Finite Group Databasebeta

← C2×#10612728 #12967376

Subgroups — C2×#10612728 #12967376

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×#713C211yesmaximal, characteristic
3C2×C5×#475C211yesmaximal, characteristic, complemented
4C2×#65376C211yesmaximal, characteristic, complemented
5C2×#65320C211yesmaximal, characteristic, complemented
6#10612728C2116yesmaximal, direct product factor
7C2×#65156C212yesmaximal, complemented
8C2×#65141C212yesmaximal, complemented
9C2×#65130C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic, complemented
11C2×#65099C212yesmaximal
12C2×#65079C212yesmaximal, complemented
13C22×#713C2211yescharacteristic
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#4092—21no—
17C2×Q8×Dic5—21no—
18C2×Q8×Dic5C2211yescharacteristic
19C22×#689C2212yes—
20C22×#677C2212yes—
21#66062—24no—
22#66062C2218yescomplemented
23C4×#713—24no—
24C4×#713C2218yes—
25C2×#4043C2212yes—
26C2×#4039—21no—
27C2×#4039C2211yescharacteristic
28C2×#4038—21no—
29C2×#4038—21no—
30C2×#4038C2211yescharacteristic
31C2×#4036—21no—
32C2×#4033C2212yes—
33C2×C4×D20—21no—
34C2×C4×D20C2211yescharacteristic
35C2×C42×D5—21no—
36C2×C42×D5C2211yescharacteristic
37C2×#4015C2211yescharacteristic
38C5×#475C2218yescomplemented
39C2×C5×#106C2212yes—
40C2×C5×C4⋊C8C2211yescharacteristic
41C2×C5×#103C2212yes—
42C2×#3952C2212yes—
43C2×#3948C2212yes—
44C2×C8×Dic5C2212yes—
45#65376C2218yescomplemented
46#65320C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3941C2212yes—
49C2×#3939C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65156C22116yescomplemented
52#65141C22116yescomplemented
53#65130C2218yescomplemented
54#65122C2218yescomplemented
55#65099C22116yes—
56#65079C22116yescomplemented
57C2×#3107—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8C2311yescharacteristic
60C2×#713—24no—
61C2×#713—24no—
62C2×#713—22no—
63C2×#713C2314yes—
64C22×D20—21no—
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×C4×D5—21no—
68C22×C4×D5—21no—
69C22×C4×D5D411yescharacteristic
70C22×C4×D5D411yescharacteristic
71C22×C4×D5C2311yescharacteristic
72C22×Dic10C2312yes—
73C22×C5×C8C2312yes—
74C5×C4⋊Q8—22no—
75C5×C4⋊Q8C2314yes—
76C4×C5×Q8—22no—
77C4×C5×Q8C2314yes—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C42×C5—21no—
83C2×C42×C5C2311yescharacteristic
84#4092—28no—
85Q8×Dic5—28no—
86Q8×Dic5—22no—
87Q8×Dic5C2314yes—
88C2×#689—28no—
89C2×#689C2318yes—
90C2×#685—21no—
91C2×#685—22no—
92C2×#685—21no—
93C2×#685—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5—22no—
98C2×C4⋊Dic5C2311yescharacteristic
99C2×#683—22no—
100C2×C4×Dic5—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5—22no—
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5C2311yescharacteristic
106C22×#168C2312yes—
107C2×#677—28no—
108C2×#677C2318yes—
109#4043—24no—
110#4043C2318yes—
111#4039—28no—
112#4039—22no—
113#4039C2314yes—
114#4038—28no—
115#4038—22no—
116#4038—28no—
117#4038C2314yes—
118#4036—28no—
119#4033—24no—
120#4033C2318yes—
121C4×D20—28no—
122C4×D20—22no—
123C4×D20C2314yes—
124C42×D5—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127#4015—22no—
128#4015C2314yes—
129C5×#106C2318yes—
130C5×C4⋊C8C2314yes—
131C5×#103C2318yes—
132#3952C2318yes—
133#3948C2318yes—
134C8×Dic5C2318yes—
135#3942C2318yes—
136#3941C2318yes—
137#3939C2318yes—
138C4⋊#168C2314yes—
139C2×#533—51no—
140C2×C4×#41—51no—
141#3107—516no—
142C2×#485—52no—
143C2×#475D511yescharacteristic, complemented
144C2×#474—51no—
145C2×#462—51no—
146C2×#461—51no—
147C2×#452—52no—
148C2×C4×Q16—52no—
149C2×C4×SD16—52no—
150C2×#431—51no—
151C23×D5—21no—
152C23×D5C2×D411yescharacteristic
153C2×C5×Q8—21no—
154C2×C5×Q8—24no—
155C2×C5×Q8—22no—
156C2×C5×Q8—44no—
157C2×C5×Q8#4114yes—
158C2×C5×Q8C2412yes—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—22no—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170#713—216no—
171#713—28no—
172#713—28no—
173C2×D20—28no—
174C2×D20—24no—
175C2×D20—21no—
176C2×D20—22no—
177C2×D20#4114yes—
178C2×D20C2412yes—
179C2×C4×D5—28no—
180C2×C4×D5—24no—
181C2×C4×D5—22no—
182C2×C4×D5—22no—
183C2×C4×D5—24no—
184C2×C4×D5—24no—
185C2×C4×D5—24no—
186C2×C4×D5—24no—
187C2×C4×D5—44no—
188C2×C4×D5—44no—
189C2×C4×D5C2×D414yes—
190C2×C4×D5C2×D414yes—
191C2×C4×D5C2×D412yes—
192C2×C4×D5C2412yes—
193C2×Dic10—24no—
194C2×Dic10—24no—
195C2×Dic10—22no—
196C2×Dic10C2414yes—
197C2×C5×C8—24no—
198C2×C5×C8C2414yes—
199C5×C4⋊C4—24no—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4C2412yes—
204C5×C4⋊C4C2414yes—
205C42×C5—24no—
206C42×C5C2×D412yes—
207C42×C5C2412yes—
208#689—216no—
209#685—24no—
210#685—28no—
211#685—24no—
212#685—24no—
213C4⋊Dic5—24no—
214C4⋊Dic5—24no—
215C4⋊Dic5—21no—
216C4⋊Dic5—24no—
217C4⋊Dic5—28no—
218C4⋊Dic5C2412yes—
219#683—28no—
220C4×Dic5—24no—
221C4×Dic5—24no—
222C4×Dic5—28no—
223C4×Dic5C2×D414yes—
224C4×Dic5C2×D412yes—
225C4×Dic5C2×D414yes—
226C4×Dic5C2412yes—
227C2×#168—24no—
228C2×#168C2414yes—
229#677—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—101no—
245C2×C4×Q8C2×D511yescharacteristic
246C2×C4×D4—51no—
247C2×C4×D4—51no—
248C2×C4×D4—101no—
249C2×C4×D4—101no—
250C2×#117—51no—
251C2×#117—101no—
252C2×#117—101no—
253C22×C42—51no—
254C22×C42—101no—
255#485—516no—
256#475C2×D518yescomplemented
257#474—58no—
258#462—58no—
259#461—58no—
260#452—516no—
261C4×Q16—516no—
262C4×SD16—516no—
263#431—58no—
264C2×#107—52no—
265C2×#106C2×D512yes—
266C2×C4⋊C8—51no—
267C2×C4⋊C8C2×D511yescharacteristic
268C2×#103—52no—
269C2×#103—52no—
270C2×#103C2×D512yes—
271C2×#102—52no—
272C2×C22⋊C8—52no—
273C2×C4×C8—52no—
274C23×C5C22×D411yescharacteristic
275C22×D5—22no—
276C22×D5—24no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—42no—
281C22×D5—42no—
282C22×D5C22×D412yes—
283C5×Q8—24no—
284C5×Q8—22no—
285C5×Q8—48no—
286C5×Q8C2×#4114yes—
287C2×C4×C5—24no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5—22no—
292C2×C4×C5—44no—
293C2×C4×C5—42no—
294C2×C4×C5#13514yes—
295C2×C4×C5C22×D412yes—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D412yes—
298C2×C4×C5C2×#4112yes—
299C2×C4×C5C2×#4112yes—
300C2×C4×C5C2511yescharacteristic
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—24no—
305C2×Dic5—44no—
306C2×Dic5—42no—
307C2×Dic5#13714yes—
308C2×Dic5C22×D412yes—
309C2×Dic5C22×D412yes—
310C2×Dic5C22×D412yes—
311D20—216no—
312D20—24no—
313D20—22no—
314D20C2×#4114yes—
315C4×D5—216no—
316C4×D5—24no—
317C4×D5—28no—
318C4×D5—22no—
319C4×D5—22no—
320C4×D5—28no—
321C4×D5—48no—
322C4×D5—48no—
323Dic10—24no—
324Dic10—24no—
325Dic10—44no—
326C5×C8—24no—
327#168—24no—
328C2×#41—54no—
329C2×#41—102no—
330C2×#41—104no—
331C2×#41—104no—
332C22×Q8—21no—
333C22×Q8—52no—
334C22×Q8C22×D511yescharacteristic
335C22×D4—51no—
336C22×D4—101no—
337C23×C4—51no—
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—101no—
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—22no—
358C4×Q8—54no—
359C4×Q8—58no—
360C4×Q8—104no—
361C4×Q8—102no—
362C4×Q8—108no—
363C4×Q8C22×D514yes—
364C4×D4—54no—
365C4×D4—54no—
366C4×D4—108no—
367C4×D4—102no—
368C4×D4—108no—
369C4×D4—102no—
370#117—54no—
371#117—108no—
372#117—102no—
373#117—108no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—51no—
377C2×C4⋊C4—101no—
378C2×C4⋊C4—102no—
379C2×C4⋊C4—102no—
380C2×C4⋊C4—101no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4C22×D511yescharacteristic
383C2×C4⋊C4C22×D512yes—
384C2×C22⋊C4—101no—
385C2×C22⋊C4—101no—
386C2×C22⋊C4—102no—
387C2×C22⋊C4—101no—
388C2×C42—21no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—54no—
393C2×C42—54no—
394C2×C42—108no—
395C2×C42—101no—
396C2×C42—102no—
397C2×C42—101no—
398C2×C42C22×D511yescharacteristic
399#107—58no—
400#106C22×D518yes—
401C4⋊C8—54no—
402C4⋊C8C22×D514yes—
403#103—58no—
404#103—58no—
405#103C22×D518yes—
406#102—58no—
407C22⋊C8—58no—
408C4×C8—58no—
409C22×C5#52512yes—
410C22×C5C2×#13512yes—
411C22×C5C2×#13712yes—
412C22×C5C23×D411yescharacteristic
413C2×D5—22no—
414C2×D5—24no—
415C2×D5—22no—
416C2×D5—22no—
417C2×D5—44no—
418C2×D5—48no—
419C2×D5—42no—
420C2×D5—42no—
421C4×C5—24no—
422C4×C5—22no—
423C4×C5—44no—
424C4×C5—42no—
425C4×C5#52512yes—
426C4×C5C2×#13514yes—
427C4×C5C22×#4111yescharacteristic
428C4×C5C22×#4111yescharacteristic
429Dic5—22no—
430Dic5—22no—
431Dic5—44no—
432Dic5—42no—
433Dic5C2×#13714yes—
434C24—51no—
435C24—101no—
436#41—108no—
437#41—1016no—
438#41—108no—
439C2×Q8—21no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—54no—
444C2×Q8—102no—
445C2×Q8—104no—
446C2×Q8—104no—
447C2×Q8#71314yes—
448C2×Q8C23×D512yes—
449C2×D4—54no—
450C2×D4—52no—
451C2×D4—104no—
452C2×D4—108no—
453C2×D4—101no—
454C2×D4—102no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—51no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—54no—
462C22×C4—52no—
463C22×C4—54no—
464C22×C4—52no—
465C22×C4—104no—
466C22×C4—104no—
467C22×C4—104no—
468C22×C4—108no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—104no—
472C22×C4—101no—
473C22×C4—102no—
474C22×C4—104no—
475C22×C4—204no—
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—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—52no—
491C4⋊C4—104no—
492C4⋊C4—108no—
493C4⋊C4—108no—
494C4⋊C4—101no—
495C4⋊C4—104no—
496C4⋊C4—104no—
497C4⋊C4C23×D514yes—
498C4⋊C4C23×D512yes—
499C22⋊C4—104no—
500C22⋊C4—104no—
501C22⋊C4—108no—
502C22⋊C4—104no—
503C42—24no—
504C42—54no—
505C42—54no—
506C42—52no—
507C42—52no—
508C42—104no—
509C42—104no—
510C42—108no—
511C42D4×D512yes—
512C42C23×D512yes—
513C2×C5#310714yescomplemented
514C2×C5C2×#52511yescharacteristic
515C2×C5C22×#13511yescharacteristic
516C2×C5C22×#13711yescharacteristic
517D5—42no—
518D5—44no—
519C23—52no—
520C23—102no—
521C23—102no—
522C23—104no—
523C23—102no—
524C23—102no—
525C23—202no—
526C23—202no—
527C23C2×D4×D511yescharacteristic
528Q8—24no—
529Q8—22no—
530Q8—48no—
531Q8—104no—
532Q8—104no—
533Q8—204no—
534Q8C2×#71314yes—
535D4—54no—
536D4—1016no—
537D4—104no—
538D4—102no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—54no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—108no—
551C2×C4—1016no—
552C2×C4—102no—
553C2×C4—102no—
554C2×C4—108no—
555C2×C4—104no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—104no—
560C2×C4—208no—
561C2×C4—202no—
562C2×C4—204no—
563C2×C4—208no—
564C2×C4#406214yes—
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—104no—
575C22—102no—
576C22—102no—
577C22—102no—
578C22—202no—
579C22—202no—
580C22—204no—
581C22—208no—
582C22#6596112yes—
583C22C2×#406212yes—
584C22D5×#13712yes—
585C22C22×D4×D511yescharacteristic
586C4—24no—
587C4—22no—
588C4—44no—
589C4—42no—
590C4—54no—
591C4—102no—
592C4—102no—
593C4—204no—
594C4—202no—
595C4#6596112yes—
596C4C2×#406214yes—
597C4C22×#71311yescharacteristic
598C4C2×D5×#4111yescharacteristic
599C2—202no—
600C2—204no—
601C2#1061272814yesdirect product factor
602C2C2×#6596111yescharacteristic
603C2C22×#406211yescharacteristic
604C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#475C2
socleC23×C5C22×D4
prime radicalC22×#713C22
hypercenterC2×#475D5
hypocenterC5C2×#3107
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×#66062C2semidirect
3. C2×C5×#475C2semidirect
4. C2×#65376C2semidirect
5. C2×#65320C2semidirect
6. #10612728C2direct (N × H)
7. C2×#65156C2semidirect
8. C2×#65141C2semidirect
9. C2×#65130C2semidirect
10. C2×#65122C2semidirect
12. C2×#65079C2semidirect
22. #66062C22semidirect
38. C5×#475C22semidirect
45. #65376C22semidirect
46. #65320C22semidirect
51. #65156C22semidirect
52. #65141C22semidirect
53. #65130C22semidirect
54. #65122C22semidirect
56. #65079C22semidirect
143. C2×#475D5semidirect
256. #475C2×D5semidirect
513. C2×C5#3107semidirect
573. C5C2×#3107semidirect
601. C2#10612728direct (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×#7132
C2×C5×#4752
C2×#653762
C2×#653202
#106127282
C2×#651562
C2×#651412
C2×#651302
C2×#651222
C2×#650992
C2×#650792
C2×#31075