Finite Group Databasebeta

← C2×#10600762 #12065245

Subgroups — C2×#10600762 #12065245

Order 1280.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4075C211yesmaximal, characteristic
2C2×C5×#407C211yesmaximal, characteristic, complemented
3C22×C22⋊#168C211yesmaximal, characteristic, complemented
4#10600762C218yesmaximal, direct product factor
5C2×#64689C212yesmaximal, complemented
6C2×#64688C212yesmaximal, complemented
7C24×C4×C5C2211yescharacteristic
8C23×C22⋊D5C411yescharacteristic
9C22×C5×M16—21no—
10C22×C22⋊Dic5C411yescharacteristic
11C2×#4075—21no—
12C2×#4075—21no—
13C2×#4075—22no—
14C2×#4075—21no—
15C2×#4075—21no—
16C2×#4075—22no—
17C2×#4075C414yes—
18C2×#4075C2214yes—
19C22×#685C411yescharacteristic
20C22×#685C2211yescharacteristic
21C22×#683C411yescharacteristic
22C22×#683C2211yescharacteristic
23C23×#168—21no—
24C5×#407—22no—
25C5×#407C2214yescomplemented
26C2×C5×C22⋊C8—21no—
27C2×C5×C22⋊C8C2212yes—
28C2×C22⋊#168—22no—
29C2×C22⋊#168—21no—
30C2×C22⋊#168C2212yes—
31C2×C22⋊#168C2214yescomplemented
32#64689C2218yescomplemented
33#64688C2218yescomplemented
34C2×#1468—51nomaximal, Sylow
35C25×C5C2×C411yescharacteristic
36C24×D5C2×C411yescharacteristic
37C23×C4×C5—44no—
38C23×C4×C5C2×C411yescharacteristic
39C23×C4×C5C2×C412yes—
40C23×C4×C5D411yescharacteristic
41C23×C4×C5D411yescharacteristic
42C23×C4×C5D412yes—
43C23×C4×C5D411yescharacteristic
44C23×C4×C5D411yescharacteristic
45C23×C4×C5D412yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2312yes—
48C22×C22⋊D5—21no—
49C22×C22⋊D5—22no—
50C22×C22⋊D5—21no—
51C22×C22⋊D5—44no—
52C22×C22⋊D5C2×C414yes—
53C23×Dic5C2×C411yescharacteristic
54C23×Dic5C2×C411yescharacteristic
55C23×Dic5C2×C411yescharacteristic
56C2×C5×M16—24no—
57C2×C5×M16—44no—
58C22×C5×C8—22no—
59C2×C22⋊Dic5—21no—
60C2×C22⋊Dic5—22no—
61C2×C22⋊Dic5—21no—
62C2×C22⋊Dic5C2×C414yes—
63#4075—24no—
64#4075—24no—
65#4075—24no—
66#4075—24no—
67#4075—24no—
68#4075—24no—
69#4075—24no—
70#4075—24no—
71C2×#685—21no—
72C2×#685—21no—
73C2×#685—22no—
74C2×#685—21no—
75C2×#685—21no—
76C2×#685—21no—
77C2×#685—21no—
78C2×#685C2×C414yes—
79C2×#685C2×C412yes—
80C2×#685C2312yes—
81C2×#683—21no—
82C2×#683—21no—
83C2×#683—22no—
84C2×#683—21no—
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683C2×C414yes—
89C2×#683C2×C412yes—
90C2×#683C2312yes—
91C22×#168—22no—
92C22×#168—24no—
93C22×#168—44no—
94C5×C22⋊C8—24no—
95C5×C22⋊C8—22no—
96C5×C22⋊C8C2314yes—
97C22⋊#168—22no—
98C22⋊#168—24no—
99C22⋊#168C2314yes—
100C22×#123—51no—
101C2×#407D511yescharacteristic, complemented
102C22×C22⋊C8—51no—
103#1468—58no—
104C2×#324—52no—
105C2×#323—52no—
106C24×C5—24no—
107C24×C5—24no—
108C24×C5—42no—
109C24×C5C22⋊C411yescharacteristic
110C24×C5C22⋊C412yes—
111C24×C5C22⋊C411yescharacteristic
112C24×C5C22×C412yes—
113C24×C5C22×C411yescharacteristic
114C23×D5—21no—
115C23×D5—21no—
116C23×D5—24no—
117C23×D5—42no—
118C23×D5—42no—
119C23×D5C22×C412yes—
120C22×C4×C5—24no—
121C22×C4×C5—24no—
122C22×C4×C5—42no—
123C22×C4×C5—44no—
124C22×C4×C5—44no—
125C22×C4×C5—42no—
126C22×C4×C5—42no—
127C22×C4×C5—44no—
128C22×C4×C5—42no—
129C22×C4×C5—44no—
130C22×C4×C5C22⋊C411yescharacteristic
131C22×C4×C5C22⋊C411yescharacteristic
132C22×C4×C5C22⋊C412yes—
133C22×C4×C5C22⋊C412yes—
134C22×C4×C5C22⋊C412yes—
135C22×C4×C5C22⋊C412yes—
136C22×C4×C5C22⋊C412yes—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C22×C411yescharacteristic
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×D412yes—
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D412yes—
147C22×C4×C5C2411yescharacteristic
148C2×C22⋊D5—24no—
149C2×C22⋊D5—24no—
150C2×C22⋊D5—48no—
151C2×C22⋊D5—44no—
152C2×C22⋊D5—44no—
153C2×C22⋊D5—48no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—24no—
161C22×Dic5—42no—
162C22×Dic5—42no—
163C22×Dic5C22×C412yes—
164C22×Dic5C22×C412yes—
165C22×Dic5C22×C412yes—
166C5×M16—48no—
167C2×C5×C8—24no—
168C2×C5×C8—44no—
169C22⋊Dic5—24no—
170C22⋊Dic5—24no—
171#685—24no—
172#685—24no—
173#685—22no—
174#685—22no—
175#685—22no—
176#685—22no—
177#683—24no—
178#683—24no—
179#683—22no—
180#683—22no—
181#683—22no—
182#683—22no—
183C2×#168—24no—
184C2×#168—44no—
185C2×#168—48no—
186C23×D4—51no—
187C24×C4C2×D511yescharacteristic
188C22×M16—21no—
189C23×C8—101no—
190C2×#123—54no—
191C2×#123—54no—
192C2×#123—101no—
193C2×#123—102no—
194C2×#123—101no—
195C2×#123—101no—
196C2×#123—101no—
197C2×#123—102no—
198C22×C4⋊C4—51no—
199C22×C4⋊C4—51no—
200C22×C22⋊C4—51no—
201C22×C22⋊C4—51no—
202C22×C22⋊C4—51no—
203#407—22no—
204#407C2×D514yescomplemented
205C2×C22⋊C8—21no—
206C2×C22⋊C8—52no—
207C2×C22⋊C8—54no—
208C2×C22⋊C8—102no—
209C2×C22⋊C8—101no—
210C2×C22⋊C8C2×D512yes—
211#324—58no—
212#323—58no—
213C23×C5—22no—
214C23×C5—22no—
215C23×C5—22no—
216C23×C5—22no—
217C23×C5—24no—
218C23×C5—24no—
219C23×C5—22no—
220C23×C5—24no—
221C23×C5—22no—
222C23×C5—24no—
223C23×C5—22no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5—416no—
227C23×C5#9914yes—
228C23×C5C2×C22⋊C412yes—
229C23×C5C2×C22⋊C411yescharacteristic
230C23×C5C2×C22⋊C412yes—
231C23×C5C2×C22⋊C411yescharacteristic
232C23×C5#13114yes—
233C23×C5C23×C411yescharacteristic
234C22×D5—22no—
235C22×D5—24no—
236C22×D5—42no—
237C22×D5—42no—
238C22×D5—42no—
239C22×D5—42no—
240C22×D5—42no—
241C22×D5—42no—
242C22×D5—42no—
243C22×D5—42no—
244C22×D5—88no—
245C2×C4×C5—24no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—24no—
250C2×C4×C5—24no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—41no—
257C2×C4×C5—42no—
258C2×C4×C5—41no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5C2×C22⋊C412yes—
264C2×C4×C5C2×C22⋊C412yes—
265C2×C4×C5C2×C22⋊C411yescharacteristic
266C2×C4×C5C2×C22⋊C411yescharacteristic
267C2×C4×C5C22×D411yescharacteristic
268C2×C4×C5C22×D411yescharacteristic
269C22⋊D5—48no—
270C22⋊D5—48no—
271C2×Dic5—22no—
272C2×Dic5—22no—
273C2×Dic5—22no—
274C2×Dic5—24no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C5×C8—44no—
284#168—44no—
285C25—51no—
286C25C4×D511yescharacteristic
287C22×D4—54no—
288C22×D4—102no—
289C22×D4—101no—
290C22×D4—101no—
291C22×D4—204no—
292C23×C4—44no—
293C23×C4—51no—
294C23×C4—51no—
295C23×C4—51no—
296C23×C4C4×D511yescharacteristic
297C23×C4C4×D512yes—
298C23×C4D2011yescharacteristic
299C23×C4D2012yes—
300C23×C4D2011yescharacteristic
301C23×C4C22⋊D511yescharacteristic
302C23×C4C22⋊D511yescharacteristic
303C23×C4C22⋊D512yes—
304C23×C4C22×D511yescharacteristic
305C23×C4C22×D512yes—
306C2×M16—24no—
307C2×M16—44no—
308C22×C8—22no—
309C22×C8—104no—
310C22×C8—102no—
311C22×C8—204no—
312#123—104no—
313#123—104no—
314#123—104no—
315#123—104no—
316#123—104no—
317#123—104no—
318#123—104no—
319#123—104no—
320C2×C4⋊C4—52no—
321C2×C4⋊C4—54no—
322C2×C4⋊C4—52no—
323C2×C4⋊C4—102no—
324C2×C4⋊C4—101no—
325C2×C4⋊C4—101no—
326C2×C4⋊C4—101no—
327C2×C4⋊C4—101no—
328C2×C4⋊C4—101no—
329C2×C4⋊C4—101no—
330C2×C22⋊C4—54no—
331C2×C22⋊C4—54no—
332C2×C22⋊C4—52no—
333C2×C22⋊C4—52no—
334C2×C22⋊C4—102no—
335C2×C22⋊C4—101no—
336C2×C22⋊C4—101no—
337C2×C22⋊C4—102no—
338C2×C22⋊C4—101no—
339C2×C22⋊C4—101no—
340C2×C22⋊C4—101no—
341C2×C22⋊C4—101no—
342C2×C22⋊C4—101no—
343C2×C22⋊C4—101no—
344C22⋊C8—24no—
345C22⋊C8—22no—
346C22⋊C8—54no—
347C22⋊C8—102no—
348C22⋊C8—104no—
349C22⋊C8C22×D514yes—
350C22×C5—22no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—22no—
356C22×C5—28no—
357C22×C5—28no—
358C22×C5—22no—
359C22×C5—22no—
360C22×C5—22no—
361C22×C5—48no—
362C22×C5—48no—
363C22×C5—44no—
364C22×C5#40812yes—
365C22×C5#40812yes—
366C22×C5C2×#9914yes—
367C22×C5C2×#9912yes—
368C22×C5#41312yes—
369C22×C5C22×C22⋊C411yescharacteristic
370C22×C5C2×#13112yes—
371C2×D5—42no—
372C2×D5—42no—
373C2×D5—42no—
374C2×D5—84no—
375C2×D5—84no—
376C2×D5—84no—
377C4×C5—21no—
378C4×C5—21no—
379C4×C5—22no—
380C4×C5—21no—
381C4×C5—21no—
382C4×C5—22no—
383Dic5—42no—
384Dic5—42no—
385Dic5—42no—
386C24—24no—
387C24—24no—
388C24—42no—
389C24—52no—
390C24—101no—
391C24—101no—
392C24—104no—
393C24—202no—
394C24—202no—
395C24#68511yescharacteristic
396C24#68512yes—
397C24#68511yescharacteristic
398C24C2×C4×D512yes—
399C24C2×C4×D511yescharacteristic
400C2×D4—104no—
401C2×D4—104no—
402C2×D4—208no—
403C2×D4—204no—
404C2×D4—204no—
405C2×D4—208no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—52no—
417C22×C4—52no—
418C22×C4—52no—
419C22×C4—104no—
420C22×C4—101no—
421C22×C4—101no—
422C22×C4—101no—
423C22×C4—101no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—202no—
427C22×C4—202no—
428C22×C4#68512yes—
429C22×C4#68512yes—
430C22×C4#68511yescharacteristic
431C22×C4#68511yescharacteristic
432C22×C4#68512yes—
433C22×C4#68512yes—
434C22×C4#68512yes—
435C22×C4C2×C4×D512yes—
436C22×C4C2×C4×D511yescharacteristic
437C22×C4C2×D2012yes—
438C22×C4C2×D2011yescharacteristic
439C22×C4C2×D2011yescharacteristic
440C22×C4C2×D2012yes—
441C22×C4C2×C22⋊D512yes—
442C22×C4C2×C22⋊D511yescharacteristic
443C22×C4C2×C22⋊D511yescharacteristic
444C22×C4C2×C22⋊D512yes—
445C22×C4C23×D511yescharacteristic
446M16—48no—
447C2×C8—24no—
448C2×C8—44no—
449C2×C8—104no—
450C2×C8—208no—
451C2×C8—204no—
452C4⋊C4—102no—
453C4⋊C4—102no—
454C4⋊C4—104no—
455C4⋊C4—104no—
456C4⋊C4—102no—
457C4⋊C4—102no—
458C22⋊C4—104no—
459C22⋊C4—104no—
460C22⋊C4—102no—
461C22⋊C4—102no—
462C22⋊C4—104no—
463C22⋊C4—104no—
464C22⋊C4—102no—
465C22⋊C4—102no—
466C2×C5—22no—
467C2×C5—22no—
468C2×C5—24no—
469C2×C5—42no—
470C2×C5#146814yescomplemented
471C2×C5C2×#40811yescharacteristic
472C2×C5C22×#9911yescharacteristic
473C2×C5C2×#41311yescharacteristic
474D5—82no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—416no—
489C23—102no—
490C23—104no—
491C23—202no—
492C23—202no—
493C23—202no—
494C23—202no—
495C23—202no—
496C23—202no—
497C23—202no—
498C23—202no—
499C23—408no—
500C23#393814yes—
501C23#405314yes—
502C23C2×#68512yes—
503C23C2×#68511yescharacteristic
504C23C2×#68512yes—
505C23C2×#68511yescharacteristic
506C23C22×C4×D511yescharacteristic
507D4—208no—
508D4—208no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—104no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—102no—
531C2×C4—202no—
532C2×C4—202no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—202no—
539C2×C4C2×#68511yescharacteristic
540C2×C4C2×#68511yescharacteristic
541C2×C4C2×#68512yes—
542C2×C4C2×#68512yes—
543C2×C4C22×D2011yescharacteristic
544C2×C4C22×C22⋊D511yescharacteristic
545C8—44no—
546C8—204no—
547C5C2×#146811yescharacteristic, complemented, Sylow
548C22—28no—
549C22—28no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—202no—
563C22—202no—
564C22—202no—
565C22—404no—
566C22—404no—
567C22—404no—
568C22C2×#393814yes—
569C22C2×#393812yes—
570C22#6541812yes—
571C22#6541812yes—
572C22#6542112yes—
573C22C2×#405312yes—
574C22C22×#68511yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—202no—
582C4—202no—
583C4—202no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—402no—
589C2#1060076214yesdirect product factor
590C2C22×#393811yescharacteristic
591C2C2×#6541811yescharacteristic
592C2C2×#6542111yescharacteristic
subgroupG/H
centerC23C2×#685
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#407C2
socleC23×C5C2×C22⋊C4
prime radicalC23×C22⋊D5C4
cosocleC22×C4×C5C24
hypercenterC2×#407D5
hypocenterC5C2×#1468
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×C5×#407C2semidirect
3. C22×C22⋊#168C2semidirect
4. #10600762C2direct (N × H)
5. C2×#64689C2semidirect
6. C2×#64688C2semidirect
25. C5×#407C22semidirect
31. C2×C22⋊#168C22semidirect
32. #64689C22semidirect
33. #64688C22semidirect
101. C2×#407D5semidirect
204. #407C2×D5semidirect
470. C2×C5#1468semidirect
547. C5C2×#1468semidirect
589. C2#10600762direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40752
C2×C5×#4072
C22×C22⋊#1682
#106007622
C2×#646892
C2×#646882
C2×#14685