Finite Group Databasebeta

← C2×#10612727 #12967331

Subgroups — C2×#10612727 #12967331

Order 1280.

2418 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66059C211yesmaximal, characteristic
2C2×C4×Q8×D5C211yesmaximal, characteristic, complemented
3C2×C5×#475C211yesmaximal, characteristic, complemented
4C2×#65383C211yesmaximal, characteristic, complemented
5C2×#65316C211yesmaximal, characteristic, complemented
6#10612727C2116yesmaximal, direct product factor
7C2×#65152C212yesmaximal, complemented
8C2×#65141C212yesmaximal, complemented
9C2×#65137C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic, complemented
11C2×#65095C212yesmaximal, complemented
12C2×#65077C212yesmaximal, complemented
13C22×#713—21no—
14C22×Q8×D5C2211yescharacteristic
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C4×C5×Q8C2211yescharacteristic
17C2×#4093—21no—
18C2×Q8×Dic5C2211yescharacteristic
19C22×#688C2212yes—
20C22×#677C2212yes—
21#66059—24no—
22#66059C2218yes—
23C4×Q8×D5—24no—
24C4×Q8×D5C2218yescomplemented
25C2×#4041—21no—
26C2×#4041C2212yes—
27C2×#4039—21no—
28C2×#4039C2212yes—
29C2×D5×C4⋊C4—21no—
30C2×D5×C4⋊C4C2211yescharacteristic
31C2×#4033—21no—
32C2×#4033C2211yescharacteristic
33C2×#4020C2211yescharacteristic
34C2×C42×D5—21no—
35C2×C42×D5C2211yescharacteristic
36C2×C4×Dic10—21no—
37C2×C4×Dic10C2211yescharacteristic
38C5×#475C2218yescomplemented
39C2×C5×#106C2212yes—
40C2×C5×C4⋊C8C2211yescharacteristic
41C2×C5×#103C2212yes—
42C2×#3953C2212yes—
43C2×#3952C2212yes—
44C2×C8×Dic5C2212yes—
45#65383C2218yescomplemented
46#65316C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3941C2212yes—
49C2×#3940C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65152C22116yescomplemented
52#65141C22116yescomplemented
53#65137C2218yescomplemented
54#65122C2218yescomplemented
55#65095C22116yescomplemented
56#65077C22116yescomplemented
57C2×#3103—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8C2311yescharacteristic
60C2×#713—28no—
61C2×#713—48no—
62C2×Q8×D5—24no—
63C2×Q8×D5—22no—
64C2×Q8×D5—24no—
65C2×Q8×D5C2314yes—
66C22×D20—22no—
67C22×D20—21no—
68C22×D20—21no—
69C22×D20C2312yes—
70C22×C4×D5—21no—
71C22×C4×D5—21no—
72C22×C4×D5—22no—
73C22×C4×D5D411yescharacteristic
74C22×C4×D5D411yescharacteristic
75C22×C4×D5C2311yescharacteristic
76C22×Dic10—21no—
77C22×Dic10C2311yescharacteristic
78C22×C5×C8C2312yes—
79C5×C4⋊Q8—22no—
80C5×C4⋊Q8C2314yes—
81C4×C5×Q8—22no—
82C4×C5×Q8C2314yes—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2312yes—
87C2×C42×C5—21no—
88C2×C42×C5C2311yescharacteristic
89#4093—28no—
90Q8×Dic5—22no—
91Q8×Dic5C2314yes—
92C2×#688—28no—
93C2×#688C2318yes—
94C2×#685—22no—
95C2×#685—22no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5C2311yescharacteristic
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5—21no—
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5D411yescharacteristic
106C2×C4×Dic5C2311yescharacteristic
107C22×#168C2312yes—
108C2×#677—28no—
109C2×#677C2318yes—
110#4041—24no—
111#4041—28no—
112#4041C2318yes—
113#4039—24no—
114#4039—28no—
115#4039C2318yes—
116D5×C4⋊C4—28no—
117D5×C4⋊C4—22no—
118D5×C4⋊C4C2314yes—
119#4033—28no—
120#4033—22no—
121#4033C2314yes—
122#4020—22no—
123#4020C2314yes—
124C42×D5—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127C4×Dic10—22no—
128C4×Dic10—28no—
129C4×Dic10C2314yes—
130C5×#106C2318yes—
131C5×C4⋊C8C2314yes—
132C5×#103C2318yes—
133#3953C2318yes—
134#3952C2318yes—
135C8×Dic5C2318yes—
136#3942C2318yes—
137#3941C2318yes—
138#3940C2318yes—
139C4⋊#168C2314yes—
140C2×#532—51no—
141C22×C4×Q8—51no—
142#3103—516no—
143C2×#481—52no—
144C2×#475—51no—
145C2×#475D511yescharacteristic, complemented
146C2×#460—51no—
147C2×#460—51no—
148C2×#448—52no—
149C2×C4×SD16—52no—
150C2×C4×SD16—52no—
151C2×#431—51no—
152C23×D5—22no—
153C23×D5C2×D411yescharacteristic
154C2×C5×Q8—21no—
155C2×C5×Q8—24no—
156C2×C5×Q8—22no—
157C2×C5×Q8—44no—
158C2×C5×Q8#4114yescomplemented
159C2×C5×Q8C2412yes—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—22no—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2411yescharacteristic
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171#713—432no—
172Q8×D5—216no—
173Q8×D5—28no—
174Q8×D5—28no—
175C2×D20—24no—
176C2×D20—24no—
177C2×D20—28no—
178C2×D20—24no—
179C2×D20—24no—
180C2×D20—22no—
181C2×D20—44no—
182C2×D20—48no—
183C2×D20—44no—
184C2×D20C2414yes—
185C2×C4×D5—28no—
186C2×C4×D5—24no—
187C2×C4×D5—22no—
188C2×C4×D5—22no—
189C2×C4×D5—24no—
190C2×C4×D5—28no—
191C2×C4×D5—42no—
192C2×C4×D5—44no—
193C2×C4×D5—48no—
194C2×C4×D5SD1618yescomplemented
195C2×C4×D5C2×D412yes—
196C2×C4×D5C2×D414yes—
197C2×C4×D5C2×D414yes—
198C2×C4×D5C2412yes—
199C2×Dic10—28no—
200C2×Dic10—21no—
201C2×Dic10—22no—
202C2×Dic10—24no—
203C2×Dic10#4114yescomplemented
204C2×Dic10C2412yes—
205C2×C5×C8—24no—
206C2×C5×C8C2414yes—
207C5×C4⋊C4—24no—
208C5×C4⋊C4—24no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—22no—
211C5×C4⋊C4C2412yes—
212C5×C4⋊C4C2414yes—
213C42×C5—24no—
214C42×C5C2×D412yes—
215C42×C5C2412yes—
216#688—216no—
217#685—28no—
218#685—28no—
219C4⋊Dic5—21no—
220C4⋊Dic5—24no—
221C4⋊Dic5C2412yes—
222#683—24no—
223#683—24no—
224#683—24no—
225C4×Dic5—24no—
226C4×Dic5—24no—
227C4×Dic5—24no—
228C4×Dic5C2×D414yes—
229C4×Dic5C2×D412yes—
230C4×Dic5C2×D414yes—
231C4×Dic5C2412yes—
232C2×#168—24no—
233C2×#168C2414yes—
234#677—216no—
235C22×#41—101no—
236C23×Q8—51no—
237C22×SD16—52no—
238C22×SD16—52no—
239#532—58no—
240#532—104no—
241C2×C4⋊Q8C2×D511yescharacteristic
242C2×#127—51no—
243C2×#124—101no—
244C2×#121—52no—
245C2×#121—101no—
246C2×C4×Q8—51no—
247C2×C4×Q8—58no—
248C2×C4×Q8—51no—
249C2×C4×Q8—51no—
250C2×C4×Q8—101no—
251C2×C4×Q8—101no—
252C2×C4×Q8—104no—
253C2×C4×Q8C2×D511yescharacteristic
254C2×C4×D4—52no—
255C2×C4×D4—101no—
256C22×C4⋊C4—51no—
257C22×C4⋊C4—101no—
258C22×C42—51no—
259C22×C42—101no—
260#481—516no—
261#475—58no—
262#475C2×D518yescomplemented
263#460—58no—
264#460—58no—
265#448—516no—
266C4×SD16—516no—
267C4×SD16—516no—
268#431—58no—
269C2×#106—52no—
270C2×#106C2×D512yes—
271C2×C4⋊C8—51no—
272C2×C4⋊C8C2×D511yescharacteristic
273C2×#103—52no—
274C2×#103C2×D512yes—
275C2×#102—52no—
276C2×#102—52no—
277C2×C22⋊C8—52no—
278C2×C4×C8—52no—
279C23×C5C22×D411yescharacteristic
280C22×D5—22no—
281C22×D5—22no—
282C22×D5—24no—
283C22×D5—44no—
284C22×D5—44no—
285C22×D5—44no—
286C22×D5C2×SD1614yes—
287C22×D5C22×D412yes—
288C5×Q8—24no—
289C5×Q8—22no—
290C5×Q8—48no—
291C5×Q8C2×#4114yescomplemented
292C2×C4×C5—24no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—24no—
297C2×C4×C5—44no—
298C2×C4×C5—42no—
299C2×C4×C5C2×SD1614yes—
300C2×C4×C5C22×D412yes—
301C2×C4×C5C22×D411yescharacteristic
302C2×C4×C5C22×D412yes—
303C2×C4×C5C2×#4112yes—
304C2×C4×C5C2×#4112yes—
305C2×C4×C5C2511yescharacteristic
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—42no—
312C2×Dic5C2×SD1614yes—
313C2×Dic5C22×D412yes—
314C2×Dic5C22×D412yes—
315C2×Dic5C22×D412yes—
316D20—24no—
317D20—24no—
318D20—48no—
319D20—416no—
320D20—44no—
321D20—48no—
322C4×D5—216no—
323C4×D5—24no—
324C4×D5—22no—
325C4×D5—22no—
326C4×D5—44no—
327C4×D5—48no—
328C4×D5—416no—
329C4×D5C2×SD16116yescomplemented
330Dic10—216no—
331Dic10—24no—
332Dic10—22no—
333Dic10C2×#4114yescomplemented
334C5×C8—24no—
335#168—24no—
336C2×#41—108no—
337C2×#41—208no—
338C22×Q8—21no—
339C22×Q8—54no—
340C22×Q8—51no—
341C22×Q8—101no—
342C22×Q8—102no—
343C22×Q8—104no—
344C22×Q8—104no—
345C22×Q8C22×D511yescharacteristic
346C22×D4—52no—
347C22×D4—102no—
348C22×D4—101no—
349C22×D4—101no—
350C23×C4—51no—
351C23×C4—51no—
352C23×C4—51no—
353C23×C4—102no—
354C23×C4—101no—
355C23×C4—101no—
356C2×SD16—58no—
357C2×SD16—58no—
358C2×SD16—108no—
359C2×SD16—108no—
360C22×C8—52no—
361C22×C8C22×D512yes—
362C4⋊Q8—22no—
363C4⋊Q8C22×D514yes—
364#127—54no—
365#127—102no—
366#124—108no—
367#121—58no—
368#121—104no—
369#121—108no—
370C4×Q8—22no—
371C4×Q8—54no—
372C4×Q8—54no—
373C4×Q8—54no—
374C4×Q8—108no—
375C4×Q8—108no—
376C4×Q8—102no—
377C4×Q8—102no—
378C4×Q8—102no—
379C4×Q8C22×D514yes—
380C4×D4—58no—
381C4×D4—104no—
382C4×D4—108no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—51no—
386C2×C4⋊C4—54no—
387C2×C4⋊C4—102no—
388C2×C4⋊C4—108no—
389C2×C4⋊C4—101no—
390C2×C4⋊C4—101no—
391C2×C4⋊C4—101no—
392C2×C4⋊C4—101no—
393C2×C4⋊C4C22×D511yescharacteristic
394C2×C4⋊C4C22×D512yes—
395C2×C22⋊C4—102no—
396C2×C22⋊C4—102no—
397C2×C42—21no—
398C2×C42—51no—
399C2×C42—51no—
400C2×C42—51no—
401C2×C42—54no—
402C2×C42—54no—
403C2×C42—108no—
404C2×C42—101no—
405C2×C42—101no—
406C2×C42—101no—
407C2×C42C22×D511yescharacteristic
408#106—58no—
409#106C22×D518yes—
410C4⋊C8—54no—
411C4⋊C8C22×D514yes—
412#103—58no—
413#103C22×D518yes—
414#102—58no—
415#102—58no—
416C22⋊C8—58no—
417C4×C8—58no—
418C22×C5#52512yes—
419C22×C5C22×SD1612yes—
420C22×C5#57512yes—
421C22×C5C23×D411yescharacteristic
422C2×D5—28no—
423C2×D5—22no—
424C2×D5—22no—
425C2×D5—44no—
426C2×D5—44no—
427C2×D5—44no—
428C2×D5—88no—
429C2×D5C22×SD1614yes—
430C4×C5—24no—
431C4×C5—22no—
432C4×C5—44no—
433C4×C5—42no—
434C4×C5#52512yes—
435C4×C5C22×SD1614yes—
436C4×C5C22×#4111yescharacteristic
437C4×C5C22×#4111yescharacteristic
438Dic5—22no—
439Dic5—24no—
440Dic5—22no—
441Dic5—42no—
442Dic5C22×SD1614yes—
443C24—51no—
444C24—102no—
445#41—2032no—
446C2×Q8—21no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×Q8—54no—
451C2×Q8—52no—
452C2×Q8—104no—
453C2×Q8—108no—
454C2×Q8—108no—
455C2×Q8—1016no—
456C2×Q8—108no—
457C2×Q8—101no—
458C2×Q8—102no—
459C2×Q8#71314yescomplemented
460C2×Q8C23×D512yes—
461C2×D4—54no—
462C2×D4—108no—
463C2×D4—104no—
464C2×D4—102no—
465C2×D4—104no—
466C2×D4—104no—
467C2×D4—104no—
468C2×D4—204no—
469C2×D4—208no—
470C2×D4—204no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—51no—
475C22×C4—54no—
476C22×C4—58no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—52no—
480C22×C4—52no—
481C22×C4—54no—
482C22×C4—108no—
483C22×C4—104no—
484C22×C4—108no—
485C22×C4—104no—
486C22×C4—101no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4—202no—
491C22×C4—208no—
492C22×C4—204no—
493C22×C4D4×D511yescharacteristic
494C22×C4D4×D511yescharacteristic
495C22×C4C23×D511yescharacteristic
496SD16—1016no—
497SD16—1016no—
498C2×C8—24no—
499C2×C8—54no—
500C2×C8—104no—
501C2×C8C23×D514yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—52no—
507C4⋊C4—104no—
508C4⋊C4—104no—
509C4⋊C4—104no—
510C4⋊C4—101no—
511C4⋊C4—104no—
512C4⋊C4C23×D512yes—
513C4⋊C4C23×D514yes—
514C22⋊C4—108no—
515C22⋊C4—108no—
516C42—24no—
517C42—52no—
518C42—54no—
519C42—54no—
520C42—52no—
521C42—104no—
522C42—104no—
523C42—104no—
524C42D4×D512yes—
525C42C23×D512yes—
526C2×C5#310314yescomplemented
527C2×C5C2×#52511yescharacteristic
528C2×C5C23×SD1611yescharacteristic
529C2×C5C2×#57511yescharacteristic
530D5—24no—
531D5—84no—
532C23—54no—
533C23—52no—
534C23—102no—
535C23—102no—
536C23—104no—
537C23—204no—
538C23—204no—
539C23—204no—
540C23C2×D4×D511yescharacteristic
541Q8—24no—
542Q8—22no—
543Q8—48no—
544Q8—54no—
545Q8—1016no—
546Q8—104no—
547Q8—102no—
548Q8C2×#71314yescomplemented
549D4—104no—
550D4—104no—
551D4—208no—
552D4—2016no—
553D4—208no—
554D4—204no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—54no—
563C2×C4—52no—
564C2×C4—516no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—1016no—
568C2×C4—102no—
569C2×C4—102no—
570C2×C4—104no—
571C2×C4—102no—
572C2×C4—104no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—204no—
577C2×C4—2016no—
578C2×C4—208no—
579C2×C4—202no—
580C2×C4D5×SD1614yes—
581C2×C4C2×D4×D512yes—
582C2×C4C2×D4×D512yes—
583C2×C4C2×D4×D511yescharacteristic
584C2×C4C2×#71312yes—
585C2×C4D5×#4112yes—
586C2×C4C24×D511yescharacteristic
587C8—24no—
588C8—104no—
589C5C2×#310311yescharacteristic, complemented, Sylow
590C22—54no—
591C22—102no—
592C22—102no—
593C22—108no—
594C22—204no—
595C22—204no—
596C22—204no—
597C22—408no—
598C22#6596112yes—
599C22C2×D5×SD1612yes—
600C22#6611112yes—
601C22C22×D4×D511yescharacteristic
602C4—24no—
603C4—22no—
604C4—44no—
605C4—42no—
606C4—54no—
607C4—102no—
608C4—104no—
609C4—102no—
610C4—202no—
611C4#6596112yes—
612C4C2×D5×SD1614yes—
613C4C22×#71311yescharacteristic
614C4C2×D5×#4111yescharacteristic
615C2—104no—
616C2—404no—
617C2#1061272714yesdirect product factor
618C2C2×#6596111yescharacteristic
619C2C22×D5×SD1611yescharacteristic
620C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#475C2
socleC23×C5C22×D4
prime radicalC2×#66059C2
hypercenterC2×#475D5
hypocenterC5C2×#3103
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
2. C2×C4×Q8×D5C2semidirect
3. C2×C5×#475C2semidirect
4. C2×#65383C2semidirect
5. C2×#65316C2semidirect
6. #10612727C2direct (N × H)
7. C2×#65152C2semidirect
8. C2×#65141C2semidirect
9. C2×#65137C2semidirect
10. C2×#65122C2semidirect
11. C2×#65095C2semidirect
12. C2×#65077C2semidirect
24. C4×Q8×D5C22semidirect
38. C5×#475C22semidirect
45. #65383C22semidirect
46. #65316C22semidirect
51. #65152C22semidirect
52. #65141C22semidirect
53. #65137C22semidirect
54. #65122C22semidirect
55. #65095C22semidirect
56. #65077C22semidirect
145. C2×#475D5semidirect
158. C2×C5×Q8#41semidirect
194. C2×C4×D5SD16semidirect
203. C2×Dic10#41semidirect
262. #475C2×D5semidirect
291. C5×Q8C2×#41semidirect
329. C4×D5C2×SD16semidirect
333. Dic10C2×#41semidirect
459. C2×Q8#713semidirect
526. C2×C5#3103semidirect
548. Q8C2×#713semidirect
589. C5C2×#3103semidirect
617. C2#10612727direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660592
C2×C4×Q8×D52
C2×C5×#4752
C2×#653832
C2×#653162
#106127272
C2×#651522
C2×#651412
C2×#651372
C2×#651222
C2×#650952
C2×#650772
C2×#31035