Finite Group Databasebeta

← C2×#10610970 #12966810

Subgroups — C2×#10610970 #12966810

Order 1280.

2530 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65851C211yesmaximal, characteristic, complemented
2C22×#4039C211yesmaximal, characteristic
3C2×C5×#428C211yesmaximal, characteristic, complemented
4C2×#65420C211yesmaximal, characteristic
5C2×#65407C211yesmaximal, characteristic, complemented
6C2×#65290C211yesmaximal, characteristic
7C2×#65253C211yesmaximal, characteristic, complemented
8#10610970C2116yesmaximal, direct product factor
9C2×#65180C212yesmaximal, complemented
10C2×#65179C212yesmaximal, complemented
11C2×#65160C212yesmaximal, complemented
12C2×#65159C212yesmaximal, complemented
13C23×D20C2211yescharacteristic
14C23×C4×D5—21no—
15C22×C5×M16C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×#685—21no—
19C2×#4072C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×#681C2211yescharacteristic
22#65851—24no—
23#65851C2218yescomplemented
24C2×#4039—21no—
25C2×#4039—21no—
26C2×#4039—24no—
27C2×#4039C2218yes—
28C2×#4039C2212yes—
29C2×#4039C2212yes—
30C2×#4036C2211yescharacteristic, complemented
31C2×#4036C2211yescharacteristic, complemented
32C2×#4034C2211yescharacteristic, complemented
33C2×#4034C2211yescharacteristic, complemented
34C2×#4024—21no—
35C2×#4024—21no—
36C2×C4×Dic10—21no—
37C2×C4×Dic10—21no—
38C5×#428C2218yescomplemented
39C2×C5×#107C2212yescomplemented
40C2×C5×#106C2212yescomplemented
41#65420C2218yes—
42#65407C2218yescomplemented
43C2×#3953C2212yes—
44C2×#3953C2212yes—
45C2×#3946C2212yescomplemented
46C2×#3946C2212yescomplemented
47#65290C2218yes—
48C2×#3941C2212yes—
49C2×#3941C2212yes—
50#65253C2218yescomplemented
51C2×#3940C2212yescomplemented
52C2×#3939C2212yescomplemented
53#65180C22116yescomplemented
54#65179C22116yescomplemented
55#65160C22116yescomplemented
56#65159C22116yescomplemented
57C2×#3077—51nomaximal, Sylow
58C24×D5—21no—
59C23×C4×C5—21no—
60C23×C4×C5C2311yescharacteristic
61C23×Dic5D411yescharacteristic
62C23×Dic5D411yescharacteristic
63C22×D20—21no—
64C22×D20—22no—
65C22×D20—24no—
66C22×D20Q818yescomplemented
67C22×D20C2312yes—
68C22×D20C2314yes—
69C22×C4×D5—28no—
70C22×C4×D5—22no—
71C22×C4×D5—22no—
72C22×C4×D5—48no—
73C22×Dic10—21no—
74C22×Dic10—21no—
75C2×C5×M16—24no—
76C2×C5×M16C2314yes—
77C22×C5×C8C2312yes—
78C5×#117—22no—
79C5×#117C2314yes—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—22no—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2312yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C22⋊C4—21no—
87C2×C42×C5—21no—
88C2×C22⋊Dic5—21no—
89C2×#685—22no—
90C2×#685—22no—
91C2×#685—28no—
92#4072—22no—
93#4072C2314yes—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5C2311yescharacteristic
97C2×C4⋊Dic5C2311yescharacteristic
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C2×#683—21no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5D414yes—
106C2×C4×Dic5C2311yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C2×C4×Dic5C2314yes—
109C2×#681—24no—
110C2×#681C2314yes—
111C22×#168C2312yes—
112#4039—28no—
113#4039—28no—
114#4039—24no—
115#4039—24no—
116#4039C2318yes—
117#4039C2318yes—
118#4036—22no—
119#4036—22no—
120#4036C2314yescomplemented
121#4036C2314yescomplemented
122#4034—22no—
123#4034—22no—
124#4034C2314yescomplemented
125#4034C2314yescomplemented
126#4024—28no—
127#4024—28no—
128C4×Dic10—28no—
129C4×Dic10—28no—
130C5×#107C2318yescomplemented
131C5×#106C2318yescomplemented
132#3953C2318yes—
133#3953C2318yes—
134#3946C2318yescomplemented
135#3946C2318yescomplemented
136#3941C2318yes—
137#3941C2318yes—
138#3940C2318yescomplemented
139#3939C2318yescomplemented
140C2×#533—51no—
141C22×C4×D4—51no—
142#3077—516no—
143C2×#478—52no—
144C2×#478—52no—
145C2×#476—52no—
146C2×#474—52no—
147C2×#428—51no—
148C2×#428D511yescharacteristic, complemented
149C2×#423—51no—
150C2×#418—51no—
151C2×#418—51no—
152C24×C5C2×D411yescharacteristic
153C23×D5—28no—
154C23×D5—24no—
155C23×D5—22no—
156C23×D5—44no—
157C23×D5—44no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—24no—
161C22×C4×C5—22no—
162C22×C4×C5—44no—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D412yes—
165C22×C4×C5C2×Q814yes—
166C22×C4×C5#4114yes—
167C22×C4×C5C2411yescharacteristic
168C22×C4×C5C2412yes—
169C22×Dic5—24no—
170C22×Dic5—21no—
171C22×Dic5—21no—
172C22×Dic5—42no—
173C22×Dic5C2×D411yescharacteristic
174C22×Dic5C2×D414yes—
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D414yes—
178C22×Dic5C2×D411yescharacteristic
179C2×D20—216no—
180C2×D20—28no—
181C2×D20—22no—
182C2×D20—24no—
183C2×D20—24no—
184C2×D20—28no—
185C2×D20C2×Q818yes—
186C2×D20C2×Q8116yescomplemented
187C2×D20C2414yes—
188C2×C4×D5—28no—
189C2×C4×D5—28no—
190C2×C4×D5—48no—
191C2×C4×D5—432no—
192C2×C4×D5—48no—
193C2×Dic10—24no—
194C2×Dic10—24no—
195C2×Dic10—44no—
196C2×Dic10—44no—
197C5×M16—28no—
198C2×C5×C8—24no—
199C2×C5×C8C2414yes—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—22no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C2414yes—
205C5×C4⋊C4C2412yes—
206C5×C4⋊C4C2412yes—
207C5×C22⋊C4—24no—
208C42×C5—24no—
209C22⋊Dic5—24no—
210#685—28no—
211#685—28no—
212C4⋊Dic5—21no—
213C4⋊Dic5—21no—
214C4⋊Dic5—24no—
215C4⋊Dic5—24no—
216C4⋊Dic5C2412yes—
217C4⋊Dic5C2412yes—
218#683—24no—
219#683—24no—
220#683—24no—
221#683—24no—
222C4×Dic5—24no—
223C4×Dic5C2×D414yes—
224C4×Dic5C2×D414yes—
225C4×Dic5C2×D412yes—
226C4×Dic5C2×D412yes—
227C4×Dic5C2412yes—
228C4×Dic5C2412yes—
229#681—28no—
230C2×#168—24no—
231C2×#168C2414yes—
232C23×D4—51no—
233C24×C4—101no—
234C22×M16—51no—
235C22×M16C2×D511yescharacteristic
236#533—58no—
237#533—104no—
238C2×C4⋊Q8—51no—
239C2×C4⋊Q8—51no—
240C2×#125—51no—
241C2×#125—51no—
242C2×C22⋊Q8—101no—
243C2×C22⋊Q8—101no—
244C2×C4×Q8—101no—
245C2×C4×Q8—101no—
246C2×C4×D4—52no—
247C2×C4×D4—58no—
248C2×C4×D4—52no—
249C2×C4×D4—101no—
250C2×C4×D4—101no—
251C2×C4×D4—104no—
252C2×#117—51no—
253C2×#117C2×D511yescharacteristic
254C22×C4⋊C4C2×D511yescharacteristic
255C22×C22⋊C4—101no—
256C22×C42—51no—
257#478—516no—
258#478—516no—
259#476—516no—
260#474—516no—
261#428—58no—
262#428C2×D518yescomplemented
263C2×#107—52no—
264C2×#107C2×D512yescomplemented
265C2×#106—52no—
266C2×#106C2×D512yescomplemented
267#423—58no—
268C2×C4⋊C8—52no—
269C2×C4⋊C8—52no—
270#418—58no—
271#418—58no—
272C2×#102—52no—
273C2×#102—52no—
274C2×#102—52no—
275C2×#102—52no—
276C23×C5—22no—
277C23×C5—22no—
278C23×C5C22⋊Q814yes—
279C23×C5C22×D412yes—
280C23×C5C22×D411yescharacteristic
281C22×D5—28no—
282C22×D5—216no—
283C22×D5—24no—
284C22×D5—48no—
285C22×D5—432no—
286C22×D5—48no—
287C22×D5—44no—
288C22×D5—44no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5—48no—
293C2×C4×C5—42no—
294C2×C4×C5—44no—
295C2×C4×C5—42no—
296C2×C4×C5C22⋊Q818yes—
297C2×C4×C5C22⋊Q814yes—
298C2×C4×C5C22×D412yes—
299C2×C4×C5C22×D411yescharacteristic
300C2×C4×C5C22×Q812yes—
301C2×C4×C5C22×Q814yes—
302C2×C4×C5C2×#4112yes—
303C2×C4×C5C2×#4114yes—
304C2×C4×C5C2511yescharacteristic
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—28no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—44no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5AGL(1,ℤ/8)14yes—
315C2×Dic5C22×D412yes—
316C2×Dic5C22×D412yes—
317C2×Dic5C22×D412yes—
318C2×Dic5C22×D412yes—
319D20—28no—
320D20—216no—
321D20—24no—
322D20C22×Q818yes—
323C4×D5—416no—
324C4×D5—416no—
325Dic10—48no—
326Dic10—48no—
327C5×C8—24no—
328#168—24no—
329C25—101no—
330C22×Q8—101no—
331C22×Q8—101no—
332C22×D4—58no—
333C22×D4—52no—
334C22×D4—54no—
335C22×D4—104no—
336C22×D4—102no—
337C22×D4—101no—
338C23×C4—21no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—108no—
342C23×C4—102no—
343C23×C4—102no—
344C23×C4—208no—
345C23×C4C22×D511yescharacteristic
346C2×M16—24no—
347C2×M16—54no—
348C2×M16—104no—
349C2×M16C22×D514yes—
350C22×C8—52no—
351C22×C8C22×D512yes—
352C4⋊Q8—54no—
353C4⋊Q8—54no—
354C4⋊Q8—102no—
355C4⋊Q8—102no—
356#125—54no—
357#125—54no—
358#125—102no—
359#125—102no—
360C22⋊Q8—108no—
361C22⋊Q8—108no—
362C4×Q8—108no—
363C4×Q8—108no—
364C4×D4—58no—
365C4×D4—58no—
366C4×D4—108no—
367C4×D4—108no—
368C4×D4—104no—
369C4×D4—104no—
370#117—22no—
371#117—54no—
372#117—102no—
373#117C22×D514yes—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—51no—
377C2×C4⋊C4—51no—
378C2×C4⋊C4—101no—
379C2×C4⋊C4—101no—
380C2×C4⋊C4—101no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4—101no—
383C2×C4⋊C4—101no—
384C2×C4⋊C4C22×D512yes—
385C2×C4⋊C4C22×D514yes—
386C2×C4⋊C4C22×D511yescharacteristic
387C2×C4⋊C4C22×D511yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—108no—
390C2×C22⋊C4—102no—
391C2×C22⋊C4—102no—
392C2×C22⋊C4—101no—
393C2×C42—21no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—54no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—54no—
400C2×C42—101no—
401#107—58no—
402#107C22×D518yescomplemented
403#106—58no—
404#106C22×D518yescomplemented
405C4⋊C8—58no—
406C4⋊C8—58no—
407#102—58no—
408#102—58no—
409#102—58no—
410#102—58no—
411C22×C5—28no—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5C2×C22⋊Q814yes—
415C22×C5C2×C22⋊Q812yes—
416C22×C5C2×AGL(1,ℤ/8)12yes—
417C22×C5#57512yes—
418C22×C5C23×D411yescharacteristic
419C2×D5—28no—
420C2×D5—432no—
421C2×D5—416no—
422C2×D5—44no—
423C2×D5—44no—
424C4×C5—42no—
425C4×C5—44no—
426C4×C5—42no—
427C4×C5C2×C22⋊Q812yes—
428C4×C5C2×C22⋊Q814yes—
429C4×C5C23×Q811yescharacteristic
430C4×C5C22×#4111yescharacteristic
431Dic5—22no—
432Dic5—22no—
433Dic5—22no—
434Dic5—42no—
435Dic5—42no—
436Dic5C2×AGL(1,ℤ/8)14yes—
437C24—104no—
438C24—102no—
439C24—108no—
440C24—204no—
441C24—204no—
442C24D4×D511yescharacteristic
443C2×Q8—104no—
444C2×Q8—104no—
445C2×Q8—204no—
446C2×Q8—204no—
447C2×D4—58no—
448C2×D4—516no—
449C2×D4—54no—
450C2×D4—104no—
451C2×D4—108no—
452C2×D4—104no—
453C2×D4—102no—
454C2×D4—1016no—
455C2×D4—108no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—51no—
462C22×C4—54no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—54no—
467C22×C4—108no—
468C22×C4—108no—
469C22×C4—104no—
470C22×C4—101no—
471C22×C4—101no—
472C22×C4—208no—
473C22×C4—2032no—
474C22×C4—208no—
475C22×C4—202no—
476C22×C4#70914yes—
477C22×C4D4×D512yes—
478C22×C4D4×D511yescharacteristic
479C22×C4Q8×D514yes—
480C22×C4C23×D512yes—
481C22×C4C23×D511yescharacteristic
482M16—28no—
483M16—108no—
484C2×C8—24no—
485C2×C8—54no—
486C2×C8—104no—
487C2×C8C23×D514yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—52no—
493C4⋊C4—52no—
494C4⋊C4—104no—
495C4⋊C4—104no—
496C4⋊C4—104no—
497C4⋊C4—104no—
498C4⋊C4—104no—
499C4⋊C4—104no—
500C4⋊C4—101no—
501C4⋊C4—101no—
502C4⋊C4C23×D512yes—
503C4⋊C4C23×D512yes—
504C4⋊C4C23×D514yes—
505C22⋊C4—24no—
506C22⋊C4—108no—
507C22⋊C4—108no—
508C22⋊C4—104no—
509C42—24no—
510C42—54no—
511C42—54no—
512C42—52no—
513C42—52no—
514C42—52no—
515C42—52no—
516C42—104no—
517C2×C5—24no—
518C2×C5#307714yescomplemented
519C2×C5C22×C22⋊Q811yescharacteristic
520C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
521C2×C5C2×#57511yescharacteristic
522D5—48no—
523C23—22no—
524C23—22no—
525C23—1016no—
526C23—108no—
527C23—104no—
528C23—208no—
529C23—2032no—
530C23—208no—
531C23—204no—
532C23—204no—
533C23#404214yes—
534C23C2×D4×D512yes—
535C23C2×D4×D511yescharacteristic
536Q8—208no—
537Q8—208no—
538D4—58no—
539D4—108no—
540D4—1016no—
541D4—104no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—54no—
550C2×C4—52no—
551C2×C4—52no—
552C2×C4—52no—
553C2×C4—52no—
554C2×C4—108no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—2016no—
561C2×C4—2016no—
562C2×C4—204no—
563C2×C4—202no—
564C2×C4—202no—
565C2×C4#404218yes—
566C2×C4#404214yes—
567C2×C4C2×#70912yes—
568C2×C4C2×#70914yes—
569C2×C4C2×D4×D512yes—
570C2×C4C2×D4×D511yescharacteristic
571C2×C4C2×Q8×D514yes—
572C2×C4C2×Q8×D512yes—
573C2×C4C24×D511yescharacteristic
574C8—24no—
575C8—104no—
576C5C2×#307711yescharacteristic, complemented, Sylow
577C22—28no—
578C22—22no—
579C22—22no—
580C22—108no—
581C22—2032no—
582C22—2016no—
583C22—204no—
584C22—204no—
585C22C2×#404214yes—
586C22C2×#404212yes—
587C22D5×AGL(1,ℤ/8)12yes—
588C22#6611112yes—
589C22C22×D4×D511yescharacteristic
590C4—44no—
591C4—42no—
592C4—42no—
593C4—54no—
594C4—102no—
595C4—102no—
596C4—102no—
597C4—202no—
598C4—202no—
599C4C2×#404212yes—
600C4C2×#404214yes—
601C4C22×#70911yescharacteristic
602C4C22×Q8×D511yescharacteristic
603C2—24no—
604C2—208no—
605C2#1061097014yesdirect product factor
606C2C22×#404211yescharacteristic
607C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
608C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC23×D20C22
hypercenterC2×#428D5
hypocenterC5C2×#3077
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#65851C2semidirect
3. C2×C5×#428C2semidirect
5. C2×#65407C2semidirect
7. C2×#65253C2semidirect
8. #10610970C2direct (N × H)
9. C2×#65180C2semidirect
10. C2×#65179C2semidirect
11. C2×#65160C2semidirect
12. C2×#65159C2semidirect
23. #65851C22semidirect
30. C2×#4036C22semidirect
31. C2×#4036C22semidirect
32. C2×#4034C22semidirect
33. C2×#4034C22semidirect
38. C5×#428C22semidirect
39. C2×C5×#107C22semidirect
40. C2×C5×#106C22semidirect
42. #65407C22semidirect
45. C2×#3946C22semidirect
46. C2×#3946C22semidirect
50. #65253C22semidirect
51. C2×#3940C22semidirect
52. C2×#3939C22semidirect
53. #65180C22semidirect
54. #65179C22semidirect
55. #65160C22semidirect
56. #65159C22semidirect
66. C22×D20Q8semidirect
120. #4036C23semidirect
121. #4036C23semidirect
124. #4034C23semidirect
125. #4034C23semidirect
130. C5×#107C23semidirect
131. C5×#106C23semidirect
134. #3946C23semidirect
135. #3946C23semidirect
138. #3940C23semidirect
139. #3939C23semidirect
148. C2×#428D5semidirect
186. C2×D20C2×Q8semidirect
262. #428C2×D5semidirect
264. C2×#107C2×D5semidirect
266. C2×#106C2×D5semidirect
402. #107C22×D5semidirect
404. #106C22×D5semidirect
518. C2×C5#3077semidirect
576. C5C2×#3077semidirect
605. C2#10610970direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658512
C22×#40392
C2×C5×#4282
C2×#654202
C2×#654072
C2×#652902
C2×#652532
#106109702
C2×#651802
C2×#651792
C2×#651602
C2×#651592
C2×#30775