Finite Group Databasebeta

← C2×#10601473 #12100151

Subgroups — C2×#10601473 #12100151

Order 1280.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C22×#3957C211yesmaximal, characteristic
4C2×#65265C211yesmaximal, characteristic, complemented
5C2×#65255C211yesmaximal, characteristic
6C2×#64774C211yesmaximal, characteristic, complemented
7C2×#64752C211yesmaximal, characteristic, complemented
8#10601473C218yesmaximal, direct product factor
9C22×#709C2211yescharacteristic
10C22×C5×M16C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic5C2211yescharacteristic
14C22×#683—22no—
15C22×C4×Dic5C2211yescharacteristic
16C22×#681C2211yescharacteristic
17#65857—22no—
18#65857—22no—
19#65857—22no—
20#65857C2214yes—
21C2×#4038—21no—
22C2×D5×C4⋊C4—21no—
23C2×#4027—22no—
24C2×C4×D20—21no—
25C2×C4×Dic10—21no—
26C5×#428—22no—
27C5×#428C2214yescomplemented
28C2×C5×#107—21no—
29C2×C5×#106—21no—
30C2×#3957C418yescomplemented
31C2×#3957C2214yes—
32#65265—22no—
33#65265C2214yescomplemented
34C2×#3942—21no—
35#65255—22no—
36#65255C2214yes—
37C2×#3941—21no—
38#64774C2214yescomplemented
39#64752C2214yescomplemented
40C2×#1859—51nomaximal, Sylow
41C23×C4×C5—21no—
42C23×C4×C5C2311yescharacteristic
43C22×C22⋊D5—21no—
44C23×Dic5—21no—
45C23×Dic5D412yes—
46C2×#709—21no—
47C2×#709—21no—
48C2×#709—22no—
49C2×#709—22no—
50C2×#709C2×C414yes—
51C2×#709D412yes—
52C2×#709Q814yes—
53C2×#709C2312yes—
54C22×D20D411yescharacteristic
55C22×C4×D5—21no—
56C22×C4×D5—22no—
57C22×Dic10D411yescharacteristic
58C2×C5×M16—21no—
59C2×C5×M16—22no—
60C2×C5×M16C2×C414yescomplemented
61C2×C5×M16C2312yes—
62C22×C5×C8—21no—
63C5×#117—21no—
64C5×#117—21no—
65C5×#117D412yes—
66C5×#117C2312yes—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4D411yescharacteristic
73C2×C5×C4⋊C4D411yescharacteristic
74C2×C5×C4⋊C4C2312yes—
75C2×C5×C22⋊C4—21no—
76C2×C42×C5—21no—
77C2×#685—22no—
78C2×C4⋊Dic5—21no—
79C2×C4⋊Dic5—21no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5D412yes—
83C2×C4⋊Dic5C2312yes—
84C2×#683—22no—
85C2×#683—28no—
86C2×#683—48no—
87C2×C4×Dic5—22no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—22no—
90C2×C4×Dic5C2×C414yescomplemented
91C2×C4×Dic5C2312yes—
92C2×#681—21no—
93C2×#681—22no—
94C2×#681—22no—
95C2×#681C2312yes—
96C22×#168—21no—
97#4038—24no—
98#4038—24no—
99D5×C4⋊C4—24no—
100D5×C4⋊C4—24no—
101#4027—28no—
102#4027—28no—
103C4×D20—24no—
104C4×D20—24no—
105C4×Dic10—24no—
106C4×Dic10—24no—
107C5×#107—24no—
108C5×#106—24no—
109#3957C2×C4116yescomplemented
110#3942—24no—
111#3941—24no—
112C2×#520—51no—
113C2×#428D511yescharacteristic, complemented
114C22×#104—51no—
115C2×#417—51no—
116C2×#384—51no—
117#1859—58no—
118C2×#343—51no—
119C2×#339—51no—
120C24×C5C2×D411yescharacteristic
121C23×D5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5—44no—
126C22×C4×C5—41no—
127C22×C4×C5—41no—
128C22×C4×C5C22×C412yes—
129C22×C4×C5C2×D411yescharacteristic
130C22×C4×C5C2×D411yescharacteristic
131C22×C4×C5C2×D411yescharacteristic
132C22×C4×C5C2×D411yescharacteristic
133C22×C4×C5C2×D411yescharacteristic
134C22×C4×C5C2×Q812yes—
135C22×C4×C5#4112yes—
136C22×C4×C5#4112yes—
137C22×C4×C5C2411yescharacteristic
138C2×C22⋊D5—24no—
139C2×C22⋊D5—22no—
140C2×C22⋊D5—24no—
141C2×C22⋊D5—22no—
142C22×Dic5—24no—
143C22×Dic5—24no—
144C22×Dic5—24no—
145C22×Dic5—21no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—44no—
150C22×Dic5C2×D414yes—
151#709—24no—
152#709—24no—
153#709—24no—
154#709—24no—
155#709—24no—
156#709—22no—
157#709—22no—
158#709C4⋊C418yescomplemented
159#709C22×C414yes—
160#709C2×Q814yes—
161C2×D20—21no—
162C2×D20—22no—
163C2×D20C4⋊C414yescomplemented
164C2×D20C2×D412yes—
165C2×C4×D5—24no—
166C2×C4×D5—22no—
167C2×C4×D5—24no—
168C2×C4×D5—22no—
169C2×C4×D5—28no—
170C2×C4×D5—84no—
171C2×Dic10—21no—
172C2×Dic10—22no—
173C2×Dic10C4⋊C414yescomplemented
174C2×Dic10C2×D412yes—
175C5×M16—24no—
176C5×M16—22no—
177C5×M16C22×C414yescomplemented
178C2×C5×C8—24no—
179C2×C5×C8—22no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—41no—
185C5×C4⋊C4—41no—
186C5×C4⋊C4C2×D412yes—
187C5×C4⋊C4C2×D412yes—
188C5×C22⋊C4—22no—
189C5×C22⋊C4—22no—
190C42×C5—22no—
191C42×C5—22no—
192#685—28no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195C4⋊Dic5—22no—
196C4⋊Dic5—22no—
197#683—28no—
198#683—416no—
199C4×Dic5—28no—
200C4×Dic5—24no—
201C4×Dic5—22no—
202C4×Dic5C22×C414yescomplemented
203#681—22no—
204#681—22no—
205#681—42no—
206C2×#168—22no—
207C2×#168—42no—
208C22×#41—51no—
209C22×M16—51no—
210C22×M16C2×D511yescharacteristic
211#520—54no—
212#520—102no—
213#520—102no—
214#520—102no—
215C2×C4×Q8—101no—
216C2×C4×D4—101no—
217C2×C4×D4—102no—
218C2×#117—101no—
219C2×#117C2×D511yescharacteristic
220C22×C4⋊C4—51no—
221C22×C4⋊C4—101no—
222C22×C4⋊C4—102no—
223C22×C4⋊C4C2×D511yescharacteristic
224C22×C42—51no—
225#428—22no—
226#428C2×D514yescomplemented
227C2×#107—21no—
228C2×#106—21no—
229C2×#104—58no—
230C2×#104—54no—
231#417—54no—
232#417—102no—
233C2×#103—101no—
234C2×#102—101no—
235#384—54no—
236#384—102no—
237#343—54no—
238#339—54no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5C4×D412yes—
242C23×C5#12011yescharacteristic
243C23×C5#12011yescharacteristic
244C23×C5C22⋊Q812yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—22no—
247C22×D5—24no—
248C22×D5—42no—
249C22×D5—42no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—44no—
255C2×C4×C5—42no—
256C2×C4×C5—82no—
257C2×C4×C5—82no—
258C2×C4×C5C2×C4⋊C412yes—
259C2×C4×C5C2×C4⋊C412yes—
260C2×C4×C5C4×D412yes—
261C2×C4×C5C4×D412yes—
262C2×C4×C5C4×D412yes—
263C2×C4×C5#12011yescharacteristic
264C2×C4×C5#12011yescharacteristic
265C2×C4×C5C22⋊Q812yes—
266C2×C4×C5C22⋊Q812yes—
267C2×C4×C5C22⋊Q812yes—
268C2×C4×C5#12312yes—
269C2×C4×C5#12312yes—
270C2×C4×C5C23×C411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C22×Q811yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C22⋊D5—24no—
277C22⋊D5—24no—
278C22⋊D5—28no—
279C2×Dic5—24no—
280C2×Dic5—216no—
281C2×Dic5—24no—
282C2×Dic5—24no—
283C2×Dic5—22no—
284C2×Dic5—24no—
285C2×Dic5—24no—
286C2×Dic5—44no—
287C2×Dic5—44no—
288C2×Dic5—82no—
289D20—22no—
290D20—24no—
291D20C2×C4⋊C414yescomplemented
292C4×D5—24no—
293C4×D5—24no—
294C4×D5—28no—
295C4×D5—88no—
296Dic10—22no—
297Dic10—24no—
298Dic10C2×C4⋊C414yescomplemented
299C5×C8—24no—
300#168—42no—
301C2×#41—54no—
302C2×#41—52no—
303C2×#41—54no—
304C2×#41—52no—
305C2×#41—102no—
306C2×#41—101no—
307C2×#41—102no—
308C2×#41—101no—
309C22×Q8—51no—
310C22×D4—51no—
311C22×D4—101no—
312C23×C4—21no—
313C23×C4—52no—
314C23×C4—102no—
315C23×C4—101no—
316C23×C4—101no—
317C23×C4C22×D511yescharacteristic
318C2×M16—22no—
319C2×M16—21no—
320C2×M16—52no—
321C2×M16—101no—
322C2×M16—102no—
323C2×M16—102no—
324C2×M16C4×D514yescomplemented
325C2×M16C22×D512yes—
326C22×C8—21no—
327C22×C8—101no—
328C4×Q8—104no—
329C4×Q8—104no—
330C4×D4—104no—
331C4×D4—104no—
332C4×D4—108no—
333C4×D4—108no—
334#117—21no—
335#117—21no—
336#117—104no—
337#117—104no—
338#117C22⋊D512yes—
339#117C22×D512yes—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—52no—
346C2×C4⋊C4—52no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4—102no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4—104no—
354C2×C4⋊C4—108no—
355C2×C4⋊C4—208no—
356C2×C4⋊C4C22⋊D511yescharacteristic
357C2×C4⋊C4C22⋊D511yescharacteristic
358C2×C4⋊C4C22×D512yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—102no—
361C2×C42—21no—
362C2×C42—54no—
363C2×C42—52no—
364C2×C42—102no—
365C2×C42—101no—
366C2×C42—102no—
367#107—24no—
368#106—24no—
369#104—516no—
370#103—104no—
371#102—104no—
372C22×C5—24no—
373C22×C5—24no—
374C22×C5—21no—
375C22×C5—21no—
376C22×C5—21no—
377C22×C5—21no—
378C22×C5#38512yes—
379C22×C5#38512yes—
380C22×C5#45312yes—
381C22×C5#45612yes—
382C22×C5C2×C4×D411yescharacteristic
383C22×C5C2×#12011yescharacteristic
384C22×C5C2×C22⋊Q811yescharacteristic
385C2×D5—24no—
386C2×D5—48no—
387C2×D5—42no—
388C2×D5—42no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—82no—
392C4×C5#38512yes—
393C4×C5#38512yes—
394C4×C5C22×C4⋊C411yescharacteristic
395C4×C5C2×C4×D411yescharacteristic
396C4×C5C2×C22⋊Q811yescharacteristic
397C4×C5C2×#12311yescharacteristic
398Dic5—28no—
399Dic5—24no—
400Dic5—44no—
401C24—101no—
402C24D4×D511yescharacteristic
403#41—54no—
404#41—54no—
405#41—58no—
406#41—102no—
407#41—102no—
408#41—104no—
409#41—104no—
410#41—104no—
411#41—104no—
412#41—104no—
413C2×Q8—52no—
414C2×Q8—54no—
415C2×Q8—102no—
416C2×Q8—101no—
417C2×D4—52no—
418C2×D4—54no—
419C2×D4—102no—
420C2×D4—104no—
421C2×D4—104no—
422C2×D4—102no—
423C2×D4—102no—
424C2×D4—101no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—44no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—54no—
432C22×C4—108no—
433C22×C4—104no—
434C22×C4—104no—
435C22×C4—102no—
436C22×C4—104no—
437C22×C4—101no—
438C22×C4—102no—
439C22×C4—104no—
440C22×C4—102no—
441C22×C4—104no—
442C22×C4—102no—
443C22×C4—102no—
444C22×C4—204no—
445C22×C4—404no—
446C22×C4C2×C4×D512yes—
447C22×C4#70912yes—
448C22×C4D4×D511yescharacteristic
449C22×C4D4×D511yescharacteristic
450C22×C4D4×D511yescharacteristic
451C22×C4Q8×D512yes—
452C22×C4#71312yes—
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C23×D511yescharacteristic
456M16—22no—
457M16—24no—
458M16—102no—
459M16—102no—
460M16—202no—
461M16C2×C4×D514yescomplemented
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—102no—
465C2×C8—202no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—108no—
473C4⋊C4—102no—
474C4⋊C4—102no—
475C4⋊C4—102no—
476C4⋊C4—102no—
477C4⋊C4—2016no—
478C4⋊C4C2×C22⋊D512yes—
479C4⋊C4C2×C22⋊D512yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—108no—
483C42—22no—
484C42—22no—
485C42—54no—
486C42—108no—
487C42—102no—
488C42—104no—
489C2×C5—21no—
490C2×C5—21no—
491C2×C5—22no—
492C2×C5#185914yescomplemented
493C2×C5C2×#38511yescharacteristic
494C2×C5C2×#45311yescharacteristic
495C2×C5C2×#45611yescharacteristic
496D5—44no—
497C23—22no—
498C23—22no—
499C23—102no—
500C23—104no—
501C23—202no—
502C23—202no—
503C23#403912yes—
504C23#404212yes—
505C23#408311yescharacteristic
506C23#408311yescharacteristic
507C23C2×D4×D511yescharacteristic
508Q8—54no—
509Q8—102no—
510Q8—104no—
511D4—54no—
512D4—104no—
513D4—104no—
514D4—108no—
515D4—102no—
516D4—104no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—104no—
526C2×C4—1016no—
527C2×C4—104no—
528C2×C4—104no—
529C2×C4—102no—
530C2×C4—104no—
531C2×C4—104no—
532C2×C4—104no—
533C2×C4—108no—
534C2×C4—104no—
535C2×C4—204no—
536C2×C4—204no—
537C2×C4—408no—
538C2×C4—402no—
539C2×C4D5×C4⋊C412yes—
540C2×C4D5×C4⋊C412yes—
541C2×C4#403912yes—
542C2×C4#404012yes—
543C2×C4#404012yes—
544C2×C4#404212yes—
545C2×C4C4×C22⋊D512yes—
546C2×C4C4×C22⋊D512yes—
547C2×C4#408311yescharacteristic
548C2×C4#408311yescharacteristic
549C2×C4#409212yes—
550C2×C4#409212yes—
551C2×C4C22×C4×D511yescharacteristic
552C2×C4C2×#70911yescharacteristic
553C2×C4C2×D4×D511yescharacteristic
554C2×C4C2×Q8×D511yescharacteristic
555C2×C4C2×#71311yescharacteristic
556C2×C4C22×C22⋊D511yescharacteristic
557C8—24no—
558C8—202no—
559C5C2×#185911yescharacteristic, complemented, Sylow
560C22—24no—
561C22—24no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—104no—
567C22—208no—
568C22—202no—
569C22—202no—
570C22#6527812yes—
571C22#6527812yes—
572C22#6548712yes—
573C22#6549412yes—
574C22C2×#403911yescharacteristic
575C22C2×#404211yescharacteristic
576C22C2×#408311yescharacteristic
577C4—42no—
578C4—42no—
579C4—82no—
580C4—108no—
581C4—104no—
582C4—204no—
583C4#6527812yes—
584C4#6527812yes—
585C4C2×D5×C4⋊C411yescharacteristic
586C4C2×#404011yescharacteristic
587C4C2×C4×C22⋊D511yescharacteristic
588C4C2×#409211yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—204no—
593C2#1060147314yesdirect product factor
594C2C2×#6527811yescharacteristic
595C2C2×#6548711yescharacteristic
596C2C2×#6549411yescharacteristic
subgroupG/H
centerC23#4083
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#428C2
socleC23×C5#120
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#428D5
hypocenterC5C2×#1859
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
4. C2×#65265C2semidirect
6. C2×#64774C2semidirect
7. C2×#64752C2semidirect
8. #10601473C2direct (N × H)
27. C5×#428C22semidirect
30. C2×#3957C4semidirect
33. #65265C22semidirect
38. #64774C22semidirect
39. #64752C22semidirect
60. C2×C5×M16C2×C4semidirect
90. C2×C4×Dic5C2×C4semidirect
109. #3957C2×C4semidirect
113. C2×#428D5semidirect
158. #709C4⋊C4semidirect
163. C2×D20C4⋊C4semidirect
173. C2×Dic10C4⋊C4semidirect
177. C5×M16C22×C4semidirect
202. C4×Dic5C22×C4semidirect
226. #428C2×D5semidirect
291. D20C2×C4⋊C4semidirect
298. Dic10C2×C4⋊C4semidirect
324. C2×M16C4×D5semidirect
461. M16C2×C4×D5semidirect
492. C2×C5#1859semidirect
559. C5C2×#1859semidirect
593. C2#10601473direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658572
C2×C5×#4282
C22×#39572
C2×#652652
C2×#652552
C2×#647742
C2×#647522
#106014732
C2×#18595