Finite Group Databasebeta

← C2×#10606681 #12258136

Subgroups — C2×#10606681 #12258136

Order 1280.

2526 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4034C211yesmaximal, characteristic
2C2×C5×#387C211yesmaximal, characteristic
3C2×#65263C211yesmaximal, characteristic
4C2×#65261C211yesmaximal, characteristic
5C2×#65270C211yesmaximal, characteristic
6C2×#65218C211yesmaximal, characteristic
7C2×C4×#683C211yesmaximal, characteristic
8#10606681C2116yesmaximal, direct product factor
9C2×#64935C211yesmaximal, characteristic
10C2×#64935C211yesmaximal, characteristic
11C2×#64936C211yesmaximal, characteristic
12C2×#64943C211yesmaximal, characteristic
13C2×#64942C211yesmaximal, characteristic
14C2×#64939C211yesmaximal, characteristic
15C2×#64938C211yesmaximal, characteristic
16C2×#64937C211yesmaximal, characteristic
17C23×Dic10C2211yescharacteristic
18C23×Dic10C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C42×C5C2211yescharacteristic
22C22×C4⋊Dic5C2211yescharacteristic
23C22×C4⋊Dic5C2211yescharacteristic
24C22×#683C2211yescharacteristic
25C22×#683C2211yescharacteristic
26C22×#683C2211yescharacteristic
27C22×#683C2211yescharacteristic
28C22×#683C2211yescharacteristic
29C22×#683C2211yescharacteristic
30C22×C4×Dic5C2211yescharacteristic
31C22×C4×Dic5C2211yescharacteristic
32C22×C4×Dic5C2211yescharacteristic
33C22×C4×Dic5C2211yescharacteristic
34C22×C4×Dic5C2211yescharacteristic
35C22×C4×Dic5C2211yescharacteristic
36C2×#4034C4116yescomplemented
37C2×#4034C2218yes—
38C5×#387C2218yes—
39C2×C5×#95C2211yescharacteristic
40C2×C5×#95C2211yescharacteristic
41C2×C5×#95C2211yescharacteristic
42C2×C5×#95C2211yescharacteristic
43C2×#3963C2211yescharacteristic
44C2×#3963C2211yescharacteristic
45C2×#3963C2211yescharacteristic
46C2×#3963C2211yescharacteristic
47C2×#3963C2211yescharacteristic
48C2×#3963C2211yescharacteristic
49C2×#3963C2211yescharacteristic
50C2×#3963C2211yescharacteristic
51C2×#3963C2211yescharacteristic
52C2×#3963C2211yescharacteristic
53C2×#3963C2211yescharacteristic
54C2×#3963C2211yescharacteristic
55#65270C2218yes—
56#65263C2218yes—
57#65261C2218yes—
58C4×#683C2218yes—
59#65218C2218yes—
60#64943C2218yes—
61#64942C2218yes—
62#64939C2218yes—
63#64938C2218yes—
64#64937C2218yes—
65#64936C2218yes—
66#64935C2218yes—
67#64935C2218yes—
68C2×#2331—51nomaximal, Sylow
69C23×C4×C5C2311yescharacteristic
70C23×C4×C5C2311yescharacteristic
71C23×C4×C5C2311yescharacteristic
72C23×C4×C5C2311yescharacteristic
73C23×C4×C5C2311yescharacteristic
74C23×C4×C5C2311yescharacteristic
75C23×C4×C5C2311yescharacteristic
76C23×Dic5C2311yescharacteristic
77C23×Dic5C2311yescharacteristic
78C23×Dic5C2311yescharacteristic
79C23×Dic5C2311yescharacteristic
80C23×Dic5C2311yescharacteristic
81C23×Dic5C2311yescharacteristic
82C23×Dic5C2311yescharacteristic
83C23×Dic5C2311yescharacteristic
84C22×Dic10—42no—
85C22×Dic10—42no—
86C22×Dic10—42no—
87C22×Dic10—42no—
88C22×Dic10C2×C418yes—
89C22×Dic10C2×C418yes—
90C22×Dic10C2314yes—
91C22×Dic10C2314yes—
92C2×C5×C4⋊C4—42no—
93C2×C5×C4⋊C4C2×C418yes—
94C2×C5×C4⋊C4C2314yes—
95C2×C5×C4⋊C4C2314yes—
96C2×C42×C5—42no—
97C2×C42×C5C2314yes—
98C2×C4⋊Dic5—42no—
99C2×C4⋊Dic5C2×C418yes—
100C2×C4⋊Dic5C2314yes—
101C2×C4⋊Dic5C2314yes—
102C2×#683—24no—
103C2×#683—24no—
104C2×#683—42no—
105C2×#683—42no—
106C2×#683C2×C418yes—
107C2×#683C2×C418yes—
108C2×#683C2314yes—
109C2×#683C2314yes—
110C2×#683C2314yes—
111C2×#683C2314yes—
112C2×#683C2314yes—
113C2×#683C2314yes—
114C2×C4×Dic5—24no—
115C2×C4×Dic5—42no—
116C2×C4×Dic5—42no—
117C2×C4×Dic5—42no—
118C2×C4×Dic5C2×C418yes—
119C2×C4×Dic5Q818yescomplemented
120C2×C4×Dic5C2314yes—
121C2×C4×Dic5C2314yes—
122C2×C4×Dic5C2314yes—
123C2×C4×Dic5C2314yes—
124C2×C4×Dic5C2314yes—
125C2×C4×Dic5C2314yes—
126#4034C2×C4164yescomplemented
127C5×#95C2314yes—
128C5×#95C2314yes—
129C5×#95C2314yes—
130C5×#95C2314yes—
131#3963C2314yes—
132#3963C2314yes—
133#3963C2314yes—
134#3963C2314yes—
135#3963C2314yes—
136#3963C2314yes—
137#3963C2314yes—
138#3963C2314yes—
139#3963C2314yes—
140#3963C2314yes—
141#3963C2314yes—
142#3963C2314yes—
143C22×C4⋊Q8—51no—
144#2331—516no—
145C2×#391—51no—
146C2×#391—51no—
147C2×#391—51no—
148C2×#391—51no—
149C2×#389—51no—
150C2×#389—51no—
151C2×#387—51no—
152C2×#387—51no—
153C2×#387—51no—
154C2×#387D511yescharacteristic
155C2×#382—51no—
156C2×C4×C4⋊C4—51no—
157C2×C4×C4⋊C4—51no—
158C2×#376—51no—
159C24×C5C2411yescharacteristic
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—22no—
170C22×C4×C5—22no—
171C22×C4×C5—22no—
172C22×C4×C5—22no—
173C22×C4×C5—22no—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—22no—
177C22×C4×C5C22×C414yes—
178C22×C4×C5C22×C414yes—
179C22×C4×C5C22×C414yes—
180C22×C4×C5C2×Q814yes—
181C22×C4×C5C2412yes—
182C22×C4×C5C2412yes—
183C22×C4×C5C2412yes—
184C22×C4×C5C2412yes—
185C22×C4×C5C2412yes—
186C22×C4×C5C2412yes—
187C22×C4×C5C2412yes—
188C22×Dic5—22no—
189C22×Dic5—22no—
190C22×Dic5—22no—
191C22×Dic5—22no—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5—22no—
196C22×Dic5—22no—
197C22×Dic5—22no—
198C22×Dic5—22no—
199C22×Dic5—22no—
200C22×Dic5—22no—
201C22×Dic5—22no—
202C22×Dic5—22no—
203C22×Dic5—22no—
204C22×Dic5—22no—
205C22×Dic5C22×C414yes—
206C22×Dic5C22×C414yes—
207C22×Dic5C22×C414yes—
208C22×Dic5C22×C414yes—
209C22×Dic5C2×Q814yes—
210C22×Dic5C2×Q814yes—
211C22×Dic5#4114yes—
212C22×Dic5C2412yes—
213C22×Dic5C2412yes—
214C22×Dic5C2412yes—
215C22×Dic5C2412yes—
216C22×Dic5C2412yes—
217C22×Dic5C2412yes—
218C22×Dic5C2412yes—
219C22×Dic5C2412yes—
220C2×Dic10—416no—
221C2×Dic10—416no—
222C2×Dic10—44no—
223C2×Dic10—44no—
224C2×Dic10—44no—
225C2×Dic10—44no—
226C2×Dic10C22×C4116yes—
227C2×Dic10C22×C4116yes—
228C5×C4⋊C4—44no—
229C5×C4⋊C4C22×C4116yes—
230C42×C5—44no—
231C4⋊Dic5—44no—
232C4⋊Dic5C22×C4116yes—
233#683—28no—
234#683—28no—
235#683—44no—
236#683—44no—
237#683C22×C4116yes—
238#683C22×C4116yes—
239C4×Dic5—28no—
240C4×Dic5—44no—
241C4×Dic5—44no—
242C4×Dic5—44no—
243C4×Dic5C22×C4116yes—
244C4×Dic5C2×Q8116yescomplemented
245C23×Q8—51no—
246C23×Q8—51no—
247C2×C4⋊Q8—516no—
248C2×C4⋊Q8—58no—
249C22×C4⋊C4—51no—
250C22×C4⋊C4—51no—
251C22×C4⋊C4—51no—
252C22×C4⋊C4—51no—
253C22×C4⋊C4—51no—
254C22×C4⋊C4—51no—
255C22×C4⋊C4—51no—
256C22×C4⋊C4—51no—
257C22×C4⋊C4C2×D511yescharacteristic
258C22×C4⋊C4C2×D511yescharacteristic
259C22×C42—51no—
260C22×C42—51no—
261C22×C42—51no—
262C22×C42—51no—
263C22×C42—51no—
264C22×C42—51no—
265C22×C42C2×D511yescharacteristic
266#391—58no—
267#391—58no—
268#391—58no—
269#391—58no—
270#389—58no—
271#389—58no—
272#387—58no—
273#387—58no—
274#387—58no—
275#387C2×D518yes—
276#382—58no—
277C4×C4⋊C4—58no—
278C4×C4⋊C4—58no—
279#376—58no—
280C2×#95—51no—
281C2×#95—51no—
282C2×#95—51no—
283C2×#95—51no—
284C2×#95—51no—
285C2×#95—51no—
286C2×#95—51no—
287C2×#95—51no—
288C2×#95—51no—
289C2×#95—51no—
290C2×#95—51no—
291C2×#95—51no—
292C2×#95C2×D511yescharacteristic
293C2×#95C2×D511yescharacteristic
294C2×#95C2×D511yescharacteristic
295C2×#95C2×D511yescharacteristic
296C23×C5C23×C412yes—
297C23×C5C22×Q812yes—
298C23×C5C2×#4112yes—
299C23×C5H(5,𝔽2)12yes—
300C23×C5H(5,𝔽2)12yes—
301C23×C52−1+412yes—
302C23×C52−1+412yes—
303C23×C5C2511yescharacteristic
304C2×C4×C5—28no—
305C2×C4×C5—28no—
306C2×C4×C5—28no—
307C2×C4×C5—28no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—22no—
325C2×C4×C5—44no—
326C2×C4×C5—44no—
327C2×C4×C5—44no—
328C2×C4×C5C23×C414yes—
329C2×C4×C5C23×C414yes—
330C2×C4×C5C23×C414yes—
331C2×C4×C5C22×Q814yes—
332C2×Dic5—28no—
333C2×Dic5—28no—
334C2×Dic5—28no—
335C2×Dic5—28no—
336C2×Dic5—22no—
337C2×Dic5—22no—
338C2×Dic5—22no—
339C2×Dic5—22no—
340C2×Dic5—22no—
341C2×Dic5—22no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—22no—
348C2×Dic5—22no—
349C2×Dic5—22no—
350C2×Dic5—22no—
351C2×Dic5—22no—
352C2×Dic5—22no—
353C2×Dic5—44no—
354C2×Dic5—44no—
355C2×Dic5—44no—
356C2×Dic5C4×Q8116yes—
357C2×Dic5C23×C414yes—
358C2×Dic5C23×C414yes—
359C2×Dic5C23×C414yes—
360C2×Dic5C23×C414yes—
361C2×Dic5C22×Q814yes—
362C2×Dic5C22×Q814yes—
363C2×Dic5C2×#4114yes—
364Dic10—416no—
365Dic10—416no—
366C22×Q8—58no—
367C22×Q8—58no—
368C22×Q8—54no—
369C22×Q8—54no—
370C22×Q8—202no—
371C22×Q8—202no—
372C22×Q8—202no—
373C22×Q8—202no—
374C23×C4—51no—
375C23×C4—51no—
376C23×C4—51no—
377C23×C4—51no—
378C23×C4—51no—
379C23×C4—51no—
380C23×C4—51no—
381C23×C4—51no—
382C23×C4C22×D511yescharacteristic
383C23×C4C22×D511yescharacteristic
384C23×C4C22×D511yescharacteristic
385C23×C4C22×D511yescharacteristic
386C23×C4C22×D511yescharacteristic
387C23×C4C22×D511yescharacteristic
388C23×C4C22×D511yescharacteristic
389C4⋊Q8—564no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—58no—
392C2×C4⋊C4—58no—
393C2×C4⋊C4—58no—
394C2×C4⋊C4—54no—
395C2×C4⋊C4—54no—
396C2×C4⋊C4—54no—
397C2×C4⋊C4—54no—
398C2×C4⋊C4—54no—
399C2×C4⋊C4—54no—
400C2×C4⋊C4—54no—
401C2×C4⋊C4—54no—
402C2×C4⋊C4—104no—
403C2×C4⋊C4—104no—
404C2×C4⋊C4—202no—
405C2×C4⋊C4—202no—
406C2×C4⋊C4—202no—
407C2×C4⋊C4C4×D518yes—
408C2×C4⋊C4C22×D514yes—
409C2×C4⋊C4C22×D514yes—
410C2×C42—42no—
411C2×C42—58no—
412C2×C42—58no—
413C2×C42—54no—
414C2×C42—54no—
415C2×C42—54no—
416C2×C42—54no—
417C2×C42—54no—
418C2×C42—54no—
419C2×C42—104no—
420C2×C42—202no—
421C2×C42—202no—
422C2×C42—202no—
423C2×C42C22×D514yes—
424#95—54no—
425#95—54no—
426#95—54no—
427#95—54no—
428#95—54no—
429#95—54no—
430#95—54no—
431#95—54no—
432#95—54no—
433#95—54no—
434#95—54no—
435#95—54no—
436#95C22×D514yes—
437#95C22×D514yes—
438#95C22×D514yes—
439#95C22×D514yes—
440C22×C5C2×C4×Q814yes—
441C22×C5#51814yes—
442C22×C5#51914yes—
443C22×C5#54214yes—
444C22×C5#54214yes—
445C22×C5#54414yes—
446C22×C5#54414yes—
447C22×C5C24×C411yescharacteristic
448C22×C5C23×Q811yescharacteristic
449C22×C5C22×#4111yescharacteristic
450C22×C5C2×H(5,𝔽2)11yescharacteristic
451C22×C5C2×H(5,𝔽2)11yescharacteristic
452C22×C5C2×2−1+411yescharacteristic
453C22×C5C2×2−1+411yescharacteristic
454C4×C5—24no—
455C4×C5—24no—
456C4×C5—24no—
457C4×C5—24no—
458C4×C5—42no—
459C4×C5—42no—
460C4×C5—42no—
461Dic5—24no—
462Dic5—24no—
463Dic5—24no—
464Dic5—24no—
465Dic5—42no—
466Dic5—42no—
467Dic5—42no—
468Dic5C2×C4×Q818yes—
469C24C23×D511yescharacteristic
470C2×Q8—516no—
471C2×Q8—516no—
472C2×Q8—2016no—
473C2×Q8—2016no—
474C2×Q8—204no—
475C2×Q8—204no—
476C2×Q8—204no—
477C2×Q8—204no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—54no—
496C22×C4—54no—
497C22×C4—54no—
498C22×C4—52no—
499C22×C4—54no—
500C22×C4—54no—
501C22×C4—54no—
502C22×C4—54no—
503C22×C4—52no—
504C22×C4—52no—
505C22×C4—52no—
506C22×C4—52no—
507C22×C4—52no—
508C22×C4—52no—
509C22×C4—52no—
510C22×C4—102no—
511C22×C4—102no—
512C22×C4—102no—
513C22×C4—102no—
514C22×C4—102no—
515C22×C4—102no—
516C22×C4—102no—
517C22×C4—102no—
518C22×C4—102no—
519C22×C4—102no—
520C22×C4—102no—
521C22×C4—102no—
522C22×C4—102no—
523C22×C4—102no—
524C22×C4—102no—
525C22×C4—102no—
526C22×C4—102no—
527C22×C4C2×C4×D514yes—
528C22×C4C2×C4×D514yes—
529C22×C4C2×C4×D514yes—
530C22×C4Q8×D514yes—
531C22×C4C23×D512yes—
532C22×C4C23×D512yes—
533C22×C4C23×D512yes—
534C22×C4C23×D512yes—
535C22×C4C23×D512yes—
536C22×C4C23×D512yes—
537C22×C4C23×D512yes—
538C4⋊C4—44no—
539C4⋊C4—516no—
540C4⋊C4—516no—
541C4⋊C4—516no—
542C4⋊C4—108no—
543C4⋊C4—108no—
544C4⋊C4—204no—
545C4⋊C4—204no—
546C4⋊C4—204no—
547C4⋊C4C2×C4×D5116yes—
548C42—44no—
549C42—516no—
550C42—516no—
551C42—108no—
552C42—204no—
553C42—204no—
554C42—204no—
555C2×C5#233118yescomplemented
556C2×C5C22×C4×Q811yescharacteristic
557C2×C5C2×#51811yescharacteristic
558C2×C5C2×#51911yescharacteristic
559C2×C5C2×#54211yescharacteristic
560C2×C5C2×#54211yescharacteristic
561C2×C5C2×#54411yescharacteristic
562C2×C5C2×#54411yescharacteristic
563C23C22×C4×D512yes—
564C23#414412yes—
565C23C2×Q8×D512yes—
566C23D5×#4112yes—
567C23#414912yes—
568C23#415012yes—
569C23#415012yes—
570C23C24×D511yescharacteristic
571Q8—2016no—
572Q8—2016no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—516no—
598C2×C4—54no—
599C2×C4—54no—
600C2×C4—54no—
601C2×C4—54no—
602C2×C4—54no—
603C2×C4—54no—
604C2×C4—54no—
605C2×C4—108no—
606C2×C4—108no—
607C2×C4—108no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—102no—
611C2×C4—108no—
612C2×C4—102no—
613C2×C4—102no—
614C2×C4—102no—
615C2×C4—102no—
616C2×C4—102no—
617C2×C4—102no—
618C2×C4—102no—
619C2×C4—102no—
620C2×C4—102no—
621C2×C4—102no—
622C2×C4—102no—
623C2×C4—102no—
624C2×C4—102no—
625C2×C4—102no—
626C2×C4—204no—
627C2×C4—204no—
628C2×C4—204no—
629C2×C4C22×C4×D514yes—
630C2×C4C22×C4×D514yes—
631C2×C4C22×C4×D514yes—
632C2×C4C2×Q8×D514yes—
633C5C2×#233111yescharacteristic, complemented, Sylow
634C22#6585014yes—
635C22#6587914yes—
636C22C4×Q8×D514yes—
637C22#6595714yes—
638C22#6599614yes—
639C22#6602314yes—
640C22#6604414yes—
641C22C23×C4×D511yescharacteristic
642C22C2×#414411yescharacteristic
643C22C22×Q8×D511yescharacteristic
644C22C2×D5×#4111yescharacteristic
645C22C2×#414911yescharacteristic
646C22C2×#415011yescharacteristic
647C22C2×#415011yescharacteristic
648C4—24no—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—58no—
656C4—104no—
657C4—104no—
658C4—104no—
659C4—104no—
660C4—202no—
661C4—202no—
662C4—202no—
663C2#1060668118yesdirect product factor
664C2C2×#6585011yescharacteristic
665C2C2×#6587911yescharacteristic
666C2C2×C4×Q8×D511yescharacteristic
667C2C2×#6595711yescharacteristic
668C2C2×#6599611yescharacteristic
669C2C2×#6602311yescharacteristic
670C2C2×#6604411yescharacteristic
subgroupG/H
centerC24C23×D5
commutatorC22×C5C24×C4
FrattiniC23C24×D5
FittingC2×C5×#387C2
socle, prime radicalC24×C5C24
cosocleC23×C5C25
hypercenterC2×#387D5
hypocenterC5C2×#2331
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
8. #10606681C2direct (N × H)
36. C2×#4034C4semidirect
119. C2×C4×Dic5Q8semidirect
126. #4034C2×C4semidirect
244. C4×Dic5C2×Q8semidirect
555. C2×C5#2331semidirect
633. C5C2×#2331semidirect
663. C2#10606681direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40342
C2×C5×#3872
C2×#652632
C2×#652612
C2×#652702
C2×#652182
C2×C4×#6832
#106066812
C2×#649352
C2×#649362
C2×#649432
C2×#649422
C2×#649392
C2×#649382
C2×#649372
C2×#23315