Finite Group Databasebeta

← C2×#10613045 #12971339

Subgroups — C2×#10613045 #12971339

Order 1280.

2286 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×#65930C211yesmaximal, characteristic
2C22×#4034C211yesmaximal, characteristic, complemented
3C2×C5×#481C211yesmaximal, characteristic, complemented
4C2×#65323C211yesmaximal, characteristic, complemented
5C2×#65257C211yesmaximal, characteristic, complemented
6#10613045C2116yesmaximal, direct product factor
7C2×#65159C212yesmaximal, complemented
8C2×#65144C212yesmaximal, complemented
9C2×#65049C212yesmaximal, complemented
10C2×#65050C212yesmaximal, complemented
11C2×#65024C211yesmaximal, characteristic, complemented
12C2×#65007C211yesmaximal, characteristic, complemented
13C22×#711—21no—
14C23×Dic10—21no—
15C2×C5×#121C2211yescharacteristic
16C22×C5×C4⋊C4C2211yescharacteristic
17C2×#4086C2211yescharacteristic
18C2×#4085—21no—
19C2×D4×Dic5C2211yescharacteristic
20C2×#4076C2211yescharacteristic
21C22×C4⋊Dic5C2211yescharacteristic
22C22×#683—21no—
23C22×C4×Dic5C2211yescharacteristic
24#65930—24no—
25#65930C2218yes—
26C2×#4039C2211yescharacteristic
27C2×#4034—21no—
28C2×#4034—21no—
29C2×#4034—24no—
30C2×#4034C2212yes—
31C2×#4034C2212yes—
32C2×#4034C2211yescharacteristic
33C2×#4034C2218yescomplemented
34C2×#4031—21no—
35C2×#4027—21no—
36C5×#481C2218yescomplemented
37C2×C5×#106C2212yes—
38C2×C5×#102C2212yes—
39C2×C5×C22⋊C8C2211yescharacteristic
40C2×#3964C2212yes—
41C2×C22⋊#168C2211yescharacteristic
42C2×#3953C2212yes—
43C2×#3949C2212yes—
44C2×#3946C2212yes—
45C2×C8×Dic5C2212yes—
46#65323C2218yescomplemented
47#65257C2218yescomplemented
48C2×#3941C2212yes—
49C2×#3940C2212yes—
50C2×#3940C2212yes—
51#65159C22116yescomplemented
52#65144C22116yescomplemented
53#65050C22116yescomplemented
54#65049C22116yescomplemented
55#65024C2218yescomplemented
56#65007C2218yescomplemented
57C2×#3227—51nomaximal, Sylow
58C22×C5×D4—21no—
59C22×C5×D4C2311yescharacteristic
60C23×C4×C5—21no—
61C23×C4×C5C2311yescharacteristic
62C22×C22⋊D5—21no—
63C22×C22⋊D5—21no—
64C22×C22⋊D5—21no—
65C23×Dic5—21no—
66C23×Dic5—21no—
67C23×Dic5D411yescharacteristic
68C23×Dic5D411yescharacteristic
69C2×#711—28no—
70C2×#711—48no—
71C22×D20C2311yescharacteristic
72C22×C4×D5—21no—
73C22×Dic10—21no—
74C22×Dic10—22no—
75C22×Dic10—22no—
76C22×Dic10—28no—
77C22×Dic10—48no—
78C22×C5×C8C2312yes—
79C5×#121—22no—
80C5×#121C2314yes—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—22no—
83C2×C5×C4⋊C4C2312yes—
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C22⋊C4—21no—
87C2×C22⋊Dic5—21no—
88#4086—22no—
89#4086C2314yes—
90#4085—28no—
91D4×Dic5—22no—
92D4×Dic5C2314yes—
93#4076—22no—
94#4076C2314yes—
95C2×#685—21no—
96C2×#685—21no—
97C2×C4⋊Dic5—21no—
98C2×C4⋊Dic5—22no—
99C2×C4⋊Dic5C2314yes—
100C2×C4⋊Dic5C2312yes—
101C2×C4⋊Dic5C2311yescharacteristic
102C2×#683—21no—
103C2×#683—22no—
104C2×#683—22no—
105C2×#683—28no—
106C2×C4×Dic5D411yescharacteristic
107C2×C4×Dic5D411yescharacteristic
108C2×C4×Dic5D414yes—
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2311yescharacteristic
111C2×C4×Dic5C2314yes—
112C22×#168C2312yes—
113#4039—22no—
114#4039C2314yes—
115#4034—24no—
116#4034—24no—
117#4034—28no—
118#4034—28no—
119#4034—22no—
120#4034C2318yes—
121#4034C2318yes—
122#4034C2314yes—
123#4031—28no—
124#4027—28no—
125C5×#106C2318yes—
126C5×#102C2318yes—
127C5×C22⋊C8C2314yes—
128#3964C2318yes—
129C22⋊#168C2314yes—
130#3953C2318yes—
131#3949C2318yes—
132#3946C2318yes—
133C8×Dic5C2318yes—
134#3941C2318yes—
135#3940C2318yes—
136#3940C2318yes—
137C2×#532—51no—
138C22×C4⋊Q8—51no—
139#3227—516no—
140C2×#498—52no—
141C2×#486—52no—
142C2×#481—51no—
143C2×#481—51no—
144C2×#481—51no—
145C2×#481D511yescharacteristic, complemented
146C2×#476—52no—
147C2×#476—52no—
148C2×#431—51no—
149C24×C5—21no—
150C24×C5C2×D411yescharacteristic
151C23×D5—21no—
152C2×C5×D4—24no—
153C2×C5×D4—21no—
154C2×C5×D4—24no—
155C2×C5×D4—44no—
156C2×C5×D4C2412yes—
157C22×C4×C5—24no—
158C22×C4×C5—21no—
159C22×C4×C5—24no—
160C22×C4×C5—22no—
161C22×C4×C5—44no—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D412yes—
164C22×C4×C5C2412yes—
165C22×C4×C5C2411yescharacteristic
166C2×C22⋊D5—24no—
167C2×C22⋊D5—24no—
168C2×C22⋊D5—24no—
169C2×C22⋊D5—44no—
170C2×C22⋊D5—44no—
171C2×C22⋊D5—44no—
172C22×Dic5—24no—
173C22×Dic5—21no—
174C22×Dic5—24no—
175C22×Dic5—22no—
176C22×Dic5—24no—
177C22×Dic5—44no—
178C22×Dic5—44no—
179C22×Dic5SD1618yes—
180C22×Dic5C2×D411yescharacteristic
181C22×Dic5C2×D414yes—
182C22×Dic5C2×D411yescharacteristic
183C22×Dic5C2×D411yescharacteristic
184C22×Dic5C2×D414yes—
185C22×Dic5C2×D411yescharacteristic
186#711—432no—
187C2×D20—24no—
188C2×D20—21no—
189C2×D20C2412yes—
190C2×C4×D5—24no—
191C2×C4×D5—44no—
192C2×Dic10—28no—
193C2×Dic10—28no—
194C2×Dic10—24no—
195C2×Dic10—48no—
196C2×Dic10—432no—
197C2×Dic10—44no—
198C2×Dic10—48no—
199C2×C5×C8—24no—
200C2×C5×C8C2414yes—
201C5×C4⋊C4—24no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C2414yes—
205C5×C4⋊C4C2412yes—
206C5×C22⋊C4—24no—
207C22⋊Dic5—24no—
208#685—24no—
209#685—24no—
210C4⋊Dic5—24no—
211C4⋊Dic5—22no—
212C4⋊Dic5—21no—
213C4⋊Dic5C2412yes—
214C4⋊Dic5C2414yes—
215#683—24no—
216#683—28no—
217#683—28no—
218C4×Dic5C2×D414yes—
219C4×Dic5C2×D414yes—
220C4×Dic5C2×D412yes—
221C4×Dic5C2×D412yes—
222C4×Dic5C2412yes—
223C4×Dic5C2412yes—
224C2×#168—24no—
225C2×#168C2414yes—
226C22×#41—101no—
227C23×Q8—101no—
228#532—58no—
229#532—104no—
230C2×C4⋊Q8—52no—
231C2×C4⋊Q8—51no—
232C2×C4⋊Q8—52no—
233C2×C4⋊Q8—58no—
234C2×C4⋊Q8—101no—
235C2×C4⋊Q8—101no—
236C2×C4⋊Q8—104no—
237C2×#127—51no—
238C2×#124—101no—
239C2×#121—51no—
240C2×#121—101no—
241C2×#121C2×D511yescharacteristic
242C2×C4×D4—51no—
243C2×C4×D4—51no—
244C2×C4×D4—101no—
245C22×C4⋊C4—51no—
246C22×C4⋊C4—101no—
247C22×C4⋊C4C2×D511yescharacteristic
248C22×C42—51no—
249#498—516no—
250#486—516no—
251#481—58no—
252#481—58no—
253#481—58no—
254#481C2×D518yescomplemented
255#476—516no—
256#476—516no—
257#431—58no—
258C2×#106—52no—
259C2×#106—52no—
260C2×#106—52no—
261C2×#106C2×D512yes—
262C2×C4⋊C8—52no—
263C2×#102—52no—
264C2×#102—52no—
265C2×#102—52no—
266C2×#102C2×D512yes—
267C2×C22⋊C8—51no—
268C2×C22⋊C8C2×D511yescharacteristic
269C2×C4×C8—52no—
270C23×C5—22no—
271C23×C5—22no—
272C23×C5—22no—
273C23×C5—42no—
274C23×C5—42no—
275C23×C5—42no—
276C23×C5C2×SD1614yes—
277C23×C5C22×D412yes—
278C23×C5C22×D411yescharacteristic
279C22×D5—22no—
280C22×D5—42no—
281C22×D5—42no—
282C22×D5—42no—
283C5×D4—24no—
284C5×D4—48no—
285C5×D4—42no—
286C2×C4×C5—22no—
287C2×C4×C5—24no—
288C2×C4×C5—24no—
289C2×C4×C5—24no—
290C2×C4×C5—22no—
291C2×C4×C5—48no—
292C2×C4×C5—42no—
293C2×C4×C5—44no—
294C2×C4×C5C22×D412yes—
295C2×C4×C5C22×D411yescharacteristic
296C2×C4×C52−1+414yes—
297C2×C4×C5C2511yescharacteristic
298C22⋊D5—48no—
299C22⋊D5—48no—
300C22⋊D5—48no—
301C2×Dic5—22no—
302C2×Dic5—28no—
303C2×Dic5—24no—
304C2×Dic5—22no—
305C2×Dic5—48no—
306C2×Dic5—42no—
307C2×Dic5—44no—
308C2×Dic5—48no—
309C2×Dic5C2×SD1614yes—
310C2×Dic5C2×SD1614yes—
311C2×Dic5C2×SD16116yes—
312C2×Dic5AGL(1,ℤ/8)14yes—
313C2×Dic5C22×D412yes—
314C2×Dic5C22×D412yes—
315C2×Dic5C22×D412yes—
316C2×Dic5C22×D412yes—
317D20—24no—
318D20—42no—
319C4×D5—48no—
320Dic10—416no—
321Dic10—48no—
322Dic10—416no—
323C5×C8—24no—
324#168—24no—
325C2×#41—108no—
326C2×#41—208no—
327C22×Q8—101no—
328C22×Q8—108no—
329C22×Q8—102no—
330C22×Q8—102no—
331C22×Q8—208no—
332C22×D4—21no—
333C22×D4—51no—
334C22×D4—101no—
335C22×D4—101no—
336C22×D4—101no—
337C22×D4C22×D511yescharacteristic
338C23×C4—21no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—101no—
342C23×C4—101no—
343C23×C4—101no—
344C23×C4C22×D511yescharacteristic
345C22×C8—52no—
346C22×C8C22×D512yes—
347C4⋊Q8—58no—
348C4⋊Q8—58no—
349C4⋊Q8—54no—
350C4⋊Q8—104no—
351C4⋊Q8—108no—
352C4⋊Q8—108no—
353C4⋊Q8—104no—
354C4⋊Q8—102no—
355#127—54no—
356#127—102no—
357#124—108no—
358#121—22no—
359#121—54no—
360#121—108no—
361#121—102no—
362#121C22×D514yes—
363C4×D4—54no—
364C4×D4—54no—
365C4×D4—108no—
366C4×D4—102no—
367C4×D4—102no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—52no—
371C2×C4⋊C4—54no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—108no—
374C2×C4⋊C4—102no—
375C2×C4⋊C4—102no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4—102no—
378C2×C4⋊C4—101no—
379C2×C4⋊C4C22×D512yes—
380C2×C4⋊C4C22×D514yes—
381C2×C4⋊C4C22×D511yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—101no—
384C2×C22⋊C4—101no—
385C2×C22⋊C4—101no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—54no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—54no—
392#106—58no—
393#106—58no—
394#106—58no—
395#106C22×D518yes—
396C4⋊C8—58no—
397#102—58no—
398#102—58no—
399#102—58no—
400#102C22×D518yes—
401C22⋊C8—54no—
402C22⋊C8C22×D514yes—
403C4×C8—58no—
404C22×C5—22no—
405C22×C5—28no—
406C22×C5—22no—
407C22×C5—42no—
408C22×C5—42no—
409C22×C5—42no—
410C22×C5—84no—
411C22×C5#53612yes—
412C22×C5C22×SD1614yes—
413C22×C5C22×SD1612yes—
414C22×C5C2×AGL(1,ℤ/8)12yes—
415C22×C5C23×D411yescharacteristic
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—84no—
420C4×C5—22no—
421C4×C5—42no—
422C4×C5—44no—
423C4×C5#53612yes—
424C4×C5C2×2−1+412yes—
425Dic5—22no—
426Dic5—42no—
427Dic5—44no—
428Dic5C22×SD1614yes—
429Dic5C22×SD1614yes—
430Dic5C2×AGL(1,ℤ/8)14yes—
431C24—21no—
432C24—101no—
433C24D4×D511yescharacteristic
434#41—2032no—
435C2×Q8—104no—
436C2×Q8—108no—
437C2×Q8—108no—
438C2×Q8—208no—
439C2×Q8—204no—
440C2×Q8—2032no—
441C2×Q8—208no—
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—24no—
445C2×D4—44no—
446C2×D4—52no—
447C2×D4—104no—
448C2×D4—104no—
449C2×D4—104no—
450C2×D4—101no—
451C2×D4—104no—
452C2×D4—204no—
453C2×D4—204no—
454C2×D4—204no—
455C2×D4C23×D512yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—54no—
462C22×C4—51no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—58no—
466C22×C4—54no—
467C22×C4—51no—
468C22×C4—104no—
469C22×C4—104no—
470C22×C4—104no—
471C22×C4—104no—
472C22×C4—101no—
473C22×C4—102no—
474C22×C4—204no—
475C22×C4—204no—
476C22×C4—204no—
477C22×C4D4×D512yes—
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D512yes—
480C22×C4C23×D511yescharacteristic
481C2×C8—24no—
482C2×C8—54no—
483C2×C8—104no—
484C2×C8C23×D514yes—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—54no—
489C4⋊C4—52no—
490C4⋊C4—108no—
491C4⋊C4—108no—
492C4⋊C4—104no—
493C4⋊C4—104no—
494C4⋊C4—102no—
495C4⋊C4—101no—
496C4⋊C4C23×D514yes—
497C4⋊C4C23×D512yes—
498C22⋊C4—24no—
499C22⋊C4—104no—
500C22⋊C4—104no—
501C22⋊C4—104no—
502C42—54no—
503C42—52no—
504C42—54no—
505C42—52no—
506C42—52no—
507C42—52no—
508C2×C5—24no—
509C2×C5—82no—
510C2×C5#322714yescomplemented
511C2×C5C2×#53611yescharacteristic
512C2×C5C23×SD1611yescharacteristic
513C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
514D5—82no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—102no—
522C23—202no—
523C23—202no—
524C23—202no—
525C23D5×SD1614yes—
526C23C2×D4×D512yes—
527C23C2×D4×D511yescharacteristic
528Q8—2016no—
529Q8—208no—
530Q8—2016no—
531D4—24no—
532D4—48no—
533D4—42no—
534D4—104no—
535D4—208no—
536D4—208no—
537D4—208no—
538D4—202no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—54no—
548C2×C4—54no—
549C2×C4—516no—
550C2×C4—54no—
551C2×C4—52no—
552C2×C4—52no—
553C2×C4—52no—
554C2×C4—52no—
555C2×C4—108no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—104no—
559C2×C4—208no—
560C2×C4—208no—
561C2×C4—208no—
562C2×C4—202no—
563C2×C4—204no—
564C2×C4C2×D4×D512yes—
565C2×C4C2×D4×D511yescharacteristic
566C2×C4#415014yes—
567C2×C4C24×D511yescharacteristic
568C8—24no—
569C8—104no—
570C5C2×#322711yescharacteristic, complemented, Sylow
571C22—22no—
572C22—28no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—84no—
578C22—202no—
579C22—202no—
580C22—202no—
581C22—404no—
582C22#6598212yes—
583C22C2×D5×SD1614yes—
584C22C2×D5×SD1612yes—
585C22D5×AGL(1,ℤ/8)12yes—
586C22C22×D4×D511yescharacteristic
587C4—22no—
588C4—42no—
589C4—44no—
590C4—54no—
591C4—54no—
592C4—54no—
593C4—102no—
594C4—202no—
595C4—204no—
596C4#6598212yes—
597C4C2×#415012yes—
598C2—24no—
599C2—82no—
600C2—402no—
601C2#1061304514yesdirect product factor
602C2C2×#6598211yescharacteristic
603C2C22×D5×SD1611yescharacteristic
604C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#481C2
socleC23×C5C22×D4
prime radicalC2×#65930C2
hypercenterC2×#481D5
hypocenterC5C2×#3227
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#4034C2semidirect
3. C2×C5×#481C2semidirect
4. C2×#65323C2semidirect
5. C2×#65257C2semidirect
6. #10613045C2direct (N × H)
7. C2×#65159C2semidirect
8. C2×#65144C2semidirect
9. C2×#65049C2semidirect
10. C2×#65050C2semidirect
11. C2×#65024C2semidirect
12. C2×#65007C2semidirect
33. C2×#4034C22semidirect
36. C5×#481C22semidirect
46. #65323C22semidirect
47. #65257C22semidirect
51. #65159C22semidirect
52. #65144C22semidirect
53. #65050C22semidirect
54. #65049C22semidirect
55. #65024C22semidirect
56. #65007C22semidirect
145. C2×#481D5semidirect
254. #481C2×D5semidirect
510. C2×C5#3227semidirect
570. C5C2×#3227semidirect
601. C2#10613045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659302
C22×#40342
C2×C5×#4812
C2×#653232
C2×#652572
#106130452
C2×#651592
C2×#651442
C2×#650492
C2×#650502
C2×#650242
C2×#650072
C2×#32275