C23×D512yes—
605C2×C5#324014yescomplemented
606C2×C5C2×#53611yescharacteristic
607C2×C5C2×#57511yescharacteristic
608C2×C5C2×#57511yescharacteristic
609D5—42no—
610D5—82no—
611C23—52no—
612C23—102no—
613C23—102no—
614C23—102no—
615C23—102no—
616C23—202no—
617C23—202no—
618C23—202no—
619C23C2×D4×D511yescharacteristic
620Q8—22no—
621Q8—22no—
622Q8—42no—
623Q8—208no—
624Q8—208no—
625D4—102no—
626D4—102no—
627D4—208no—
628D4—202no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—52no—
640C2×C4—52no—
641C2×C4—52no—
642C2×C4—102no—
643C2×C4—102no—
644C2×C4—102no—
645C2×C4—104no—
646C2×C4—102no—
647C2×C4—102no—
648C2×C4—102no—
649C2×C4—102no—
650C2×C4—102no—
651C2×C4—204no—
652C2×C4—204no—
653C2×C4—208no—
654C2×C4—202no—
655C2×C4—202no—
656C2×C4—202no—
657C2×C4C2×D4×D512yes—
658C2×C4C2×D4×D512yes—
659C2×C4C2×D4×D511yescharacteristic
660C2×C4#414712yes—
661C2×C4#415012yes—
662C2×C4C24×D511yescharacteristic
663C8—22no—
664C8—22no—
665C8—102no—
666C8—102no—
667C5C2×#324011yescharacteristic, complemented, Sylow
668C22—102no—
669C22—102no—
670C22—102no—
671C22—204no—
672C22—202no—
673C22—202no—
674C22—202no—
675C22—404no—
676C22#6596212yes—
677C22#6610212yes—
678C22#6611112yes—
679C22C22×D4×D511yescharacteristic
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—102no—
687C4—102no—
688C4—102no—
689C4—202no—
690C4—202no—
691C4—202no—
692C4#6596212yes—
693C4C2×#414711yescharacteristic
694C4C2×#415011yescharacteristic
695C2—202no—
696C2—402no—
697C2#1061297014yesdirect product factor
698C2C2×#6596211yescharacteristic
699C2C2×#6610211yescharacteristic
700C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#479C2
socleC23×C5C22×D4
prime radicalC2×#4041C22
hypercenterC2×#479D5
hypocenterC5C2×#3240
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
1. C2×#66022C2semidirect
3. C2×C5×#479C2semidirect
4. C2×#65355C2semidirect
5. C2×#65311C2semidirect
6. #10612970C2direct (N × H)
8. C2×#65165C2semidirect
10. C2×#65145C2semidirect
11. C2×#65136C2semidirect
12. C2×#65124C2semidirect
13. C2×#65104C2semidirect
14. C2×#65084C2semidirect
15. C2×#65084C2semidirect
16. C2×#65103C2semidirect
22. #66022C22semidirect
42. C5×#479C22semidirect
58. #65355C22semidirect
59. #65311C22semidirect
68. #65165C22semidirect
70. #65145C22semidirect
71. #65136C22semidirect
72. #65124C22semidirect
73. #65104C22semidirect
74. #65103C22semidirect
75. #65084C22semidirect
76. #65084C22semidirect
189. C2×#479D5semidirect
312. #479C2×D5semidirect
605. C2×C5#3240semidirect
667. C5C2×#3240semidirect
697. C2#10612970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660222
C2×#659142
C2×C5×#4792
C2×#653552
C2×#653112
#106129702
C2×#651742
C2×#651652
C2×#651552
C2×#651452
C2×#651362
C2×#651242
C2×#651042
C2×#650842
C2×#651032
C2×#32405