Finite Group Databasebeta

← C2×#10612680 #12967375

Subgroups — C2×#10612680 #12967375

Order 1280.

2170 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic, complemented
2C2×C4×#711C211yesmaximal, characteristic
3C2×C5×#474C211yesmaximal, characteristic, complemented
4C2×#65383C211yesmaximal, characteristic, complemented
5C2×#65308C211yesmaximal, characteristic, complemented
6#10612680C2116yesmaximal, direct product factor
7C2×#65177C212yesmaximal, complemented
8C2×#65162C212yesmaximal, complemented
9C2×#65138C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic, complemented
11C2×#65072C212yesmaximal
12C2×#65047C212yesmaximal, complemented
13C22×#711C2211yescharacteristic
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×D4C2211yescharacteristic
16C2×#4092—21no—
17C2×Q8×Dic5—21no—
18C2×D4×Dic5C2211yescharacteristic
19C22×#687C2212yes—
20C2×C4×C22⋊D5—21no—
21C22×C4×Dic5—21no—
22C22×Dic20C2212yes—
23#66062—24no—
24#66062C2218yescomplemented
25C4×#711—24no—
26C4×#711C2218yes—
27C2×#4043C2212yes—
28C2×#4038—21no—
29C2×#4038C2211yescharacteristic
30C2×#4036—21no—
31C2×#4033C2212yes—
32C2×#4033C2211yescharacteristic
33C2×#4027—21no—
34C2×#4023—21no—
35C2×C42×D5C2211yescharacteristic
36C2×#4015C2211yescharacteristic
37C2×C4×Dic10C2211yescharacteristic
38C5×#474C2218yescomplemented
39C2×C5×#107C2212yes—
40C2×C5×C4⋊C8C2211yescharacteristic
41C2×C5×#102C2212yes—
42C2×#3952C2212yes—
43C2×#3948C2212yes—
44C2×C8×Dic5C2212yes—
45#65383C2218yescomplemented
46#65308C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3942C2212yes—
49C2×#3940C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65177C22116yescomplemented
52#65162C22116yescomplemented
53#65138C2218yescomplemented
54#65122C2218yescomplemented
55#65072C22116yes—
56#65047C22116yescomplemented
57C2×#3107—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×D4C2311yescharacteristic
60C23×C4×C5—21no—
61C22×C22⋊D5—21no—
62C23×Dic5—21no—
63C23×Dic5—21no—
64C2×#711—24no—
65C2×#711—22no—
66C2×#711—24no—
67C2×#711C2314yes—
68C22×C4×D5D411yescharacteristic
69C22×C4×D5D411yescharacteristic
70C22×C4×D5C2311yescharacteristic
71C22×Dic10C2312yes—
72C22×Dic10C2311yescharacteristic
73C22×C5×C8C2312yes—
74C5×C4⋊Q8—22no—
75C5×C4⋊Q8C2314yes—
76C4×C5×D4—22no—
77C4×C5×D4C2314yes—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4C2312yes—
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C22⋊C4—21no—
82C2×C42×C5C2311yescharacteristic
83C2×C22⋊Dic5—21no—
84#4092—28no—
85Q8×Dic5—28no—
86D4×Dic5—22no—
87D4×Dic5C2314yes—
88C2×#687—28no—
89C2×#687C2318yes—
90C4×C22⋊D5—28no—
91C2×#685—21no—
92C2×#685—22no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—22no—
96C2×C4⋊Dic5C2311yescharacteristic
97C2×#683—21no—
98C2×#683—21no—
99C2×#683—22no—
100C2×C4×Dic5—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5—22no—
104C2×C4×Dic5—28no—
105C2×C4×Dic5D411yescharacteristic
106C2×C4×Dic5D411yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C22×#168C2312yes—
109C2×Dic20—28no—
110C2×Dic20C2318yes—
111#4043—24no—
112#4043C2318yes—
113#4038—28no—
114#4038—22no—
115#4038C2314yes—
116#4036—28no—
117#4033—24no—
118#4033—22no—
119#4033C2318yes—
120#4033C2314yes—
121#4027—28no—
122#4023—28no—
123C42×D5D414yes—
124C42×D5C2314yes—
125#4015—22no—
126#4015C2314yes—
127C4×Dic10—22no—
128C4×Dic10C2314yes—
129C5×#107C2318yes—
130C5×C4⋊C8C2314yes—
131C5×#102C2318yes—
132#3952C2318yes—
133#3948C2318yes—
134C8×Dic5C2318yes—
135#3942C2318yes—
136#3942C2318yes—
137#3940C2318yes—
138C4⋊#168C2314yes—
139C2×#533—51no—
140C2×C4×#41—51no—
141#3107—516no—
142C2×#485—52no—
143C2×#475—51no—
144C2×#474D511yescharacteristic, complemented
145C2×#462—51no—
146C2×#461—51no—
147C2×#452—52no—
148C2×C4×Q16—52no—
149C2×C4×SD16—52no—
150C2×#431—51no—
151C24×C5—21no—
152C23×D5C2×D411yescharacteristic
153C2×C5×Q8—24no—
154C2×C5×Q8—44no—
155C2×C5×D4—21no—
156C2×C5×D4—22no—
157C2×C5×D4#4114yes—
158C2×C5×D4C2412yes—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—24no—
162C22×C4×C5—44no—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2411yescharacteristic
166C2×C22⋊D5—28no—
167C2×C22⋊D5—24no—
168C22×Dic5—28no—
169C22×Dic5—21no—
170C22×Dic5—24no—
171C22×Dic5—21no—
172C22×Dic5—22no—
173C22×Dic5—24no—
174C22×Dic5—44no—
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D411yescharacteristic
178#711—216no—
179#711—28no—
180#711—28no—
181C2×C4×D5—24no—
182C2×C4×D5—22no—
183C2×C4×D5—22no—
184C2×C4×D5—24no—
185C2×C4×D5C2×D414yes—
186C2×C4×D5C2×D414yes—
187C2×C4×D5C2×D412yes—
188C2×C4×D5C2412yes—
189C2×Dic10—21no—
190C2×Dic10—22no—
191C2×Dic10—24no—
192C2×Dic10—24no—
193C2×Dic10—22no—
194C2×Dic10#4114yes—
195C2×Dic10C2414yes—
196C2×Dic10C2412yes—
197C2×C5×C8—24no—
198C2×C5×C8C2414yes—
199C5×C4⋊C4—24no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4C2412yes—
203C5×C4⋊C4C2414yes—
204C5×C22⋊C4—24no—
205C42×C5C2×D412yes—
206C42×C5C2412yes—
207C22⋊Dic5—24no—
208#687—216no—
209#685—24no—
210#685—28no—
211C4⋊Dic5—21no—
212C4⋊Dic5—24no—
213C4⋊Dic5—24no—
214C4⋊Dic5—28no—
215C4⋊Dic5C2412yes—
216#683—24no—
217#683—24no—
218#683—28no—
219C4×Dic5—24no—
220C4×Dic5—24no—
221C4×Dic5—24no—
222C4×Dic5—28no—
223C4×Dic5C2×D412yes—
224C4×Dic5C2×D414yes—
225C4×Dic5C2×D414yes—
226C4×Dic5C2412yes—
227C2×#168—24no—
228C2×#168C2414yes—
229Dic20—216no—
230C22×#41—51no—
231C22×Q16—52no—
232C22×SD16—52no—
233#533—58no—
234#533—104no—
235C2×C4⋊Q8—51no—
236C2×C4⋊Q8C2×D511yescharacteristic
237C2×#125—101no—
238C2×C22⋊Q8—52no—
239C2×C22⋊Q8—101no—
240C4×#41—58no—
241C4×#41—104no—
242C2×C4×Q8—51no—
243C2×C4×Q8—52no—
244C2×C4×Q8—51no—
245C2×C4×Q8—101no—
246C2×C4×D4—51no—
247C2×C4×D4—101no—
248C2×C4×D4—101no—
249C2×C4×D4C2×D511yescharacteristic
250C2×#117—51no—
251C2×#117—101no—
252C2×#117—101no—
253C22×C42—51no—
254C22×C42—101no—
255#485—516no—
256#475—58no—
257#474C2×D518yescomplemented
258#462—58no—
259#461—58no—
260#452—516no—
261C4×Q16—516no—
262C4×SD16—516no—
263#431—58no—
264C2×#107C2×D512yes—
265C2×#106—52no—
266C2×C4⋊C8—51no—
267C2×C4⋊C8C2×D511yescharacteristic
268C2×#103—52no—
269C2×#103—52no—
270C2×#103—52no—
271C2×#102C2×D512yes—
272C2×C22⋊C8—52no—
273C2×C4×C8—52no—
274C23×C5—24no—
275C23×C5—22no—
276C23×C5—42no—
277C23×C5—42no—
278C23×C5C22×D411yescharacteristic
279C22×D5—22no—
280C22×D5—22no—
281C22×D5—22no—
282C22×D5C22×D412yes—
283C5×Q8—48no—
284C5×D4—24no—
285C5×D4—22no—
286C5×D4C2×#4114yes—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—24no—
290C2×C4×C5—42no—
291C2×C4×C5—44no—
292C2×C4×C5—48no—
293C2×C4×C5#13514yes—
294C2×C4×C5C22×D412yes—
295C2×C4×C5C22×D411yescharacteristic
296C2×C4×C5C22×D412yes—
297C2×C4×C5C2×#4112yes—
298C2×C4×C5C2×#4112yes—
299C2×C4×C5C2511yescharacteristic
300C22⋊D5—216no—
301C2×Dic5—216no—
302C2×Dic5—24no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—44no—
309C2×Dic5—42no—
310C2×Dic5—48no—
311C2×Dic5#13714yes—
312C2×Dic5C22×D412yes—
313C2×Dic5C22×D412yes—
314C2×Dic5C22×D412yes—
315C4×D5—24no—
316C4×D5—28no—
317C4×D5—22no—
318C4×D5—22no—
319C4×D5—28no—
320Dic10—24no—
321Dic10—22no—
322Dic10—24no—
323Dic10—24no—
324Dic10—44no—
325Dic10C2×#4114yes—
326C5×C8—24no—
327#168—24no—
328C2×#41—54no—
329C2×#41—102no—
330C2×#41—104no—
331C2×#41—104no—
332C22×Q8—21no—
333C22×Q8—51no—
334C22×Q8—52no—
335C22×D4—101no—
336C22×D4C22×D511yescharacteristic
337C23×C4—21no—
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—101no—
342C23×C4—101no—
343C2×Q16—58no—
344C2×Q16—108no—
345C2×SD16—58no—
346C2×SD16—108no—
347C22×C8—52no—
348C22×C8C22×D512yes—
349C4⋊Q8—22no—
350C4⋊Q8—54no—
351C4⋊Q8—102no—
352C4⋊Q8C22×D514yes—
353#125—108no—
354C22⋊Q8—58no—
355C22⋊Q8—104no—
356C22⋊Q8—108no—
357C4×Q8—54no—
358C4×Q8—58no—
359C4×Q8—54no—
360C4×Q8—102no—
361C4×Q8—104no—
362C4×Q8—108no—
363C4×Q8—102no—
364C4×D4—22no—
365C4×D4—54no—
366C4×D4—102no—
367C4×D4—108no—
368C4×D4—108no—
369C4×D4C22×D514yes—
370#117—54no—
371#117—108no—
372#117—102no—
373#117—108no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—51no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4—102no—
378C2×C4⋊C4—102no—
379C2×C4⋊C4—101no—
380C2×C4⋊C4—101no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4C22×D511yescharacteristic
383C2×C4⋊C4C22×D512yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—101no—
386C2×C22⋊C4—102no—
387C2×C22⋊C4—101no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—54no—
392C2×C42—54no—
393C2×C42—108no—
394C2×C42—101no—
395C2×C42—102no—
396C2×C42—101no—
397C2×C42—101no—
398C2×C42C22×D511yescharacteristic
399#107C22×D518yes—
400#106—58no—
401C4⋊C8—54no—
402C4⋊C8C22×D514yes—
403#103—58no—
404#103—58no—
405#103—58no—
406#102C22×D518yes—
407C22⋊C8—58no—
408C4×C8—58no—
409C22×C5—24no—
410C22×C5—48no—
411C22×C5—42no—
412C22×C5—42no—
413C22×C5#52512yes—
414C22×C5C2×#13512yes—
415C22×C5C2×#13712yes—
416C22×C5C23×D411yescharacteristic
417C2×D5—22no—
418C2×D5—22no—
419C2×D5—22no—
420C2×D5—44no—
421C4×C5—22no—
422C4×C5—42no—
423C4×C5—44no—
424C4×C5#52512yes—
425C4×C5C2×#13514yes—
426C4×C5C22×#4111yescharacteristic
427C4×C5C22×#4111yescharacteristic
428Dic5—24no—
429Dic5—22no—
430Dic5—22no—
431Dic5—44no—
432Dic5—42no—
433Dic5C2×#13714yes—
434C24—21no—
435C24—51no—
436#41—108no—
437#41—1016no—
438#41—108no—
439C2×Q8—24no—
440C2×Q8—44no—
441C2×Q8—54no—
442C2×Q8—52no—
443C2×Q8—54no—
444C2×Q8—101no—
445C2×Q8—102no—
446C2×Q8—102no—
447C2×Q8—104no—
448C2×Q8—104no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—104no—
452C2×D4—108no—
453C2×D4#71314yes—
454C2×D4C23×D512yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—51no—
462C22×C4—54no—
463C22×C4—52no—
464C22×C4—54no—
465C22×C4—52no—
466C22×C4—104no—
467C22×C4—104no—
468C22×C4—104no—
469C22×C4—108no—
470C22×C4—101no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—104no—
474C22×C4—101no—
475C22×C4—102no—
476C22×C4—204no—
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480Q16—1016no—
481SD16—1016no—
482C2×C8—24no—
483C2×C8—54no—
484C2×C8—104no—
485C2×C8C23×D514yes—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—52no—
490C4⋊C4—104no—
491C4⋊C4—108no—
492C4⋊C4—108no—
493C4⋊C4—101no—
494C4⋊C4—104no—
495C4⋊C4—104no—
496C4⋊C4—104no—
497C4⋊C4C23×D514yes—
498C4⋊C4C23×D512yes—
499C22⋊C4—24no—
500C22⋊C4—104no—
501C22⋊C4—108no—
502C22⋊C4—104no—
503C42—54no—
504C42—54no—
505C42—52no—
506C42—52no—
507C42—104no—
508C42—104no—
509C42—108no—
510C42—104no—
511C42D4×D512yes—
512C42C23×D512yes—
513C2×C5—44no—
514C2×C5#310714yescomplemented
515C2×C5C2×#52511yescharacteristic
516C2×C5C22×#13511yescharacteristic
517C2×C5C22×#13711yescharacteristic
518D5—42no—
519C23—24no—
520C23—22no—
521C23—42no—
522C23—42no—
523C23—52no—
524C23—102no—
525C23—102no—
526C23—102no—
527C23C2×D4×D511yescharacteristic
528Q8—48no—
529Q8—54no—
530Q8—104no—
531Q8—102no—
532Q8—104no—
533Q8—104no—
534Q8—204no—
535D4—24no—
536D4—22no—
537D4—1016no—
538D4C2×#71314yes—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—54no—
546C2×C4—52no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—108no—
550C2×C4—1016no—
551C2×C4—102no—
552C2×C4—102no—
553C2×C4—104no—
554C2×C4—102no—
555C2×C4—108no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—104no—
561C2×C4—208no—
562C2×C4—202no—
563C2×C4—204no—
564C2×C4#405814yes—
565C2×C4C2×D4×D512yes—
566C2×C4C2×D4×D512yes—
567C2×C4C2×D4×D511yescharacteristic
568C2×C4C2×#71312yes—
569C2×C4D5×#4112yes—
570C2×C4C24×D511yescharacteristic
571C8—24no—
572C8—104no—
573C5C2×#310711yescharacteristic, complemented, Sylow
574C22—24no—
575C22—42no—
576C22—48no—
577C22—42no—
578C22—102no—
579C22—102no—
580C22—102no—
581C22—204no—
582C22#6596112yes—
583C22C2×#405812yes—
584C22D5×#13712yes—
585C22C22×D4×D511yescharacteristic
586C4—22no—
587C4—42no—
588C4—44no—
589C4—54no—
590C4—102no—
591C4—104no—
592C4—102no—
593C4—204no—
594C4—202no—
595C4#6596112yes—
596C4C2×#405814yes—
597C4C22×#71311yescharacteristic
598C4C2×D5×#4111yescharacteristic
599C2—44no—
600C2—202no—
601C2#1061268014yesdirect product factor
602C2C2×#6596111yescharacteristic
603C2C22×#405811yescharacteristic
604C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#474C2
socleC23×C5C22×D4
prime radicalC22×#711C22
hypercenterC2×#474D5
hypocenterC5C2×#3107
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×#66062C2semidirect
3. C2×C5×#474C2semidirect
4. C2×#65383C2semidirect
5. C2×#65308C2semidirect
6. #10612680C2direct (N × H)
7. C2×#65177C2semidirect
8. C2×#65162C2semidirect
9. C2×#65138C2semidirect
10. C2×#65122C2semidirect
12. C2×#65047C2semidirect
24. #66062C22semidirect
38. C5×#474C22semidirect
45. #65383C22semidirect
46. #65308C22semidirect
51. #65177C22semidirect
52. #65162C22semidirect
53. #65138C22semidirect
54. #65122C22semidirect
56. #65047C22semidirect
144. C2×#474D5semidirect
257. #474C2×D5semidirect
514. C2×C5#3107semidirect
573. C5C2×#3107semidirect
601. C2#10612680direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660622
C2×C4×#7112
C2×C5×#4742
C2×#653832
C2×#653082
#106126802
C2×#651772
C2×#651622
C2×#651382
C2×#651222
C2×#650722
C2×#650472
C2×#31075