Finite Group Databasebeta

← C2×#10612703 #12971431

Subgroups — C2×#10612703 #12971431

Order 1280.

2018 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#65891C211yesmaximal, characteristic
3C2×C5×#474C211yesmaximal, characteristic, complemented
4C2×#65383C211yesmaximal, characteristic, complemented
5C2×#65296C211yesmaximal, characteristic, complemented
6#10612703C2116yesmaximal, direct product factor
7C2×#65173C212yesmaximal, complemented
8C2×#65165C212yesmaximal, complemented
9C2×#65140C211yesmaximal, characteristic, complemented
10C2×#65124C211yesmaximal, characteristic, complemented
11C2×#65066C212yesmaximal
12C2×#65054C212yesmaximal, complemented
13C22×Q8×D5—21no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×D4C2211yescharacteristic
16C2×#4090—21no—
17C2×#4084C2211yescharacteristic
18C2×#4082C2211yescharacteristic
19C2×C4×C22⋊D5—21no—
20C2×#4072—21no—
21D5×C4⋊Q8—24no—
22D5×C4⋊Q8C2218yescomplemented
23#65891—24no—
24#65891C2218yes—
25C2×#4043C2211yescharacteristic
26C2×D5×C4⋊C4—21no—
27C2×D5×C4⋊C4C2212yes—
28C2×#4036C2211yescharacteristic
29C2×#4034—21no—
30C2×#4034C2212yes—
31C2×#4029—21no—
32C2×#4025—21no—
33C2×C42×D5C2211yescharacteristic
34C2×#4015C2211yescharacteristic
35C2×C4×Dic10C2211yescharacteristic
36C5×#474C2218yescomplemented
37C2×C5×#107C2212yes—
38C2×C5×C4⋊C8C2211yescharacteristic
39C2×C5×#102C2212yes—
40C2×#3964C2212yes—
41C2×#3952C2212yes—
42C2×#3949C2212yes—
43C2×#3948C2212yes—
44C2×#3947C2212yes—
45#65383C2218yescomplemented
46#65296C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3940C2212yes—
49C2×#3939C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65173C22116yescomplemented
52#65165C22116yescomplemented
53#65140C2218yescomplemented
54#65124C2218yescomplemented
55#65066C22116yes—
56#65054C22116yescomplemented
57C2×#3239—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×D4C2311yescharacteristic
60C23×C4×C5—21no—
61C22×C22⋊D5—21no—
62C2×Q8×D5—28no—
63C2×Q8×D5—48no—
64C22×C4×D5—22no—
65C22×C4×D5D411yescharacteristic
66C22×C4×D5D411yescharacteristic
67C22×C4×D5C2311yescharacteristic
68C22×Dic10—21no—
69C22×Dic10—22no—
70C22×Dic10C2311yescharacteristic
71C22×C5×C8C2312yes—
72C5×C4⋊Q8—22no—
73C5×C4⋊Q8C2314yes—
74C4×C5×D4—22no—
75C4×C5×D4C2314yes—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2312yes—
78C2×C5×C4⋊C4C2311yescharacteristic
79C2×C5×C22⋊C4—21no—
80C2×C42×C5C2311yescharacteristic
81C2×C22⋊Dic5—21no—
82C2×C22⋊Dic5—21no—
83C2×C22⋊Dic5—21no—
84#4090—28no—
85#4084—22no—
86#4084C2314yes—
87#4082—22no—
88#4082C2314yes—
89C4×C22⋊D5—28no—
90C2×#685—21no—
91#4072—28no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5C2311yescharacteristic
96C2×C4⋊Dic5C2312yes—
97C2×#683—21no—
98C2×#683—21no—
99C2×#683—22no—
100C2×#683—22no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5C2311yescharacteristic
105C22×#168C2312yes—
106#4043—22no—
107#4043C2314yes—
108D5×C4⋊C4—24no—
109D5×C4⋊C4—28no—
110D5×C4⋊C4C2318yes—
111#4036—22no—
112#4036C2314yes—
113#4034—24no—
114#4034—28no—
115#4034C2318yes—
116#4029—28no—
117#4025—28no—
118C42×D5D414yes—
119C42×D5C2314yes—
120#4015—22no—
121#4015C2314yes—
122C4×Dic10—22no—
123C4×Dic10C2314yes—
124C5×#107C2318yes—
125C5×C4⋊C8C2314yes—
126C5×#102C2318yes—
127#3964C2318yes—
128#3952C2318yes—
129#3949C2318yes—
130#3948C2318yes—
131#3947C2318yes—
132#3942C2318yes—
133#3940C2318yes—
134#3939C2318yes—
135C4⋊#168C2314yes—
136C22×C4⋊Q8—51no—
137C2×#529—51no—
138#3239—516no—
139C2×#501—52no—
140C2×#489—52no—
141C2×#484—52no—
142C2×#481—52no—
143C2×#475—51no—
144C2×#475—51no—
145C2×#474—51no—
146C2×#474D511yescharacteristic, complemented
147C2×#432—51no—
148C24×C5—21no—
149C23×D5C2×D411yescharacteristic
150C2×C5×Q8—24no—
151C2×C5×Q8—44no—
152C2×C5×D4—21no—
153C2×C5×D4—22no—
154C2×C5×D4—22no—
155C2×C5×D4C2412yes—
156C22×C4×C5—21no—
157C22×C4×C5—22no—
158C22×C4×C5—24no—
159C22×C4×C5—44no—
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2411yescharacteristic
163C2×C22⋊D5—24no—
164C2×C22⋊D5—44no—
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—22no—
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171Q8×D5—432no—
172C2×C4×D5—24no—
173C2×C4×D5—24no—
174C2×C4×D5—22no—
175C2×C4×D5—22no—
176C2×C4×D5—28no—
177C2×C4×D5—48no—
178C2×C4×D5C2×D414yes—
179C2×C4×D5C2×D414yes—
180C2×C4×D5C2×D412yes—
181C2×C4×D5C2412yes—
182C2×Dic10—22no—
183C2×Dic10—22no—
184C2×Dic10—24no—
185C2×Dic10—21no—
186C2×Dic10—28no—
187C2×Dic10—48no—
188C2×Dic10—44no—
189C2×Dic10C2412yes—
190C2×C5×C8—24no—
191C2×C5×C8C2414yes—
192C5×C4⋊C4—24no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4C2412yes—
196C5×C4⋊C4C2414yes—
197C5×C22⋊C4—24no—
198C42×C5C2×D412yes—
199C42×C5C2412yes—
200C22⋊Dic5—24no—
201C22⋊Dic5—24no—
202C22⋊Dic5—24no—
203#685—24no—
204C4⋊Dic5—21no—
205C4⋊Dic5—22no—
206C4⋊Dic5—24no—
207C4⋊Dic5—24no—
208C4⋊Dic5—24no—
209C4⋊Dic5C2414yes—
210C4⋊Dic5C2412yes—
211#683—24no—
212#683—24no—
213#683—28no—
214#683—28no—
215C4×Dic5—24no—
216C4×Dic5C2×D412yes—
217C4×Dic5C2×D414yes—
218C4×Dic5C2×D414yes—
219C4×Dic5C2412yes—
220C2×#168—24no—
221C2×#168C2414yes—
222C23×Q8—101no—
223C2×C4⋊Q8—58no—
224C2×C4⋊Q8—52no—
225C2×C4⋊Q8—51no—
226C2×C4⋊Q8—101no—
227C2×C4⋊Q8—104no—
228C2×C4⋊Q8—101no—
229C2×C4⋊Q8C2×D511yescharacteristic
230#529—58no—
231#529—104no—
232C2×#126—101no—
233C2×#125—51no—
234C2×#124—51no—
235C2×#123—101no—
236C2×C22⋊Q8—51no—
237C2×#121—51no—
238C2×C4×Q8—51no—
239C2×C4×D4—101no—
240C2×C4×D4C2×D511yescharacteristic
241C2×#117—101no—
242C22×C4⋊C4—52no—
243C22×C4⋊C4—101no—
244C22×C42—51no—
245#501—516no—
246#489—516no—
247#484—516no—
248#481—516no—
249#475—58no—
250#475—58no—
251#474—58no—
252#474C2×D518yescomplemented
253#432—58no—
254C2×#107—52no—
255C2×#107C2×D512yes—
256C2×#106—52no—
257C2×#106—52no—
258C2×C4⋊C8—51no—
259C2×C4⋊C8C2×D511yescharacteristic
260C2×#103—52no—
261C2×#103—52no—
262C2×#102—52no—
263C2×#102C2×D512yes—
264C2×C22⋊C8—52no—
265C2×#97—52no—
266C23×C5—22no—
267C23×C5—42no—
268C23×C5—42no—
269C23×C5—42no—
270C23×C5C22×D411yescharacteristic
271C22×D5—22no—
272C22×D5—22no—
273C22×D5#13714yes—
274C22×D5C22×D412yes—
275C5×Q8—48no—
276C5×D4—22no—
277C5×D4—22no—
278C5×D4—42no—
279C2×C4×C5—22no—
280C2×C4×C5—24no—
281C2×C4×C5—22no—
282C2×C4×C5—44no—
283C2×C4×C5—42no—
284C2×C4×C5—48no—
285C2×C4×C5AGL(1,ℤ/8)14yes—
286C2×C4×C5C22×D412yes—
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×D412yes—
289C2×C4×C52−1+412yes—
290C2×C4×C52−1+412yes—
291C2×C4×C5C2511yescharacteristic
292C22⋊D5—48no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—24no—
299C2×Dic5—42no—
300C2×Dic5—44no—
301C2×Dic5—42no—
302C2×Dic5C22×D412yes—
303C2×Dic5C22×D412yes—
304C2×Dic5C22×D412yes—
305C4×D5—28no—
306C4×D5—24no—
307C4×D5—28no—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5—416no—
311Dic10—22no—
312Dic10—22no—
313Dic10—416no—
314Dic10—42no—
315Dic10—48no—
316C5×C8—24no—
317#168—24no—
318C22×Q8—21no—
319C22×Q8—51no—
320C22×Q8—102no—
321C22×Q8—108no—
322C22×Q8—101no—
323C22×Q8—208no—
324C22×D4—101no—
325C22×D4C22×D511yescharacteristic
326C23×C4—21no—
327C23×C4—51no—
328C23×C4—51no—
329C23×C4—51no—
330C23×C4—102no—
331C22×C8—52no—
332C22×C8C22×D512yes—
333C4⋊Q8—22no—
334C4⋊Q8—58no—
335C4⋊Q8—54no—
336C4⋊Q8—104no—
337C4⋊Q8—108no—
338C4⋊Q8—108no—
339C4⋊Q8—102no—
340C4⋊Q8C22×D514yes—
341#126—108no—
342#125—54no—
343#125—102no—
344#124—54no—
345#124—102no—
346#123—108no—
347C22⋊Q8—54no—
348C22⋊Q8—102no—
349#121—54no—
350#121—102no—
351C4×Q8—54no—
352C4×Q8—102no—
353C4×D4—22no—
354C4×D4—108no—
355C4×D4C22×D514yes—
356#117—108no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—52no—
359C2×C4⋊C4—51no—
360C2×C4⋊C4—58no—
361C2×C4⋊C4—104no—
362C2×C4⋊C4—108no—
363C2×C4⋊C4—102no—
364C2×C4⋊C4—102no—
365C2×C4⋊C4—101no—
366C2×C4⋊C4—101no—
367C2×C4⋊C4—101no—
368C2×C4⋊C4—101no—
369C2×C4⋊C4—101no—
370C2×C4⋊C4C22×D512yes—
371C2×C4⋊C4C22×D511yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—101no—
374C2×C22⋊C4—101no—
375C2×C22⋊C4—101no—
376C2×C22⋊C4—101no—
377C2×C42—51no—
378C2×C42—51no—
379C2×C42—51no—
380C2×C42—54no—
381C2×C42—54no—
382C2×C42—101no—
383C2×C42C22×D511yescharacteristic
384#107—58no—
385#107C22×D518yes—
386#106—58no—
387#106—58no—
388C4⋊C8—54no—
389C4⋊C8C22×D514yes—
390#103—58no—
391#103—58no—
392#102—58no—
393#102C22×D518yes—
394C22⋊C8—58no—
395#97—58no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5—84no—
400C22×C5#53612yes—
401C22×C5C2×AGL(1,ℤ/8)12yes—
402C22×C5C2×#13712yes—
403C22×C5C23×D411yescharacteristic
404C2×D5—22no—
405C2×D5—28no—
406C2×D5—22no—
407C2×D5C2×#13714yes—
408C4×C5—22no—
409C4×C5—44no—
410C4×C5—42no—
411C4×C5#53612yes—
412C4×C5C2×AGL(1,ℤ/8)14yes—
413C4×C5C2×2−1+411yescharacteristic
414C4×C5C2×2−1+411yescharacteristic
415Dic5—22no—
416Dic5—22no—
417Dic5—22no—
418Dic5—42no—
419Dic5—44no—
420Dic5—42no—
421C24—21no—
422C24—51no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—52no—
426C2×Q8—108no—
427C2×Q8—101no—
428C2×Q8—104no—
429C2×Q8—102no—
430C2×Q8—102no—
431C2×Q8—208no—
432C2×Q8—2032no—
433C2×Q8—204no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—104no—
438C2×D4—204no—
439C2×D4C23×D512yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—51no—
445C22×C4—51no—
446C22×C4—51no—
447C22×C4—52no—
448C22×C4—54no—
449C22×C4—54no—
450C22×C4—52no—
451C22×C4—104no—
452C22×C4—108no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—104no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—102no—
459C22×C4—208no—
460C22×C4D4×D511yescharacteristic
461C22×C4D4×D511yescharacteristic
462C22×C4C23×D511yescharacteristic
463C2×C8—24no—
464C2×C8—54no—
465C2×C8—104no—
466C2×C8C23×D514yes—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—54no—
471C4⋊C4—52no—
472C4⋊C4—108no—
473C4⋊C4—108no—
474C4⋊C4—104no—
475C4⋊C4—104no—
476C4⋊C4—104no—
477C4⋊C4—102no—
478C4⋊C4—101no—
479C4⋊C4—104no—
480C4⋊C4—104no—
481C4⋊C4C23×D514yes—
482C4⋊C4C23×D512yes—
483C22⋊C4—24no—
484C22⋊C4—104no—
485C22⋊C4—104no—
486C22⋊C4—104no—
487C22⋊C4—104no—
488C42—54no—
489C42—52no—
490C42—54no—
491C42—52no—
492C42—104no—
493C42D4×D512yes—
494C42C23×D512yes—
495C2×C5—82no—
496C2×C5#323914yescomplemented
497C2×C5C2×#53611yescharacteristic
498C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
499C2×C5C22×#13711yescharacteristic
500D5—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—54no—
506C23—52no—
507C23—102no—
508C23—102no—
509C23C2×D4×D511yescharacteristic
510Q8—48no—
511Q8—102no—
512Q8—102no—
513Q8—2016no—
514Q8—208no—
515Q8—202no—
516D4—22no—
517D4—22no—
518D4—42no—
519D4—208no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—52no—
527C2×C4—52no—
528C2×C4—52no—
529C2×C4—108no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—102no—
533C2×C4—108no—
534C2×C4—104no—
535C2×C4—102no—
536C2×C4—102no—
537C2×C4—102no—
538C2×C4—102no—
539C2×C4—104no—
540C2×C4—2016no—
541C2×C4—204no—
542C2×C4—202no—
543C2×C4—202no—
544C2×C4#405714yes—
545C2×C4C2×D4×D512yes—
546C2×C4C2×D4×D512yes—
547C2×C4C2×D4×D511yescharacteristic
548C2×C4#414712yes—
549C2×C4#415012yes—
550C2×C4C24×D511yescharacteristic
551C8—24no—
552C8—104no—
553C5C2×#323911yescharacteristic, complemented, Sylow
554C22—42no—
555C22—42no—
556C22—42no—
557C22—54no—
558C22—84no—
559C22—102no—
560C22—102no—
561C22—108no—
562C22#6596212yes—
563C22C2×#405712yes—
564C22D5×#13712yes—
565C22C22×D4×D511yescharacteristic
566C4—22no—
567C4—42no—
568C4—44no—
569C4—102no—
570C4—102no—
571C4—102no—
572C4—202no—
573C4—204no—
574C4—202no—
575C4#6596212yes—
576C4C2×#405714yes—
577C4C2×#414711yescharacteristic
578C4C2×#415011yescharacteristic
579C2—82no—
580C2—104no—
581C2#1061270314yesdirect product factor
582C2C2×#6596211yescharacteristic
583C2C22×#405711yescharacteristic
584C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#474C2
socleC23×C5C22×D4
prime radicalC2×#4084C22
hypercenterC2×#474D5
hypocenterC5C2×#3239
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊Q8C2semidirect
3. C2×C5×#474C2semidirect
4. C2×#65383C2semidirect
5. C2×#65296C2semidirect
6. #10612703C2direct (N × H)
7. C2×#65173C2semidirect
8. C2×#65165C2semidirect
9. C2×#65140C2semidirect
10. C2×#65124C2semidirect
12. C2×#65054C2semidirect
22. D5×C4⋊Q8C22semidirect
36. C5×#474C22semidirect
45. #65383C22semidirect
46. #65296C22semidirect
51. #65173C22semidirect
52. #65165C22semidirect
53. #65140C22semidirect
54. #65124C22semidirect
56. #65054C22semidirect
146. C2×#474D5semidirect
252. #474C2×D5semidirect
496. C2×C5#3239semidirect
553. C5C2×#3239semidirect
581. C2#10612703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#658912
C2×C5×#4742
C2×#653832
C2×#652962
#106127032
C2×#651732
C2×#651652
C2×#651402
C2×#651242
C2×#650662
C2×#650542
C2×#32395