Finite Group Databasebeta

← C2×#10601090 #12100152

Subgroups — C2×#10601090 #12100152

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×#66161C211yesmaximal, characteristic
2C2×C5×#417C211yesmaximal, characteristic
3C22×#3969C211yesmaximal, characteristic
4C2×#65265C211yesmaximal, characteristic, complemented
5C2×#65253C211yesmaximal, characteristic, complemented
6C2×#64774C211yesmaximal, characteristic, complemented
7C2×#64775C211yesmaximal, characteristic, complemented
8#10601090C218yesmaximal, direct product factor
9C22×C5×#41C2211yescharacteristic
10C22×C5×M16C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12#66161—22no—
13#66161—22no—
14#66161—22no—
15#66161C2214yes—
16C2×Q8×Dic5—21no—
17C2×D4×Dic5—21no—
18C2×D4×Dic5—22no—
19C2×#4072—21no—
20C2×#4072C2211yescharacteristic
21C22×C4⋊Dic5—21no—
22C22×C4⋊Dic5C2211yescharacteristic
23C22×#683—22no—
24C22×C4×Dic5C2211yescharacteristic
25C22×#681C2211yescharacteristic
26C5×#417—22no—
27C5×#417C2214yes—
28C2×C5×#103—21no—
29C2×C5×#102—21no—
30C2×#3969C418yescomplemented
31C2×#3969C2214yes—
32#65265—22no—
33#65265C2214yescomplemented
34#65253—22no—
35#65253C2214yescomplemented
36C2×#3940—21no—
37C2×#3939—21no—
38#64775C2214yescomplemented
39#64774C2214yescomplemented
40C2×#1859—51nomaximal, Sylow
41C2×C5×#41—21no—
42C2×C5×#41—21no—
43C2×C5×#41—22no—
44C2×C5×#41—22no—
45C2×C5×#41C2×C414yes—
46C2×C5×#41D412yes—
47C2×C5×#41Q814yes—
48C2×C5×#41C2312yes—
49C22×C5×Q8D411yescharacteristic
50C22×C5×D4—21no—
51C22×C5×D4D411yescharacteristic
52C23×C4×C5—21no—
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C23×Dic5—21no—
56C23×Dic5—22no—
57C23×Dic5D412yes—
58C2×C5×M16—21no—
59C2×C5×M16—22no—
60C2×C5×M16—22no—
61C2×C5×M16C2312yes—
62C22×C5×C8—21no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4C2312yes—
69C2×C22⋊Dic5—22no—
70C2×C22⋊Dic5—21no—
71Q8×Dic5—24no—
72Q8×Dic5—24no—
73D4×Dic5—28no—
74D4×Dic5—24no—
75D4×Dic5—24no—
76D4×Dic5—28no—
77#4072—24no—
78#4072—21no—
79#4072—21no—
80#4072—24no—
81#4072D412yes—
82#4072C2312yes—
83C2×C4⋊Dic5—22no—
84C2×C4⋊Dic5—24no—
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—24no—
90C2×C4⋊Dic5D412yes—
91C2×C4⋊Dic5D411yescharacteristic
92C2×C4⋊Dic5D411yescharacteristic
93C2×C4⋊Dic5C2312yes—
94C2×#683—28no—
95C2×#683—48no—
96C2×C4×Dic5—22no—
97C2×C4×Dic5—21no—
98C2×C4×Dic5—21no—
99C2×C4×Dic5—22no—
100C2×C4×Dic5C2×C414yescomplemented
101C2×C4×Dic5C2312yes—
102C2×#681—21no—
103C2×#681—22no—
104C2×#681C2×C414yescomplemented
105C2×#681C2312yes—
106C22×#168—21no—
107C5×#103—24no—
108C5×#102—24no—
109#3969C2×C4116yescomplemented
110#3940—24no—
111#3939—24no—
112C2×#520—51no—
113C2×#428—51no—
114C22×#104—51no—
115C2×#417D511yescharacteristic
116C2×#384—51no—
117#1859—58no—
118C2×#343—51no—
119C2×#339—51no—
120C24×C5—21no—
121C24×C5C2×D411yescharacteristic
122C5×#41—24no—
123C5×#41—24no—
124C5×#41—24no—
125C5×#41—24no—
126C5×#41—24no—
127C5×#41—22no—
128C5×#41—22no—
129C5×#41C4⋊C418yescomplemented
130C5×#41C22×C414yes—
131C5×#41C2×Q814yes—
132C2×C5×Q8—21no—
133C2×C5×Q8—22no—
134C2×C5×Q8C4⋊C414yescomplemented
135C2×C5×Q8C2×D412yes—
136C2×C5×D4—24no—
137C2×C5×D4—22no—
138C2×C5×D4—24no—
139C2×C5×D4—22no—
140C2×C5×D4—21no—
141C2×C5×D4—22no—
142C2×C5×D4C4⋊C414yescomplemented
143C2×C5×D4C2×D412yes—
144C22×C4×C5—21no—
145C22×C4×C5—24no—
146C22×C4×C5—22no—
147C22×C4×C5—24no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—44no—
151C22×C4×C5—41no—
152C22×C4×C5—41no—
153C22×C4×C5C22×C412yes—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×Q812yes—
160C22×C4×C5#4112yes—
161C22×C4×C5#4112yes—
162C22×C4×C5C2411yescharacteristic
163C22×Dic5—24no—
164C22×Dic5—24no—
165C22×Dic5—24no—
166C22×Dic5—22no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C22×Dic5—22no—
170C22×Dic5—28no—
171C22×Dic5—22no—
172C22×Dic5—44no—
173C22×Dic5—84no—
174C22×Dic5C2×D414yes—
175C5×M16—22no—
176C5×M16—22no—
177C5×M16—42no—
178C2×C5×C8—22no—
179C2×C5×C8—42no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—41no—
183C5×C4⋊C4—41no—
184C22⋊Dic5—28no—
185C22⋊Dic5—22no—
186C22⋊Dic5—22no—
187C4⋊Dic5—28no—
188C4⋊Dic5—22no—
189C4⋊Dic5—21no—
190C4⋊Dic5—22no—
191C4⋊Dic5—21no—
192C4⋊Dic5—22no—
193C4⋊Dic5—22no—
194C4⋊Dic5C2×D412yes—
195C4⋊Dic5C2×D412yes—
196#683—416no—
197C4×Dic5—28no—
198C4×Dic5—24no—
199C4×Dic5—22no—
200C4×Dic5—22no—
201C4×Dic5—22no—
202C4×Dic5C22×C414yescomplemented
203#681—24no—
204#681—22no—
205#681C22×C414yescomplemented
206C2×#168—24no—
207C2×#168—22no—
208C22×#41C2×D511yescharacteristic
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—102no—
217C2×C4×D4—101no—
218C2×#117—51no—
219C2×#117—101no—
220C22×C4⋊C4—51no—
221C22×C4⋊C4—101no—
222C22×C4⋊C4—102no—
223C22×C4⋊C4C2×D511yescharacteristic
224C22×C42—51no—
225#428—54no—
226#428—102no—
227C2×#107—101no—
228C2×#106—101no—
229C2×#104—58no—
230C2×#104—54no—
231#417—22no—
232#417C2×D514yes—
233C2×#103—21no—
234C2×#102—21no—
235#384—54no—
236#384—102no—
237#343—54no—
238#339—54no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—22no—
242C23×C5—24no—
243C23×C5—42no—
244C23×C5—42no—
245C23×C5C4×D412yes—
246C23×C5#12011yescharacteristic
247C23×C5#12011yescharacteristic
248C23×C5C22⋊Q812yes—
249C23×C5C22×D411yescharacteristic
250C5×Q8—22no—
251C5×Q8—24no—
252C5×Q8C2×C4⋊C414yescomplemented
253C5×D4—24no—
254C5×D4—24no—
255C5×D4—28no—
256C5×D4—22no—
257C5×D4—24no—
258C5×D4C2×C4⋊C414yescomplemented
259C2×C4×C5—22no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—24no—
263C2×C4×C5—28no—
264C2×C4×C5—44no—
265C2×C4×C5—42no—
266C2×C4×C5—82no—
267C2×C4×C5—82no—
268C2×C4×C5C2×C4⋊C412yes—
269C2×C4×C5C2×C4⋊C412yes—
270C2×C4×C5C4×D412yes—
271C2×C4×C5C4×D412yes—
272C2×C4×C5C4×D412yes—
273C2×C4×C5#12011yescharacteristic
274C2×C4×C5#12011yescharacteristic
275C2×C4×C5C22⋊Q812yes—
276C2×C4×C5C22⋊Q812yes—
277C2×C4×C5C22⋊Q812yes—
278C2×C4×C5#12312yes—
279C2×C4×C5#12312yes—
280C2×C4×C5C23×C411yescharacteristic
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C22×D411yescharacteristic
283C2×C4×C5C22×Q811yescharacteristic
284C2×C4×C5C2×#4111yescharacteristic
285C2×C4×C5C2×#4111yescharacteristic
286C2×Dic5—24no—
287C2×Dic5—216no—
288C2×Dic5—24no—
289C2×Dic5—24no—
290C2×Dic5—22no—
291C2×Dic5—24no—
292C2×Dic5—22no—
293C2×Dic5—44no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—44no—
297C2×Dic5—82no—
298C2×Dic5—88no—
299C5×C8—42no—
300#168—24no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41Dic1014yes—
306C2×#41C4×D514yes—
307C2×#41C22⋊D512yes—
308C2×#41C22×D512yes—
309C22×Q8C22⋊D511yescharacteristic
310C22×D4—21no—
311C22×D4C22⋊D511yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—52no—
315C23×C4—101no—
316C23×C4—102no—
317C23×C4C22×D511yescharacteristic
318C2×M16—21no—
319C2×M16—22no—
320C2×M16—22no—
321C2×M16—54no—
322C2×M16—52no—
323C2×M16—101no—
324C2×M16—102no—
325C2×M16C22×D512yes—
326C22×C8—21no—
327C22×C8—101no—
328C4×Q8—104no—
329C4×Q8—104no—
330C4×D4—108no—
331C4×D4—108no—
332C4×D4—104no—
333C4×D4—104no—
334#117—52no—
335#117—52no—
336#117—104no—
337#117—104no—
338#117—101no—
339#117—101no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—52no—
348C2×C4⋊C4—52no—
349C2×C4⋊C4—104no—
350C2×C4⋊C4—102no—
351C2×C4⋊C4—104no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4—101no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4—101no—
356C2×C4⋊C4—108no—
357C2×C4⋊C4—208no—
358C2×C4⋊C4C22×D512yes—
359C2×C22⋊C4—102no—
360C2×C22⋊C4—101no—
361C2×C42—54no—
362C2×C42—52no—
363C2×C42—102no—
364C2×C42—101no—
365C2×C42—102no—
366C2×C42—101no—
367#107—104no—
368#106—104no—
369#104—516no—
370#103—24no—
371#102—24no—
372C22×C5—24no—
373C22×C5—24no—
374C22×C5—21no—
375C22×C5—21no—
376C22×C5—21no—
377C22×C5—21no—
378C22×C5—24no—
379C22×C5—48no—
380C22×C5—42no—
381C22×C5—42no—
382C22×C5#38512yes—
383C22×C5#38512yes—
384C22×C5#45312yes—
385C22×C5#45612yes—
386C22×C5C2×C4×D411yescharacteristic
387C22×C5C2×#12011yescharacteristic
388C22×C5C2×C22⋊Q811yescharacteristic
389C4×C5—24no—
390C4×C5—82no—
391C4×C5#38512yes—
392C4×C5#38512yes—
393C4×C5C22×C4⋊C411yescharacteristic
394C4×C5C2×C4×D411yescharacteristic
395C4×C5C2×C22⋊Q811yescharacteristic
396C4×C5C2×#12311yescharacteristic
397Dic5—28no—
398Dic5—44no—
399Dic5—42no—
400Dic5—42no—
401C24—21no—
402C24D4×D511yescharacteristic
403#41—22no—
404#41—22no—
405#41—24no—
406#41—24no—
407#41—24no—
408#41—24no—
409#41—24no—
410#41#68318yescomplemented
411#41C2×Dic1014yes—
412#41C2×C4×D514yes—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8#68314yescomplemented
416C2×Q8C2×C22⋊D512yes—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4#68314yescomplemented
424C2×D4C2×C22⋊D512yes—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—54no—
435C22×C4—104no—
436C22×C4—104no—
437C22×C4—102no—
438C22×C4—104no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—102no—
442C22×C4—108no—
443C22×C4—102no—
444C22×C4—204no—
445C22×C4—404no—
446C22×C4C2×Dic1012yes—
447C22×C4C2×C4×D512yes—
448C22×C4D4×D511yescharacteristic
449C22×C4D4×D511yescharacteristic
450C22×C4D4×D511yescharacteristic
451C22×C4#71112yes—
452C22×C4#71112yes—
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C23×D511yescharacteristic
456M16—22no—
457M16—22no—
458M16—42no—
459M16—54no—
460M16—102no—
461M16—104no—
462C2×C8—22no—
463C2×C8—42no—
464C2×C8—102no—
465C2×C8—104no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—52no—
471C4⋊C4—52no—
472C4⋊C4—108no—
473C4⋊C4—102no—
474C4⋊C4—102no—
475C4⋊C4—101no—
476C4⋊C4—102no—
477C4⋊C4—102no—
478C4⋊C4—101no—
479C4⋊C4—2016no—
480C22⋊C4—108no—
481C22⋊C4—102no—
482C22⋊C4—102no—
483C42—54no—
484C42—108no—
485C42—102no—
486C42—102no—
487C42—104no—
488C42—102no—
489C2×C5—21no—
490C2×C5—21no—
491C2×C5—22no—
492C2×C5—44no—
493C2×C5#185914yescomplemented
494C2×C5C2×#38511yescharacteristic
495C2×C5C2×#45311yescharacteristic
496C2×C5C2×#45611yescharacteristic
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23#402412yes—
504C23#402712yes—
505C23#408311yescharacteristic
506C23#408311yescharacteristic
507C23C2×D4×D511yescharacteristic
508Q8—22no—
509Q8—24no—
510Q8C2×#68314yescomplemented
511D4—24no—
512D4—24no—
513D4—28no—
514D4—22no—
515D4—24no—
516D4C2×#68314yescomplemented
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—104no—
527C2×C4—1016no—
528C2×C4—104no—
529C2×C4—104no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—104no—
533C2×C4—204no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—204no—
537C2×C4—402no—
538C2×C4—408no—
539C2×C4#402412yes—
540C2×C4#402412yes—
541C2×C4#402412yes—
542C2×C4#402712yes—
543C2×C4#402712yes—
544C2×C4#402712yes—
545C2×C4C2×#68312yes—
546C2×C4C2×#68312yes—
547C2×C4#408112yes—
548C2×C4#408112yes—
549C2×C4#408311yescharacteristic
550C2×C4#408311yescharacteristic
551C2×C4C22×Dic1011yescharacteristic
552C2×C4C22×C4×D511yescharacteristic
553C2×C4C2×D4×D511yescharacteristic
554C2×C4C2×#71111yescharacteristic
555C2×C4C2×#71111yescharacteristic
556C2×C4C22×C22⋊D511yescharacteristic
557C8—42no—
558C8—104no—
559C5C2×#185911yescharacteristic, complemented, Sylow
560C22—24no—
561C22—24no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—24no—
567C22—48no—
568C22—42no—
569C22—42no—
570C22#6523512yes—
571C22#6523512yes—
572C22#6548712yes—
573C22#6549412yes—
574C22C2×#402411yescharacteristic
575C22C2×#402711yescharacteristic
576C22C2×#408311yescharacteristic
577C4—24no—
578C4—82no—
579C4—108no—
580C4—204no—
581C4—202no—
582C4—202no—
583C4#6523512yes—
584C4#6523512yes—
585C4C2×#402411yescharacteristic
586C4C2×#402711yescharacteristic
587C4C22×#68311yescharacteristic
588C4C2×#408111yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—44no—
593C2#1060109014yesdirect product factor
594C2C2×#6523511yescharacteristic
595C2C2×#6548711yescharacteristic
596C2C2×#6549411yescharacteristic
subgroupG/H
centerC23#4083
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#417C2
socleC23×C5#120
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenterC2×#417D5
hypocenterC5C2×#1859
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×#65265C2semidirect
5. C2×#65253C2semidirect
6. C2×#64774C2semidirect
7. C2×#64775C2semidirect
8. #10601090C2direct (N × H)
30. C2×#3969C4semidirect
33. #65265C22semidirect
35. #65253C22semidirect
38. #64775C22semidirect
39. #64774C22semidirect
100. C2×C4×Dic5C2×C4semidirect
104. C2×#681C2×C4semidirect
109. #3969C2×C4semidirect
129. C5×#41C4⋊C4semidirect
134. C2×C5×Q8C4⋊C4semidirect
142. C2×C5×D4C4⋊C4semidirect
202. C4×Dic5C22×C4semidirect
205. #681C22×C4semidirect
252. C5×Q8C2×C4⋊C4semidirect
258. C5×D4C2×C4⋊C4semidirect
410. #41#683semidirect
415. C2×Q8#683semidirect
423. C2×D4#683semidirect
493. C2×C5#1859semidirect
510. Q8C2×#683semidirect
516. D4C2×#683semidirect
559. C5C2×#1859semidirect
593. C2#10601090direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#661612
C2×C5×#4172
C22×#39692
C2×#652652
C2×#652532
C2×#647742
C2×#647752
#106010902
C2×#18595