Finite Group Databasebeta

← C2×#10613231 #12971366

Subgroups — C2×#10613231 #12971366

Order 1280.

1778 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65956C211yesmaximal, characteristic
2C2×#65851C211yesmaximal, characteristic
3C2×C5×#485C211yesmaximal, characteristic
4C2×#65334C211yesmaximal, characteristic
5C2×#65285C211yesmaximal, characteristic
6#10613231C2116yesmaximal, direct product factor
7C2×#65166C211yesmaximal, characteristic, complemented
8C2×#65163C211yesmaximal, characteristic, complemented
9C2×#65146C211yesmaximal, characteristic, complemented
10C2×#65144C211yesmaximal, characteristic, complemented
11C2×#65081C211yesmaximal, characteristic, complemented
12C2×#65081C211yesmaximal, characteristic, complemented
13C2×#65085C211yesmaximal, characteristic, complemented
14C2×#65088C211yesmaximal, characteristic, complemented
15C2×#65012C211yesmaximal, characteristic
16C2×#65007C211yesmaximal, characteristic
17C2×C5×C22⋊Q8C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C2×Q8×Dic5C2211yescharacteristic
20C2×#4090C2211yescharacteristic
21C2×#4072C2211yescharacteristic
22C2×#4070C2211yescharacteristic
23C22×C4×Dic5C2211yescharacteristic
24#65956—24no—
25#65956C2218yes—
26#65851—24no—
27#65851C2218yes—
28C2×#4036C2211yescharacteristic
29C2×#4036C2211yescharacteristic
30C2×#4035—21no—
31C2×#4034C2211yescharacteristic
32C2×#4034C2211yescharacteristic
33C2×#4034C2211yescharacteristic
34C2×#4033—21no—
35C2×#4033C2211yescharacteristic
36C2×#4024—21no—
37C2×#4024—21no—
38C2×#4024—21no—
39C2×#4023—21no—
40C2×C4×Dic10—21no—
41C2×C4×Dic10—21no—
42C5×#485C2218yes—
43C2×C5×#107C2211yescharacteristic
44C2×C5×#106C2211yescharacteristic
45C2×C5×#103C2211yescharacteristic
46C2×C5×#103C2211yescharacteristic
47C2×C5×C22⋊C8C2211yescharacteristic
48C2×#3967C2211yescharacteristic
49C2×#3967C2211yescharacteristic
50C2×C22⋊#168C2211yescharacteristic
51C2×#3950C2211yescharacteristic
52C2×#3949C2211yescharacteristic
53C2×#3948C2211yescharacteristic
54C2×#3948C2211yescharacteristic
55C2×#3946C2211yescharacteristic
56C2×#3946C2211yescharacteristic
57C2×C8×Dic5C2211yescharacteristic
58C2×C8×Dic5C2211yescharacteristic
59#65334C2218yes—
60#65285C2218yes—
61C2×#3942C2211yescharacteristic
62C2×#3942C2211yescharacteristic
63C2×#3940C2211yescharacteristic
64C2×#3940C2211yescharacteristic
65C2×#3939C2211yescharacteristic
66C2×#3939C2211yescharacteristic
67#65166C2218yescomplemented
68#65163C2218yescomplemented
69#65146C2218yescomplemented
70#65144C2218yescomplemented
71#65088C2218yescomplemented
72#65085C2218yescomplemented
73#65081C2218yescomplemented
74#65081C2218yescomplemented
75#65012C2218yes—
76#65007C2218yes—
77C2×#3231—51nomaximal, Sylow
78C22×C5×Q8C2311yescharacteristic
79C23×C4×C5C2311yescharacteristic
80C23×Dic5D411yescharacteristic
81C23×Dic5D411yescharacteristic
82C22×Dic10—21no—
83C22×Dic10—21no—
84C22×Dic10—21no—
85C22×Dic10C2311yescharacteristic
86C22×C5×C8C2311yescharacteristic
87C22×C5×C8C2311yescharacteristic
88C5×C22⋊Q8—22no—
89C5×C22⋊Q8C2314yes—
90C5×#117—22no—
91C5×#117C2314yes—
92C2×C5×C4⋊C4—21no—
93C2×C5×C4⋊C4C2311yescharacteristic
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C4⋊C4C2311yescharacteristic
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C42×C5—21no—
99C2×C22⋊Dic5—21no—
100C2×C22⋊Dic5—21no—
101Q8×Dic5—22no—
102Q8×Dic5C2314yes—
103#4090—22no—
104#4090C2314yes—
105#4072—22no—
106#4072C2314yes—
107C2×C4⋊Dic5—21no—
108C2×C4⋊Dic5—21no—
109C2×C4⋊Dic5—21no—
110C2×C4⋊Dic5C2311yescharacteristic
111C2×C4⋊Dic5C2311yescharacteristic
112C2×C4⋊Dic5C2311yescharacteristic
113#4070—22no—
114#4070C2314yes—
115C2×#683—21no—
116C2×#683—21no—
117C2×#683—21no—
118C2×#683—21no—
119C2×#683—21no—
120C2×#683—21no—
121C2×#683—21no—
122C2×#683—21no—
123C2×#683—21no—
124C2×C4×Dic5—21no—
125C2×C4×Dic5—21no—
126C2×C4×Dic5—21no—
127C2×C4×Dic5D411yescharacteristic
128C2×C4×Dic5D411yescharacteristic
129C2×C4×Dic5D414yes—
130C2×C4×Dic5C2311yescharacteristic
131C2×C4×Dic5C2311yescharacteristic
132C2×C4×Dic5C2314yes—
133C22×#168C2311yescharacteristic
134C22×#168C2311yescharacteristic
135#4036—22no—
136#4036—22no—
137#4036C2314yes—
138#4036C2314yes—
139#4035—28no—
140#4034—22no—
141#4034—22no—
142#4034—22no—
143#4034C2314yes—
144#4034C2314yes—
145#4034C2314yes—
146#4033—28no—
147#4033—22no—
148#4033C2314yes—
149#4024—28no—
150#4024—28no—
151#4024—28no—
152#4023—28no—
153C4×Dic10—28no—
154C4×Dic10—28no—
155C5×#107C2314yes—
156C5×#106C2314yes—
157C5×#103C2314yes—
158C5×#103C2314yes—
159C5×C22⋊C8C2314yes—
160#3967C2314yes—
161#3967C2314yes—
162C22⋊#168C2314yes—
163#3950C2314yes—
164#3949C2314yes—
165#3948C2314yes—
166#3948C2314yes—
167#3946C2314yes—
168#3946C2314yes—
169C8×Dic5C2314yes—
170C8×Dic5C2314yes—
171#3942C2314yes—
172#3942C2314yes—
173#3940C2314yes—
174#3940C2314yes—
175#3939C2314yes—
176#3939C2314yes—
177C2×#533—51no—
178C2×#533—51no—
179#3231—516no—
180C2×#500—51no—
181C2×#498—51no—
182C2×#487—51no—
183C2×#487—51no—
184C2×#485—51no—
185C2×#485—51no—
186C2×#485—51no—
187C2×#485D511yescharacteristic
188C2×#479—51no—
189C2×#479—51no—
190C2×#479—51no—
191C2×#479—51no—
192C2×#431—51no—
193C24×C5C2×D411yescharacteristic
194C2×C5×Q8—21no—
195C2×C5×Q8—22no—
196C2×C5×Q8—22no—
197C2×C5×Q8C2412yes—
198C22×C4×C5—22no—
199C22×C4×C5—22no—
200C22×C4×C5—21no—
201C22×C4×C5—21no—
202C22×C4×C5—21no—
203C22×C4×C5—21no—
204C22×C4×C5C2×D411yescharacteristic
205C22×C4×C5C2×D412yes—
206C22×C4×C5C2412yes—
207C22×C4×C5C2411yescharacteristic
208C22×Dic5—24no—
209C22×Dic5—24no—
210C22×Dic5—21no—
211C22×Dic5—21no—
212C22×Dic5—21no—
213C22×Dic5—21no—
214C22×Dic5C2×D414yes—
215C22×Dic5C2×D414yes—
216C22×Dic5C2×D411yescharacteristic
217C22×Dic5C2×D411yescharacteristic
218C22×Dic5C2×D411yescharacteristic
219C22×Dic5C2×D411yescharacteristic
220C2×Dic10—22no—
221C2×Dic10—22no—
222C2×Dic10—21no—
223C2×Dic10—24no—
224C2×Dic10—24no—
225C2×Dic10—24no—
226C2×Dic10—44no—
227C2×Dic10—44no—
228C2×Dic10—44no—
229C2×Dic10C2412yes—
230C2×C5×C8—22no—
231C2×C5×C8—22no—
232C2×C5×C8C2412yes—
233C2×C5×C8C2412yes—
234C5×C4⋊C4—24no—
235C5×C4⋊C4—21no—
236C5×C4⋊C4—21no—
237C5×C4⋊C4—21no—
238C5×C4⋊C4C2412yes—
239C5×C4⋊C4C2412yes—
240C5×C4⋊C4C2412yes—
241C5×C22⋊C4—24no—
242C5×C22⋊C4—24no—
243C42×C5—24no—
244C22⋊Dic5—24no—
245C22⋊Dic5—24no—
246C4⋊Dic5—21no—
247C4⋊Dic5—21no—
248C4⋊Dic5—21no—
249C4⋊Dic5—24no—
250C4⋊Dic5—24no—
251C4⋊Dic5—24no—
252C4⋊Dic5C2412yes—
253C4⋊Dic5C2412yes—
254C4⋊Dic5C2412yes—
255#683—24no—
256#683—24no—
257#683—24no—
258#683—24no—
259#683—24no—
260#683—24no—
261#683—24no—
262#683—24no—
263#683—24no—
264C4×Dic5—24no—
265C4×Dic5—24no—
266C4×Dic5—24no—
267C4×Dic5C2×D414yes—
268C4×Dic5C2×D414yes—
269C4×Dic5C2×D412yes—
270C4×Dic5C2×D412yes—
271C4×Dic5C2412yes—
272C4×Dic5C2412yes—
273C2×#168—22no—
274C2×#168—22no—
275C2×#168C2412yes—
276C2×#168C2412yes—
277#533—58no—
278#533—58no—
279#533—104no—
280#533—104no—
281C2×C4⋊Q8—51no—
282C2×C4⋊Q8—51no—
283C2×C4⋊Q8—51no—
284C2×C4⋊Q8—51no—
285C2×#125—51no—
286C2×#125—51no—
287C2×#125—101no—
288C2×C22⋊Q8—51no—
289C2×C22⋊Q8—101no—
290C2×C22⋊Q8—101no—
291C2×C22⋊Q8—101no—
292C2×C22⋊Q8C2×D511yescharacteristic
293C2×C4×Q8—51no—
294C2×C4×Q8—51no—
295C2×C4×Q8—101no—
296C2×C4×Q8—101no—
297C2×C4×Q8—101no—
298C2×#117—51no—
299C2×#117—101no—
300C2×#117C2×D511yescharacteristic
301C22×C42—51no—
302#500—58no—
303#498—58no—
304#487—58no—
305#487—58no—
306#485—58no—
307#485—58no—
308#485—58no—
309#485C2×D518yes—
310#479—58no—
311#479—58no—
312#479—58no—
313#479—58no—
314#431—58no—
315C2×#107—51no—
316C2×#107—51no—
317C2×#107—51no—
318C2×#107C2×D511yescharacteristic
319C2×#106—51no—
320C2×#106—51no—
321C2×#106—51no—
322C2×#106C2×D511yescharacteristic
323C2×C4⋊C8—51no—
324C2×C4⋊C8—51no—
325C2×#103—51no—
326C2×#103—51no—
327C2×#103—51no—
328C2×#103—51no—
329C2×#103—51no—
330C2×#103—51no—
331C2×#103C2×D511yescharacteristic
332C2×#103C2×D511yescharacteristic
333C2×C22⋊C8—51no—
334C2×C22⋊C8C2×D511yescharacteristic
335C2×C4×C8—51no—
336C2×C4×C8—51no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5C22×D412yes—
341C23×C5C22×D411yescharacteristic
342C5×Q8—22no—
343C5×Q8—22no—
344C5×Q8—42no—
345C2×C4×C5—22no—
346C2×C4×C5—24no—
347C2×C4×C5—22no—
348C2×C4×C5—22no—
349C2×C4×C5—22no—
350C2×C4×C5—22no—
351C2×C4×C5—22no—
352C2×C4×C5—22no—
353C2×C4×C5—42no—
354C2×C4×C5—42no—
355C2×C4×C5—42no—
356C2×C4×C5—42no—
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—28no—
365C2×Dic5—28no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—42no—
371C2×Dic5—42no—
372C2×Dic5—42no—
373C2×Dic5—42no—
374C2×Dic5#13514yes—
375C2×Dic5#13714yes—
376C2×Dic5C22×D412yes—
377C2×Dic5C22×D412yes—
378C2×Dic5C22×D412yes—
379C2×Dic5C22×D412yes—
380Dic10—22no—
381Dic10—22no—
382Dic10—42no—
383Dic10—48no—
384Dic10—48no—
385Dic10—48no—
386C5×C8—22no—
387C5×C8—22no—
388#168—22no—
389#168—22no—
390C22×Q8—51no—
391C22×Q8—101no—
392C22×Q8—101no—
393C22×Q8—101no—
394C22×Q8C22×D511yescharacteristic
395C23×C4—51no—
396C23×C4—51no—
397C23×C4C22×D511yescharacteristic
398C22×C8—51no—
399C22×C8—51no—
400C22×C8C22×D511yescharacteristic
401C22×C8C22×D511yescharacteristic
402C4⋊Q8—54no—
403C4⋊Q8—54no—
404C4⋊Q8—54no—
405C4⋊Q8—54no—
406C4⋊Q8—102no—
407C4⋊Q8—102no—
408C4⋊Q8—102no—
409C4⋊Q8—102no—
410#125—54no—
411#125—54no—
412#125—102no—
413#125—102no—
414#125—108no—
415C22⋊Q8—22no—
416C22⋊Q8—54no—
417C22⋊Q8—108no—
418C22⋊Q8—108no—
419C22⋊Q8—108no—
420C22⋊Q8—102no—
421C22⋊Q8C22×D514yes—
422C4×Q8—54no—
423C4×Q8—54no—
424C4×Q8—108no—
425C4×Q8—108no—
426C4×Q8—102no—
427C4×Q8—108no—
428C4×Q8—102no—
429#117—22no—
430#117—54no—
431#117—108no—
432#117—102no—
433#117C22×D514yes—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—51no—
436C2×C4⋊C4—51no—
437C2×C4⋊C4—51no—
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⋊C4—101no—
450C2×C4⋊C4C22×D511yescharacteristic
451C2×C4⋊C4C22×D511yescharacteristic
452C2×C4⋊C4C22×D511yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—101no—
456C2×C22⋊C4—101no—
457C2×C42—21no—
458C2×C42—51no—
459C2×C42—51no—
460C2×C42—54no—
461C2×C42—51no—
462C2×C42—51no—
463C2×C42—54no—
464C2×C42—101no—
465C2×C42—101no—
466C2×C42—101no—
467#107—54no—
468#107—54no—
469#107—54no—
470#107C22×D514yes—
471#106—54no—
472#106—54no—
473#106—54no—
474#106C22×D514yes—
475C4⋊C8—54no—
476C4⋊C8—54no—
477#103—54no—
478#103—54no—
479#103—54no—
480#103—54no—
481#103—54no—
482#103—54no—
483#103C22×D514yes—
484#103C22×D514yes—
485C22⋊C8—54no—
486C22⋊C8C22×D514yes—
487C4×C8—54no—
488C4×C8—54no—
489C22×C5—22no—
490C22×C5—22no—
491C22×C5—22no—
492C22×C5—44no—
493C22×C5#53612yes—
494C22×C5C2×#13512yes—
495C22×C5C2×#13712yes—
496C22×C5C23×D411yescharacteristic
497C4×C5—22no—
498C4×C5—42no—
499C4×C5—42no—
500C4×C5—42no—
501C4×C5—42no—
502C4×C5#53612yes—
503C4×C5C2×2−1+411yescharacteristic
504C4×C5C2×2−1+411yescharacteristic
505Dic5—22no—
506Dic5—22no—
507Dic5—42no—
508Dic5—42no—
509Dic5—42no—
510Dic5—42no—
511Dic5C2×#13514yes—
512Dic5C2×#13714yes—
513C24D4×D511yescharacteristic
514C2×Q8—21no—
515C2×Q8—22no—
516C2×Q8—22no—
517C2×Q8—52no—
518C2×Q8—104no—
519C2×Q8—104no—
520C2×Q8—104no—
521C2×Q8—101no—
522C2×Q8—102no—
523C2×Q8—102no—
524C2×Q8—204no—
525C2×Q8—204no—
526C2×Q8—204no—
527C2×Q8C23×D512yes—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—51no—
535C22×C4—51no—
536C22×C4—54no—
537C22×C4—54no—
538C22×C4—51no—
539C22×C4—51no—
540C22×C4—104no—
541C22×C4—104no—
542C22×C4—101no—
543C22×C4—101no—
544C22×C4—101no—
545C22×C4—101no—
546C22×C4D4×D512yes—
547C22×C4D4×D511yescharacteristic
548C22×C4C23×D512yes—
549C22×C4C23×D511yescharacteristic
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—52no—
553C2×C8—52no—
554C2×C8—102no—
555C2×C8—102no—
556C2×C8C23×D512yes—
557C2×C8C23×D512yes—
558C4⋊C4—24no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—52no—
563C4⋊C4—52no—
564C4⋊C4—52no—
565C4⋊C4—104no—
566C4⋊C4—104no—
567C4⋊C4—104no—
568C4⋊C4—104no—
569C4⋊C4—104no—
570C4⋊C4—104no—
571C4⋊C4—104no—
572C4⋊C4—104no—
573C4⋊C4—104no—
574C4⋊C4—104no—
575C4⋊C4—104no—
576C4⋊C4—101no—
577C4⋊C4—101no—
578C4⋊C4—101no—
579C4⋊C4—104no—
580C4⋊C4C23×D512yes—
581C4⋊C4C23×D512yes—
582C4⋊C4C23×D512yes—
583C22⋊C4—24no—
584C22⋊C4—24no—
585C22⋊C4—104no—
586C22⋊C4—104no—
587C42—24no—
588C42—54no—
589C42—54no—
590C42—52no—
591C42—52no—
592C42—52no—
593C42—52no—
594C42—104no—
595C42—104no—
596C42—104no—
597C2×C5—42no—
598C2×C5#323114yescomplemented
599C2×C5C2×#53611yescharacteristic
600C2×C5C22×#13511yescharacteristic
601C2×C5C22×#13711yescharacteristic
602C23—22no—
603C23—22no—
604C23—22no—
605C23C2×D4×D512yes—
606C23C2×D4×D511yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—42no—
610Q8—102no—
611Q8—102no—
612Q8—208no—
613Q8—208no—
614Q8—208no—
615Q8—202no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—54no—
629C2×C4—54no—
630C2×C4—52no—
631C2×C4—52no—
632C2×C4—52no—
633C2×C4—52no—
634C2×C4—108no—
635C2×C4—108no—
636C2×C4—102no—
637C2×C4—102no—
638C2×C4—102no—
639C2×C4—102no—
640C2×C4—102no—
641C2×C4—102no—
642C2×C4—202no—
643C2×C4—202no—
644C2×C4—202no—
645C2×C4—202no—
646C2×C4C2×D4×D512yes—
647C2×C4C2×D4×D511yescharacteristic
648C2×C4#415012yes—
649C2×C4#415012yes—
650C2×C4C24×D511yescharacteristic
651C8—22no—
652C8—22no—
653C8—102no—
654C8—102no—
655C5C2×#323111yescharacteristic, complemented, Sylow
656C22—22no—
657C22—22no—
658C22—22no—
659C22—44no—
660C22#6598212yes—
661C22D5×#13512yes—
662C22D5×#13712yes—
663C22C22×D4×D511yescharacteristic
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—54no—
670C4—54no—
671C4—102no—
672C4—102no—
673C4—202no—
674C4—202no—
675C4—202no—
676C4—202no—
677C4#6598212yes—
678C4C2×#415011yescharacteristic
679C4C2×#415011yescharacteristic
680C2—42no—
681C2#1061323114yesdirect product factor
682C2C2×#6598211yescharacteristic
683C2C2×D5×#13511yescharacteristic
684C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#485C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenterC2×#485D5
hypocenterC5C2×#3231
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
6. #10613231C2direct (N × H)
7. C2×#65166C2semidirect
8. C2×#65163C2semidirect
9. C2×#65146C2semidirect
10. C2×#65144C2semidirect
11. C2×#65081C2semidirect
12. C2×#65081C2semidirect
13. C2×#65085C2semidirect
14. C2×#65088C2semidirect
67. #65166C22semidirect
68. #65163C22semidirect
69. #65146C22semidirect
70. #65144C22semidirect
71. #65088C22semidirect
72. #65085C22semidirect
73. #65081C22semidirect
74. #65081C22semidirect
598. C2×C5#3231semidirect
655. C5C2×#3231semidirect
681. C2#10613231direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659562
C2×#658512
C2×C5×#4852
C2×#653342
C2×#652852
#106132312
C2×#651662
C2×#651632
C2×#651462
C2×#651442
C2×#650812
C2×#650852
C2×#650882
C2×#650122
C2×#650072
C2×#32315