Finite Group Databasebeta

← C2×#10612790 #12966802

Subgroups — C2×#10612790 #12966802

Order 1280.

2162 conjugacy classes of subgroups.

The subgroup lattice has 596 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×#476C211yesmaximal, characteristic, complemented
4C2×#65380C211yesmaximal, characteristic, complemented
5C2×#65296C211yesmaximal, characteristic, complemented
6#10612790C2116yesmaximal, direct product factor
7C2×#65146C212yesmaximal, complemented
8C2×#65149C212yesmaximal, complemented
9C2×#65140C211yesmaximal, characteristic, complemented
10C2×D5×C4⋊C8C211yesmaximal, characteristic, complemented
11C2×#65061C212yesmaximal
12C2×#65052C212yesmaximal, complemented
13C22×#711C2211yescharacteristic
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×D4C2211yescharacteristic
16C2×#4092—21no—
17C2×Q8×Dic5—21no—
18C2×D4×Dic5C2211yescharacteristic
19C2×C4×C22⋊D5—21no—
20C22×C4×Dic5—21no—
21C22×C8×D5C2212yes—
22#66062—24no—
23#66062C2218yescomplemented
24C4×#711—24no—
25C4×#711C2218yes—
26C2×#4043—21no—
27C2×#4038C2212yes—
28C2×#4038C2211yescharacteristic
29C2×#4036C2212yes—
30C2×#4033—21no—
31C2×#4033C2211yescharacteristic
32C2×#4027—21no—
33C2×#4023—21no—
34C2×C42×D5C2211yescharacteristic
35C2×#4015C2211yescharacteristic
36C2×C4×Dic10C2211yescharacteristic
37C5×#476C2218yescomplemented
38C2×C5×#106C2212yes—
39C2×C5×C4⋊C8C2211yescharacteristic
40C2×C5×#102C2212yes—
41C2×#3964C2212yes—
42C2×#3949C2212yes—
43C2×#3948C2212yes—
44C2×#3946C2212yes—
45#65380C2218yescomplemented
46#65296C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3939C2212yes—
49C2×#3939C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65149C22116yescomplemented
52#65146C22116yescomplemented
53#65140C2218yescomplemented
54D5×C4⋊C8C2218yescomplemented
55#65061C22116yes—
56#65052C22116yescomplemented
57C2×#3076—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—22no—
65C2×#711—24no—
66C2×#711Q818yescomplemented
67C2×#711C2314yes—
68C22×C4×D5D411yescharacteristic
69C22×C4×D5D411yescharacteristic
70C22×C4×D5C2311yescharacteristic
71C22×Dic10—21no—
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⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C22⋊C4—21no—
82C2×C42×C5C2311yescharacteristic
83C2×C22⋊Dic5—21no—
84#4092—28no—
85Q8×Dic5—28no—
86D4×Dic5—22no—
87D4×Dic5C2314yes—
88C4×C22⋊D5—28no—
89C2×#685—21no—
90C2×#685—22no—
91C2×C4⋊Dic5—21no—
92C2×C4⋊Dic5—22no—
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4⋊Dic5C2312yes—
95C2×#683—21no—
96C2×#683—21no—
97C2×#683—22no—
98C2×C4×Dic5—21no—
99C2×C4×Dic5—21no—
100C2×C4×Dic5—21no—
101C2×C4×Dic5—22no—
102C2×C4×Dic5—28no—
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5C2311yescharacteristic
106C22×#168C2312yes—
107C2×C8×D5—28no—
108C2×C8×D5C2318yes—
109#4043—28no—
110#4038—24no—
111#4038—22no—
112#4038C2318yes—
113#4038C2314yes—
114#4036—24no—
115#4036C2318yes—
116#4033—28no—
117#4033—22no—
118#4033C2314yes—
119#4027—28no—
120#4023—28no—
121C42×D5D414yes—
122C42×D5C2314yes—
123#4015—22no—
124#4015C2314yes—
125C4×Dic10—22no—
126C4×Dic10C2314yes—
127C5×#106C2318yes—
128C5×C4⋊C8C2314yes—
129C5×#102C2318yes—
130#3964C2318yes—
131#3949C2318yes—
132#3948C2318yes—
133#3946C2318yes—
134#3942C2318yes—
135#3939C2318yes—
136#3939C2318yes—
137C4⋊#168C2314yes—
138C2×#533—51no—
139C2×C4×#41—51no—
140#3076—516no—
141C2×#479—52no—
142C2×#477—51no—
143C2×#475—51no—
144C2×#478—52no—
145C2×#476D511yescharacteristic, complemented
146C2×#474—51no—
147C2×#427—52no—
148C22×C4⋊C8—51no—
149C2×#416—52no—
150C24×C5—21no—
151C23×D5C2×D411yescharacteristic
152C2×C5×Q8—24no—
153C2×C5×Q8—44no—
154C2×C5×D4—21no—
155C2×C5×D4—22no—
156C2×C5×D4C2×Q814yes—
157C2×C5×D4C2412yes—
158C22×C4×C5—21no—
159C22×C4×C5—22no—
160C22×C4×C5—24no—
161C22×C4×C5—44no—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C2×C22⋊D5—28no—
166C2×C22⋊D5—24no—
167C22×Dic5—21no—
168C22×Dic5—28no—
169C22×Dic5—24no—
170C22×Dic5—21no—
171C22×Dic5—22no—
172C22×Dic5—24no—
173C22×Dic5—44no—
174C22×Dic5C2×D411yescharacteristic
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177#711—216no—
178#711—28no—
179#711C2×Q8116yescomplemented
180C2×C4×D5—24no—
181C2×C4×D5—42no—
182C2×C4×D5C2×D414yes—
183C2×C4×D5C2×D414yes—
184C2×C4×D5C2×D412yes—
185C2×C4×D5C2×Q814yes—
186C2×C4×D5#4114yes—
187C2×C4×D5C2412yes—
188C2×Dic10—21no—
189C2×Dic10—22no—
190C2×Dic10—24no—
191C2×Dic10—44no—
192C2×Dic10C2×Q814yes—
193C2×Dic10C2412yes—
194C2×C5×C8—24no—
195C2×C5×C8C2414yes—
196C5×C4⋊C4—24no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4C2412yes—
200C5×C4⋊C4C2414yes—
201C5×C22⋊C4—24no—
202C42×C5C2×D412yes—
203C42×C5C2412yes—
204C22⋊Dic5—24no—
205#685—24no—
206#685—28no—
207C4⋊Dic5—21no—
208C4⋊Dic5—22no—
209C4⋊Dic5—24no—
210C4⋊Dic5—28no—
211C4⋊Dic5C2412yes—
212C4⋊Dic5C2414yes—
213#683—24no—
214#683—24no—
215#683—28no—
216C4×Dic5—24no—
217C4×Dic5—24no—
218C4×Dic5—24no—
219C4×Dic5—28no—
220C4×Dic5C2×D412yes—
221C4×Dic5C2×D414yes—
222C4×Dic5C2×D414yes—
223C4×Dic5C2412yes—
224C2×#168—24no—
225C2×#168C2414yes—
226C8×D5—216no—
227C22×#41—51no—
228C23×C8—52no—
229#533—58no—
230#533—104no—
231C2×C4⋊Q8—51no—
232C2×C4⋊Q8C2×D511yescharacteristic
233C2×#125—52no—
234C2×C22⋊Q8—101no—
235C2×C22⋊Q8—101no—
236C4×#41—58no—
237C4×#41—104no—
238C2×C4×Q8—51no—
239C2×C4×Q8—51no—
240C2×C4×Q8—101no—
241C2×C4×Q8—101no—
242C2×C4×D4—51no—
243C2×C4×D4—101no—
244C2×C4×D4—101no—
245C2×C4×D4C2×D511yescharacteristic
246C2×#117—52no—
247C2×#117—51no—
248C2×#117—101no—
249C22×C42—51no—
250C22×C42—101no—
251#479—516no—
252#478—516no—
253#477—58no—
254#476C2×D518yescomplemented
255#475—58no—
256#474—58no—
257#427—516no—
258C2×#107—52no—
259C2×#107—52no—
260C2×#106—52no—
261C2×#106C2×D512yes—
262C2×C4⋊C8—52no—
263C2×C4⋊C8—51no—
264C2×C4⋊C8—58no—
265C2×C4⋊C8C2×D511yescharacteristic
266#416—516no—
267C2×#103—52no—
268C2×#103—52no—
269C2×#102—52no—
270C2×#102C2×D512yes—
271C23×C5—24no—
272C23×C5—22no—
273C23×C5—42no—
274C23×C5—42no—
275C23×C5C22×D411yescharacteristic
276C22×D5—22no—
277C22×D5—22no—
278C22×D5C22⋊Q814yes—
279C22×D5C22×D412yes—
280C5×Q8—48no—
281C5×D4—24no—
282C5×D4—22no—
283C5×D4C22×Q814yes—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—24no—
287C2×C4×C5—42no—
288C2×C4×C5—44no—
289C2×C4×C5—48no—
290C2×C4×C5#13514yes—
291C2×C4×C5C22×D412yes—
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D412yes—
294C2×C4×C5C22×Q812yes—
295C2×C4×C5C2×#4112yes—
296C2×C4×C5C2511yescharacteristic
297C22⋊D5—216no—
298C2×Dic5—24no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—216no—
303C2×Dic5—22no—
304C2×Dic5—24no—
305C2×Dic5—48no—
306C2×Dic5—44no—
307C2×Dic5—42no—
308C2×Dic5C22⋊Q814yes—
309C2×Dic5C22×D412yes—
310C2×Dic5C22×D412yes—
311C2×Dic5C22×D412yes—
312C4×D5—28no—
313C4×D5—44no—
314C4×D5C22⋊Q818yes—
315C4×D5C22×Q814yes—
316C4×D5C2×#4114yes—
317Dic10—24no—
318Dic10—22no—
319Dic10—48no—
320Dic10C22×Q814yes—
321C5×C8—24no—
322#168—24no—
323C2×#41—58no—
324C2×#41—54no—
325C2×#41—102no—
326C2×#41—104no—
327C22×Q8—21no—
328C22×Q8—51no—
329C22×Q8—101no—
330C22×D4—101no—
331C22×D4C22×D511yescharacteristic
332C23×C4—21no—
333C23×C4—51no—
334C23×C4—51no—
335C23×C4—51no—
336C23×C4—101no—
337C23×C4—101no—
338C22×C8—52no—
339C22×C8—58no—
340C22×C8—108no—
341C22×C8C22×D512yes—
342C4⋊Q8—22no—
343C4⋊Q8—54no—
344C4⋊Q8—102no—
345C4⋊Q8C22×D514yes—
346#125—58no—
347#125—104no—
348C22⋊Q8—108no—
349C22⋊Q8—108no—
350C4×Q8—54no—
351C4×Q8—54no—
352C4×Q8—102no—
353C4×Q8—108no—
354C4×Q8—108no—
355C4×Q8—102no—
356C4×D4—22no—
357C4×D4—54no—
358C4×D4—102no—
359C4×D4—108no—
360C4×D4—108no—
361C4×D4C22×D514yes—
362#117—54no—
363#117—58no—
364#117—108no—
365#117—102no—
366#117—104no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4—102no—
372C2×C4⋊C4—102no—
373C2×C4⋊C4—101no—
374C2×C4⋊C4—101no—
375C2×C4⋊C4C22×D511yescharacteristic
376C2×C4⋊C4C22×D512yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—101no—
379C2×C22⋊C4—102no—
380C2×C22⋊C4—101no—
381C2×C42—51no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—54no—
385C2×C42—54no—
386C2×C42—108no—
387C2×C42—101no—
388C2×C42—102no—
389C2×C42—101no—
390C2×C42—101no—
391C2×C42C22×D511yescharacteristic
392#107—58no—
393#107—58no—
394#106—58no—
395#106C22×D518yes—
396C4⋊C8—58no—
397C4⋊C8—54no—
398C4⋊C8C22×D514yes—
399#103—58no—
400#103—58no—
401#102—58no—
402#102C22×D518yes—
403C22×C5—24no—
404C22×C5—48no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5C2×C22⋊Q812yes—
408C22×C5C2×#13512yes—
409C22×C5#57512yes—
410C22×C5C23×D411yescharacteristic
411C2×D5—28no—
412C2×D5—22no—
413C2×D5—22no—
414C2×D5C2×C22⋊Q814yes—
415C4×C5—22no—
416C4×C5—42no—
417C4×C5—44no—
418C4×C5C2×C22⋊Q812yes—
419C4×C5C2×#13514yes—
420C4×C5C23×Q811yescharacteristic
421C4×C5C22×#4111yescharacteristic
422Dic5—24no—
423Dic5—22no—
424Dic5—22no—
425Dic5—44no—
426Dic5—42no—
427Dic5C2×C22⋊Q814yes—
428C24—21no—
429C24—51no—
430#41—516no—
431#41—1016no—
432#41—108no—
433C2×Q8—24no—
434C2×Q8—44no—
435C2×Q8—54no—
436C2×Q8—52no—
437C2×Q8—101no—
438C2×Q8—104no—
439C2×Q8—102no—
440C2×Q8—204no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—104no—
444C2×D4—108no—
445C2×D4Q8×D514yes—
446C2×D4C23×D512yes—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—51no—
452C22×C4—52no—
453C22×C4—54no—
454C22×C4—54no—
455C22×C4—52no—
456C22×C4—51no—
457C22×C4—51no—
458C22×C4—54no—
459C22×C4—54no—
460C22×C4—104no—
461C22×C4—108no—
462C22×C4—104no—
463C22×C4—101no—
464C22×C4—104no—
465C22×C4—101no—
466C22×C4—102no—
467C22×C4—202no—
468C22×C4—204no—
469C22×C4D4×D511yescharacteristic
470C22×C4D4×D511yescharacteristic
471C22×C4C23×D511yescharacteristic
472C2×C8—24no—
473C2×C8—54no—
474C2×C8—1016no—
475C2×C8—104no—
476C2×C8C23×D514yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—54no—
481C4⋊C4—52no—
482C4⋊C4—104no—
483C4⋊C4—108no—
484C4⋊C4—108no—
485C4⋊C4—101no—
486C4⋊C4—102no—
487C4⋊C4—104no—
488C4⋊C4—104no—
489C4⋊C4C23×D514yes—
490C4⋊C4C23×D512yes—
491C22⋊C4—24no—
492C22⋊C4—104no—
493C22⋊C4—108no—
494C22⋊C4—104no—
495C42—54no—
496C42—54no—
497C42—52no—
498C42—52no—
499C42—104no—
500C42—104no—
501C42—108no—
502C42—104no—
503C42D4×D512yes—
504C42C23×D512yes—
505C2×C5—44no—
506C2×C5#307614yescomplemented
507C2×C5C22×C22⋊Q811yescharacteristic
508C2×C5C22×#13511yescharacteristic
509C2×C5C2×#57511yescharacteristic
510D5—24no—
511C23—24no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—54no—
516C23—52no—
517C23—102no—
518C23—102no—
519C23C2×D4×D511yescharacteristic
520Q8—48no—
521Q8—54no—
522Q8—104no—
523Q8—102no—
524Q8—208no—
525D4—24no—
526D4—22no—
527D4—1016no—
528D4C2×Q8×D514yes—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—54no—
536C2×C4—52no—
537C2×C4—58no—
538C2×C4—54no—
539C2×C4—54no—
540C2×C4—52no—
541C2×C4—52no—
542C2×C4—1016no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—104no—
546C2×C4—102no—
547C2×C4—108no—
548C2×C4—102no—
549C2×C4—104no—
550C2×C4—204no—
551C2×C4—208no—
552C2×C4—204no—
553C2×C4—202no—
554C2×C4#406214yes—
555C2×C4C2×D4×D512yes—
556C2×C4C2×D4×D512yes—
557C2×C4C2×D4×D511yescharacteristic
558C2×C4C2×Q8×D512yes—
559C2×C4D5×#4112yes—
560C2×C4C24×D511yescharacteristic
561C8—24no—
562C8—104no—
563C5C2×#307611yescharacteristic, complemented, Sylow
564C22—24no—
565C22—48no—
566C22—42no—
567C22—42no—
568C22—54no—
569C22—108no—
570C22—102no—
571C22—102no—
572C22D5×C22⋊Q812yes—
573C22C2×#406212yes—
574C22#6610712yes—
575C22C22×D4×D511yescharacteristic
576C4—22no—
577C4—42no—
578C4—44no—
579C4—54no—
580C4—104no—
581C4—102no—
582C4—102no—
583C4—204no—
584C4—202no—
585C4D5×C22⋊Q812yes—
586C4C2×#406214yes—
587C4C22×Q8×D511yescharacteristic
588C4C2×D5×#4111yescharacteristic
589C2—44no—
590C2—104no—
591C2#1061279014yesdirect product factor
592C2C2×D5×C22⋊Q811yescharacteristic
593C2C22×#406211yescharacteristic
594C2C2×#6610711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#476C2
socleC23×C5C22×D4
prime radicalC22×#711C22
hypercenterC2×#476D5
hypocenterC5C2×#3076
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#66062C2semidirect
3. C2×C5×#476C2semidirect
4. C2×#65380C2semidirect
5. C2×#65296C2semidirect
6. #10612790C2direct (N × H)
7. C2×#65146C2semidirect
8. C2×#65149C2semidirect
9. C2×#65140C2semidirect
10. C2×D5×C4⋊C8C2semidirect
12. C2×#65052C2semidirect
23. #66062C22semidirect
37. C5×#476C22semidirect
45. #65380C22semidirect
46. #65296C22semidirect
51. #65149C22semidirect
52. #65146C22semidirect
53. #65140C22semidirect
54. D5×C4⋊C8C22semidirect
56. #65052C22semidirect
66. C2×#711Q8semidirect
145. C2×#476D5semidirect
179. #711C2×Q8semidirect
254. #476C2×D5semidirect
506. C2×C5#3076semidirect
563. C5C2×#3076semidirect
591. C2#10612790direct (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×#4762
C2×#653802
C2×#652962
#106127902
C2×#651462
C2×#651492
C2×#651402
C2×D5×C4⋊C82
C2×#650612
C2×#650522
C2×#30765