Finite Group Databasebeta

← C2×#10612317 #12979101

Subgroups — C2×#10612317 #12979101

Order 1280.

1810 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66005C211yesmaximal, characteristic
2C2×#65903C211yesmaximal, characteristic, complemented
3C2×C5×#464C211yesmaximal, characteristic
4C2×#65471C211yesmaximal, characteristic, complemented
5C2×#65473C211yesmaximal, characteristic, complemented
6C2×#65455C211yesmaximal, characteristic
7C2×#65453C211yesmaximal, characteristic
8C2×#65345C211yesmaximal, characteristic
9C2×C4×#689C211yesmaximal, characteristic, complemented
10#10612317C2116yesmaximal, direct product factor
11C2×#65118C211yesmaximal, characteristic, complemented
12C2×#65120C211yesmaximal, characteristic, complemented
13C2×#65086C211yesmaximal, characteristic, complemented
14C2×#65083C211yesmaximal, characteristic, complemented
15C2×#65054C211yesmaximal, characteristic
16C2×#65052C211yesmaximal, characteristic
17C22×C5×Q16C2211yescharacteristic
18C22×C5×SD16C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23C2×Q8×Dic5C2211yescharacteristic
24C2×#4090C2211yescharacteristic
25C22×#689C2211yescharacteristic
26C2×#4082C2211yescharacteristic
27C2×D4×Dic5C2211yescharacteristic
28#66005—24no—
29#66005C2218yes—
30#65903—24no—
31#65903C2218yescomplemented
32C2×#4036—21no—
33C2×#4036C2211yescharacteristic
34C2×#4034—21no—
35C2×#4034C2211yescharacteristic
36C2×#4025—21no—
37C2×#4025—21no—
38C2×#4024—21no—
39C2×#4023—21no—
40C2×#4016—21no—
41C2×#4016C2211yescharacteristic
42C2×C4×Dic10—21no—
43C2×C4×Dic10C2211yescharacteristic
44C2×C4×Dic10C2211yescharacteristic
45C5×#464C2218yes—
46C2×C5×C4⋊C8C2211yescharacteristic
47C2×C5×#103C2211yescharacteristic
48C2×C5×#102C2211yescharacteristic
49C2×#3967C2211yescharacteristic
50C2×#3967C2211yescharacteristic
51C2×#3967C2211yescharacteristic
52C2×#3964C2211yescharacteristic
53C2×#3964C2211yescharacteristic
54#65473C2218yescomplemented
55#65471C2218yescomplemented
56#65455C2218yes—
57#65453C2218yes—
58C2×#3950C2211yescharacteristic
59C2×#3949C2211yescharacteristic
60C2×#3948C2211yescharacteristic
61C2×#3948C2211yescharacteristic
62C2×#3947C2211yescharacteristic
63C2×#3947C2211yescharacteristic
64C2×#3946C2211yescharacteristic
65C2×#3946C2211yescharacteristic
66#65345C2218yes—
67C4×#689C2218yescomplemented
68C2×#3942C2211yescharacteristic
69C2×#3939C2211yescharacteristic
70C2×C4×#168C2211yescharacteristic
71#65120C2218yescomplemented
72#65118C2218yescomplemented
73#65086C2218yescomplemented
74#65083C2218yescomplemented
75#65054C2218yes—
76#65052C2218yes—
77C2×#3377—51nomaximal, Sylow
78C22×C5×Q8C2311yescharacteristic
79C22×C5×Q8C2311yescharacteristic
80C22×C5×D4C2311yescharacteristic
81C23×Dic5—21no—
82C22×Dic10—21no—
83C22×Dic10D411yescharacteristic
84C22×Dic10C2311yescharacteristic
85C2×C5×Q16—24no—
86C2×C5×Q16C2314yes—
87C2×C5×SD16—24no—
88C2×C5×SD16C2314yes—
89C22×C5×C8C2311yescharacteristic
90C22×C5×C8C2311yescharacteristic
91C5×#124—22no—
92C5×#124C2314yes—
93C4×C5×Q8—22no—
94C4×C5×Q8C2314yes—
95C2×C5×C4⋊C4—21no—
96C2×C5×C4⋊C4C2311yescharacteristic
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C42×C5—21no—
100C2×C42×C5C2311yescharacteristic
101C2×C22⋊Dic5—21no—
102C2×C22⋊Dic5—21no—
103C2×C22⋊Dic5—21no—
104Q8×Dic5—22no—
105Q8×Dic5—22no—
106Q8×Dic5—22no—
107Q8×Dic5C2314yes—
108Q8×Dic5C2314yes—
109Q8×Dic5C2314yes—
110#4090—22no—
111#4090C2314yes—
112C2×#689—24no—
113C2×#689C2314yes—
114#4082—22no—
115#4082C2314yes—
116D4×Dic5—22no—
117D4×Dic5C2314yes—
118C2×C4⋊Dic5—21no—
119C2×C4⋊Dic5—21no—
120C2×C4⋊Dic5—21no—
121C2×C4⋊Dic5—21no—
122C2×C4⋊Dic5D411yescharacteristic
123C2×C4⋊Dic5C2311yescharacteristic
124C2×C4⋊Dic5C2311yescharacteristic
125C2×C4⋊Dic5C2311yescharacteristic
126C2×#683—21no—
127C2×#683—21no—
128C2×#683—21no—
129C2×#683—21no—
130C2×#683—21no—
131C2×#683—21no—
132C2×#683—21no—
133C2×#683D411yescharacteristic
134C2×#683D411yescharacteristic
135C2×C4×Dic5—21no—
136C2×C4×Dic5—21no—
137C2×C4×Dic5—21no—
138C2×C4×Dic5—21no—
139C2×C4×Dic5C2311yescharacteristic
140C2×C4×Dic5C2311yescharacteristic
141C22×#168C2311yescharacteristic
142C22×#168C2311yescharacteristic
143#4036—28no—
144#4036—22no—
145#4036C2314yes—
146#4034—28no—
147#4034—22no—
148#4034C2314yes—
149#4025—28no—
150#4025—28no—
151#4024—28no—
152#4023—28no—
153#4016—22no—
154#4016—28no—
155#4016C2314yes—
156C4×Dic10—22no—
157C4×Dic10—28no—
158C4×Dic10D414yes—
159C4×Dic10C2314yes—
160C4×Dic10C2314yes—
161C5×C4⋊C8C2314yes—
162C5×#103C2314yes—
163C5×#102C2314yes—
164#3967C2314yes—
165#3967C2314yes—
166#3967C2314yes—
167#3964C2314yes—
168#3964C2314yes—
169#3950C2314yes—
170#3949C2314yes—
171#3948C2314yes—
172#3948C2314yes—
173#3947C2314yes—
174#3947C2314yes—
175#3946C2314yes—
176#3946C2314yes—
177#3942C2314yes—
178#3939C2314yes—
179C4×#168C2314yes—
180C2×#557—51no—
181C2×#556—51no—
182#3377—516no—
183C2×#491—51no—
184C2×#489—51no—
185C2×#488—51no—
186C2×#479—51no—
187C2×#477—51no—
188C2×#478—51no—
189C2×#464—51no—
190C2×#464D511yescharacteristic
191C2×#462—51no—
192C2×#446—51no—
193C2×#441—51no—
194C2×#440—51no—
195C2×C4×Q16—51no—
196C24×C5—21no—
197C2×C5×Q8—21no—
198C2×C5×Q8—21no—
199C2×C5×Q8—22no—
200C2×C5×Q8—22no—
201C2×C5×Q8—22no—
202C2×C5×Q8#4114yes—
203C2×C5×Q8C2412yes—
204C2×C5×Q8C2412yes—
205C2×C5×D4—21no—
206C2×C5×D4—22no—
207C2×C5×D4—22no—
208C2×C5×D4C2412yes—
209C22×C4×C5—21no—
210C22×C4×C5—21no—
211C22×C4×C5—21no—
212C22×C4×C5C2×D411yescharacteristic
213C22×C4×C5C2×D411yescharacteristic
214C22×C4×C5C2411yescharacteristic
215C22×Dic5—21no—
216C22×Dic5—21no—
217C22×Dic5—21no—
218C22×Dic5—21no—
219C22×Dic5—24no—
220C22×Dic5—44no—
221C22×Dic5C2×D411yescharacteristic
222C22×Dic5C2×D411yescharacteristic
223C22×Dic5C2×D411yescharacteristic
224C22×Dic5C2×D411yescharacteristic
225C2×Dic10—22no—
226C2×Dic10—22no—
227C2×Dic10—21no—
228C2×Dic10—24no—
229C2×Dic10—42no—
230C2×Dic10—44no—
231C2×Dic10C2×D414yes—
232C2×Dic10C2412yes—
233C5×Q16—28no—
234C5×SD16—28no—
235C2×C5×C8—22no—
236C2×C5×C8—22no—
237C2×C5×C8C2412yes—
238C2×C5×C8C2412yes—
239C5×C4⋊C4—24no—
240C5×C4⋊C4—21no—
241C5×C4⋊C4C2412yes—
242C5×C22⋊C4—24no—
243C5×C22⋊C4—24no—
244C42×C5—24no—
245C42×C5C2×D412yes—
246C42×C5C2412yes—
247C22⋊Dic5—24no—
248C22⋊Dic5—24no—
249C22⋊Dic5—24no—
250#689—28no—
251C4⋊Dic5—21no—
252C4⋊Dic5—21no—
253C4⋊Dic5—21no—
254C4⋊Dic5—24no—
255C4⋊Dic5—24no—
256C4⋊Dic5—24no—
257C4⋊Dic5—24no—
258C4⋊Dic5C2×D414yes—
259C4⋊Dic5C2412yes—
260C4⋊Dic5C2412yes—
261C4⋊Dic5C2412yes—
262#683—24no—
263#683—24no—
264#683—24no—
265#683—24no—
266#683—24no—
267#683—24no—
268#683—24no—
269#683C2×D414yes—
270#683C2×D414yes—
271C4×Dic5—24no—
272C4×Dic5—24no—
273C4×Dic5—24no—
274C4×Dic5—24no—
275C4×Dic5C2×D412yes—
276C4×Dic5C2×D412yes—
277C4×Dic5C2412yes—
278C4×Dic5C2412yes—
279C2×#168—22no—
280C2×#168—22no—
281C2×#168C2412yes—
282C2×#168C2412yes—
283C22×Q16—51no—
284C22×Q16C2×D511yescharacteristic
285C22×SD16C2×D511yescharacteristic
286#557—58no—
287#557—104no—
288#556—58no—
289#556—104no—
290C2×C4⋊Q8—51no—
291C2×C4⋊Q8—51no—
292C2×C4⋊Q8—101no—
293C2×#126—101no—
294C2×#126—101no—
295C2×#125—51no—
296C2×#125—51no—
297C2×#125—101no—
298C2×#125—101no—
299C2×#124—51no—
300C2×#124C2×D511yescharacteristic
301C2×C22⋊Q8—101no—
302C2×C4×Q8—51no—
303C2×C4×Q8—51no—
304C2×C4×Q8—51no—
305C2×C4×Q8—51no—
306C2×C4×Q8—51no—
307C2×C4×Q8—101no—
308C2×C4×Q8C2×D511yescharacteristic
309C2×C4×D4—51no—
310C2×#117—101no—
311#491—58no—
312#489—58no—
313#488—58no—
314#479—58no—
315#478—58no—
316#477—58no—
317#464—58no—
318#464C2×D518yes—
319#462—58no—
320#446—58no—
321#441—58no—
322#440—58no—
323C4×Q16—58no—
324C2×#107—51no—
325C2×#107—51no—
326C2×#106—51no—
327C2×C4⋊C8—51no—
328C2×C4⋊C8—51no—
329C2×C4⋊C8C2×D511yescharacteristic
330C2×#103—51no—
331C2×#103—51no—
332C2×#103—51no—
333C2×#103—51no—
334C2×#103—51no—
335C2×#103—51no—
336C2×#103C2×D511yescharacteristic
337C2×#102—51no—
338C2×#102—51no—
339C2×#102C2×D511yescharacteristic
340C2×#97—51no—
341C2×#97—51no—
342C2×C4×C8—51no—
343C23×C5—22no—
344C23×C5—42no—
345C23×C5—42no—
346C23×C5—42no—
347C23×C5C22×D411yescharacteristic
348C5×Q8—24no—
349C5×Q8—22no—
350C5×Q8—22no—
351C5×Q8—22no—
352C5×Q8—42no—
353C5×Q8C2×#4114yes—
354C5×D4—22no—
355C5×D4—22no—
356C5×D4—42no—
357C2×C4×C5—24no—
358C2×C4×C5—22no—
359C2×C4×C5—22no—
360C2×C4×C5—22no—
361C2×C4×C5—22no—
362C2×C4×C5—22no—
363C2×C4×C5—42no—
364C2×C4×C5—42no—
365C2×C4×C5C22×D412yes—
366C2×C4×C5C22×D412yes—
367C2×C4×C5C22×D411yescharacteristic
368C2×C4×C5C2×#4112yes—
369C2×C4×C52−1+412yes—
370C2×C4×C5C2511yescharacteristic
371C2×Dic5—22no—
372C2×Dic5—22no—
373C2×Dic5—22no—
374C2×Dic5—22no—
375C2×Dic5—22no—
376C2×Dic5—22no—
377C2×Dic5—22no—
378C2×Dic5—22no—
379C2×Dic5—48no—
380C2×Dic5—42no—
381C2×Dic5—42no—
382C2×Dic5—42no—
383C2×Dic5—42no—
384C2×Dic5C22×D412yes—
385C2×Dic5C22×D412yes—
386C2×Dic5C22×D412yes—
387C2×Dic5C22×D412yes—
388Dic10—22no—
389Dic10—22no—
390Dic10—44no—
391Dic10—48no—
392Dic10—42no—
393C5×C8—22no—
394C5×C8—22no—
395#168—22no—
396#168—22no—
397C22×Q8—51no—
398C22×Q8—51no—
399C22×Q8—101no—
400C22×Q8C22×D511yescharacteristic
401C22×Q8C22×D511yescharacteristic
402C22×D4C22×D511yescharacteristic
403C23×C4—101no—
404C2×Q16—24no—
405C2×Q16—54no—
406C2×Q16—104no—
407C2×Q16C22×D514yes—
408C2×SD16—24no—
409C2×SD16C22×D514yes—
410C22×C8—51no—
411C22×C8—51no—
412C22×C8C22×D511yescharacteristic
413C22×C8C22×D511yescharacteristic
414C4⋊Q8—54no—
415C4⋊Q8—54no—
416C4⋊Q8—102no—
417C4⋊Q8—102no—
418C4⋊Q8—108no—
419#126—108no—
420#126—108no—
421#125—54no—
422#125—54no—
423#125—108no—
424#125—102no—
425#125—108no—
426#125—102no—
427#124—22no—
428#124—54no—
429#124—102no—
430#124C22×D514yes—
431C22⋊Q8—108no—
432C4×Q8—22no—
433C4×Q8—54no—
434C4×Q8—54no—
435C4×Q8—54no—
436C4×Q8—54no—
437C4×Q8—54no—
438C4×Q8—54no—
439C4×Q8—108no—
440C4×Q8—102no—
441C4×Q8—102no—
442C4×Q8—102no—
443C4×Q8—102no—
444C4×Q8C22×D514yes—
445C4×D4—54no—
446C4×D4—102no—
447#117—108no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—51no—
450C2×C4⋊C4—51no—
451C2×C4⋊C4—51no—
452C2×C4⋊C4—51no—
453C2×C4⋊C4—51no—
454C2×C4⋊C4—51no—
455C2×C4⋊C4—101no—
456C2×C4⋊C4—101no—
457C2×C4⋊C4—101no—
458C2×C4⋊C4—101no—
459C2×C4⋊C4—101no—
460C2×C4⋊C4—101no—
461C2×C4⋊C4—101no—
462C2×C4⋊C4—101no—
463C2×C4⋊C4—101no—
464C2×C4⋊C4—101no—
465C2×C4⋊C4—101no—
466C2×C4⋊C4C22×D511yescharacteristic
467C2×C22⋊C4—21no—
468C2×C22⋊C4—21no—
469C2×C22⋊C4—101no—
470C2×C22⋊C4—101no—
471C2×C22⋊C4—101no—
472C2×C42—21no—
473C2×C42—51no—
474C2×C42—51no—
475C2×C42—101no—
476C2×C42—101no—
477C2×C42—101no—
478C2×C42—101no—
479C2×C42C22×D511yescharacteristic
480#107—54no—
481#107—54no—
482#106—54no—
483C4⋊C8—54no—
484C4⋊C8—54no—
485C4⋊C8C22×D514yes—
486#103—54no—
487#103—54no—
488#103—54no—
489#103—54no—
490#103—54no—
491#103—54no—
492#103C22×D514yes—
493#102—54no—
494#102—54no—
495#102C22×D514yes—
496#97—54no—
497#97—54no—
498C4×C8—54no—
499C22×C5—42no—
500C22×C5—42no—
501C22×C5—42no—
502C22×C5—84no—
503C22×C5#54812yes—
504C22×C5#57512yes—
505C22×C5#57812yes—
506C22×C5C23×D411yescharacteristic
507C4×C5—24no—
508C4×C5—22no—
509C4×C5—22no—
510C4×C5—42no—
511C4×C5—42no—
512C4×C5#54812yes—
513C4×C5C22×#4111yescharacteristic
514C4×C5C2×2−1+411yescharacteristic
515Dic5—22no—
516Dic5—22no—
517Dic5—22no—
518Dic5—22no—
519Dic5—42no—
520Dic5—42no—
521Dic5—42no—
522Dic5—42no—
523C24—21no—
524C2×Q8—21no—
525C2×Q8—21no—
526C2×Q8—22no—
527C2×Q8—22no—
528C2×Q8—22no—
529C2×Q8—54no—
530C2×Q8—52no—
531C2×Q8—104no—
532C2×Q8—101no—
533C2×Q8—102no—
534C2×Q8—102no—
535C2×Q8—202no—
536C2×Q8—204no—
537C2×Q8#71114yes—
538C2×Q8C23×D512yes—
539C2×Q8C23×D512yes—
540C2×D4—21no—
541C2×D4—22no—
542C2×D4—22no—
543C2×D4C23×D512yes—
544C22×C4—21no—
545C22×C4—21no—
546C22×C4—21no—
547C22×C4—51no—
548C22×C4—51no—
549C22×C4—51no—
550C22×C4—51no—
551C22×C4—104no—
552C22×C4—101no—
553C22×C4—101no—
554C22×C4—101no—
555C22×C4—101no—
556C22×C4—204no—
557C22×C4D4×D511yescharacteristic
558C22×C4D4×D511yescharacteristic
559C22×C4C23×D511yescharacteristic
560Q16—28no—
561Q16—108no—
562SD16—28no—
563C2×C8—22no—
564C2×C8—22no—
565C2×C8—52no—
566C2×C8—52no—
567C2×C8—102no—
568C2×C8—102no—
569C2×C8C23×D512yes—
570C2×C8C23×D512yes—
571C4⋊C4—24no—
572C4⋊C4—21no—
573C4⋊C4—54no—
574C4⋊C4—54no—
575C4⋊C4—54no—
576C4⋊C4—52no—
577C4⋊C4—52no—
578C4⋊C4—52no—
579C4⋊C4—104no—
580C4⋊C4—104no—
581C4⋊C4—104no—
582C4⋊C4—104no—
583C4⋊C4—104no—
584C4⋊C4—104no—
585C4⋊C4—104no—
586C4⋊C4—104no—
587C4⋊C4—101no—
588C4⋊C4—101no—
589C4⋊C4—101no—
590C4⋊C4—104no—
591C4⋊C4—104no—
592C4⋊C4—104no—
593C4⋊C4C23×D512yes—
594C22⋊C4—24no—
595C22⋊C4—24no—
596C22⋊C4—104no—
597C22⋊C4—104no—
598C22⋊C4—104no—
599C42—24no—
600C42—52no—
601C42—52no—
602C42—52no—
603C42—52no—
604C42—104no—
605C42—104no—
606C42—104no—
607C42—104no—
608C42D4×D512yes—
609C42C23×D512yes—
610C2×C5—82no—
611C2×C5#337714yescomplemented
612C2×C5C2×#54811yescharacteristic
613C2×C5C2×#57511yescharacteristic
614C2×C5C2×#57811yescharacteristic
615C23—22no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23C2×D4×D511yescharacteristic
620Q8—24no—
621Q8—22no—
622Q8—22no—
623Q8—22no—
624Q8—42no—
625Q8—102no—
626Q8—102no—
627Q8—204no—
628Q8—208no—
629Q8—202no—
630Q8C2×#71114yes—
631D4—22no—
632D4—22no—
633D4—42no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—52no—
643C2×C4—52no—
644C2×C4—52no—
645C2×C4—52no—
646C2×C4—102no—
647C2×C4—102no—
648C2×C4—102no—
649C2×C4—102no—
650C2×C4—102no—
651C2×C4—102no—
652C2×C4—102no—
653C2×C4—102no—
654C2×C4—208no—
655C2×C4—202no—
656C2×C4—202no—
657C2×C4—202no—
658C2×C4—202no—
659C2×C4C2×D4×D512yes—
660C2×C4C2×D4×D512yes—
661C2×C4C2×D4×D511yescharacteristic
662C2×C4C2×#71112yes—
663C2×C4#415012yes—
664C2×C4C24×D511yescharacteristic
665C8—22no—
666C8—22no—
667C8—102no—
668C8—102no—
669C5C2×#337711yescharacteristic, complemented, Sylow
670C22—42no—
671C22—42no—
672C22—42no—
673C22—84no—
674C22#6593212yes—
675C22#6610212yes—
676C22#6611312yes—
677C22C22×D4×D511yescharacteristic
678C4—22no—
679C4—22no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—102no—
684C4—102no—
685C4—102no—
686C4—102no—
687C4—202no—
688C4—202no—
689C4—202no—
690C4—202no—
691C4#6593212yes—
692C4C22×#71111yescharacteristic
693C4C2×#415011yescharacteristic
694C2—82no—
695C2#1061231714yesdirect product factor
696C2C2×#6593211yescharacteristic
697C2C2×#6610211yescharacteristic
698C2C2×#6611311yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#464C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
hypercenterC2×#464D5
hypocenterC5C2×#3377
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. C2×#65903C2semidirect
4. C2×#65471C2semidirect
5. C2×#65473C2semidirect
9. C2×C4×#689C2semidirect
10. #10612317C2direct (N × H)
11. C2×#65118C2semidirect
12. C2×#65120C2semidirect
13. C2×#65086C2semidirect
14. C2×#65083C2semidirect
31. #65903C22semidirect
54. #65473C22semidirect
55. #65471C22semidirect
67. C4×#689C22semidirect
71. #65120C22semidirect
72. #65118C22semidirect
73. #65086C22semidirect
74. #65083C22semidirect
611. C2×C5#3377semidirect
669. C5C2×#3377semidirect
695. C2#10612317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660052
C2×#659032
C2×C5×#4642
C2×#654712
C2×#654732
C2×#654552
C2×#654532
C2×#653452
C2×C4×#6892
#106123172
C2×#651182
C2×#651202
C2×#650862
C2×#650832
C2×#650542
C2×#650522
C2×#33775