Finite Group Databasebeta

← C2×#10612765 #12972108

Subgroups — C2×#10612765 #12972108

Order 1280.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic, complemented
2C2×#65914C211yesmaximal, characteristic
3C2×C5×#475C211yesmaximal, characteristic, complemented
4C2×#65376C211yesmaximal, characteristic, complemented
5C2×#65320C211yesmaximal, characteristic, complemented
6#10612765C2116yesmaximal, direct product factor
7C2×#65146C212yesmaximal, complemented
8C2×#65153C212yesmaximal
9C2×#65130C211yesmaximal, characteristic, complemented
10C2×D5×C4⋊C8C211yesmaximal, characteristic, complemented
11C2×#65087C212yesmaximal, complemented
12C2×#65102C212yesmaximal, complemented
13C2×C5×C4⋊Q8C2211yescharacteristic
14C2×C4×C5×Q8C2211yescharacteristic
15C2×#4093C2211yescharacteristic
16C2×#4092—21no—
17C2×#4092C2211yescharacteristic
18C2×Q8×Dic5—21no—
19C22×#689C2212yes—
20C22×#677C2212yes—
21C22×C8×D5C2212yes—
22#66062—24no—
23#66062C2218yescomplemented
24#65914—24no—
25#65914C2218yes—
26C2×#4044—21no—
27C2×#4043C2212yes—
28C2×#4041C2211yescharacteristic
29C2×#4040—21no—
30C2×#4038—21no—
31C2×#4036—21no—
32C2×#4036C2211yescharacteristic
33C2×#4033C2212yes—
34C2×C4×D20—21no—
35C2×C4×D20C2211yescharacteristic
36C2×#4018—21no—
37C2×C42×D5C2211yescharacteristic
38C2×#4015C2211yescharacteristic
39C5×#475C2218yescomplemented
40C2×C5×#106C2212yes—
41C2×C5×C4⋊C8C2211yescharacteristic
42C2×C5×#103C2212yes—
43C2×#3948C2212yes—
44C2×#3946C2212yes—
45#65376C2218yescomplemented
46#65320C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3941C2212yes—
49C2×#3939C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65153C22116yes—
52#65146C22116yescomplemented
53#65130C2218yescomplemented
54D5×C4⋊C8C2218yescomplemented
55#65102C22116yescomplemented
56#65087C22116yescomplemented
57C2×#3248—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8C2311yescharacteristic
60C22×D20—21no—
61C22×D20C2311yescharacteristic
62C22×C4×D5—21no—
63C22×C4×D5D411yescharacteristic
64C22×C4×D5D411yescharacteristic
65C22×C4×D5C2311yescharacteristic
66C22×Dic10C2312yes—
67C22×C5×C8C2312yes—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8C2314yes—
70C4×C5×Q8—22no—
71C4×C5×Q8C2314yes—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4C2312yes—
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C42×C5—21no—
77C2×C42×C5C2311yescharacteristic
78#4093—22no—
79#4093C2314yes—
80#4092—28no—
81#4092—22no—
82#4092C2314yes—
83Q8×Dic5—28no—
84C2×#689—44no—
85C2×#689C2318yes—
86C2×#685—21no—
87C2×#685—22no—
88C2×#685—21no—
89C2×#685—21no—
90C2×#685—21no—
91C2×#685—21no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—22no—
96C2×C4⋊Dic5C2311yescharacteristic
97C2×#683—21no—
98C2×#683—21no—
99C2×#683—22no—
100C2×C4×Dic5—22no—
101C2×C4×Dic5D411yescharacteristic
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C22×#168C2312yes—
105C2×#677—44no—
106C2×#677C2318yes—
107C2×C8×D5—44no—
108C2×C8×D5C2318yes—
109#4044—28no—
110#4043—24no—
111#4043C2318yes—
112#4041—22no—
113#4041C2314yes—
114#4040—28no—
115#4038—28no—
116#4036—28no—
117#4036—22no—
118#4036C2314yes—
119#4033—24no—
120#4033C2318yes—
121C4×D20—28no—
122C4×D20—22no—
123C4×D20C2314yes—
124#4018—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127#4015—22no—
128#4015C2314yes—
129C5×#106C2318yes—
130C5×C4⋊C8C2314yes—
131C5×#103C2318yes—
132#3948C2318yes—
133#3946C2318yes—
134#3942C2318yes—
135#3941C2318yes—
136#3939C2318yes—
137C4⋊#168C2314yes—
138C2×#533—51no—
139C2×#529—51no—
140#3248—516no—
141C2×#479—52no—
142C2×#475D511yescharacteristic, complemented
143C2×#474—51no—
144C2×#467—52no—
145C2×#465—52no—
146C2×#464—52no—
147C2×#462—51no—
148C2×#461—51no—
149C22×C4⋊C8—51no—
150C23×D5—21no—
151C23×D5C2×D411yescharacteristic
152C2×C5×Q8—21no—
153C2×C5×Q8—24no—
154C2×C5×Q8—22no—
155C2×C5×Q8—22no—
156C2×C5×Q8—44no—
157C2×C5×Q8C2412yes—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2411yescharacteristic
164C22×Dic5—22no—
165C22×Dic5—21no—
166C22×Dic5C2×D411yescharacteristic
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C2×D20—22no—
170C2×D20—22no—
171C2×D20—24no—
172C2×D20—21no—
173C2×D20—44no—
174C2×D20C2412yes—
175C2×C4×D5—24no—
176C2×C4×D5—24no—
177C2×C4×D5—22no—
178C2×C4×D5—22no—
179C2×C4×D5—24no—
180C2×C4×D5—44no—
181C2×C4×D5C2×D414yes—
182C2×C4×D5C2×D414yes—
183C2×C4×D5C2×D412yes—
184C2×C4×D5C2412yes—
185C2×Dic10—24no—
186C2×Dic10—24no—
187C2×Dic10—22no—
188C2×Dic10C2414yes—
189C2×C5×C8—24no—
190C2×C5×C8C2414yes—
191C5×C4⋊C4—24no—
192C5×C4⋊C4—24no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4C2412yes—
196C5×C4⋊C4C2414yes—
197C42×C5—24no—
198C42×C5C2×D412yes—
199C42×C5C2412yes—
200#689—48no—
201#685—24no—
202#685—28no—
203#685—24no—
204#685—24no—
205#685—24no—
206#685—24no—
207C4⋊Dic5—24no—
208C4⋊Dic5—24no—
209C4⋊Dic5—21no—
210C4⋊Dic5—24no—
211C4⋊Dic5—28no—
212C4⋊Dic5C2412yes—
213#683—24no—
214#683—24no—
215#683—28no—
216C4×Dic5—28no—
217C4×Dic5C2×D412yes—
218C4×Dic5C2×D414yes—
219C4×Dic5C2×D414yes—
220C4×Dic5C2412yes—
221C2×#168—24no—
222C2×#168C2414yes—
223#677—48no—
224C8×D5—48no—
225C22×Q16—52no—
226C22×SD16—52no—
227C23×C8—52no—
228#533—58no—
229#533—104no—
230C2×C4⋊Q8—51no—
231C2×C4⋊Q8C2×D511yescharacteristic
232#529—58no—
233#529—104no—
234C2×#126—101no—
235C2×#125—51no—
236C2×#125—101no—
237C2×#124—51no—
238C2×#123—101no—
239C2×C22⋊Q8—51no—
240C2×C22⋊Q8—52no—
241C2×C22⋊Q8—101no—
242C2×#121—51no—
243C2×C4×Q8—52no—
244C2×C4×Q8—101no—
245C2×C4×Q8C2×D511yescharacteristic
246C2×C4×D4—51no—
247C2×C4×D4—101no—
248C2×#117—101no—
249C2×#117—101no—
250C22×C42—51no—
251#479—516no—
252#475C2×D518yescomplemented
253#474—58no—
254#467—516no—
255#465—516no—
256#464—516no—
257#462—58no—
258#461—58no—
259C2×#107—52no—
260C2×#106C2×D512yes—
261C2×C4⋊C8—52no—
262C2×C4⋊C8—51no—
263C2×C4⋊C8—58no—
264C2×C4⋊C8C2×D511yescharacteristic
265C2×#103—52no—
266C2×#103—52no—
267C2×#103C2×D512yes—
268C2×#102—52no—
269C23×C5C22×D411yescharacteristic
270C22×D5—22no—
271C22×D5—22no—
272C22×D5—22no—
273C22×D5—42no—
274C22×D5—42no—
275C22×D5—42no—
276C22×D5#13714yes—
277C22×D5C22×D412yes—
278C5×Q8—22no—
279C5×Q8—22no—
280C5×Q8—42no—
281C5×Q8—48no—
282C2×C4×C5—22no—
283C2×C4×C5—24no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—42no—
287C2×C4×C5—44no—
288C2×C4×C5—42no—
289C2×C4×C5#13514yes—
290C2×C4×C5C22×D412yes—
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5C22×D412yes—
293C2×C4×C5H(5,𝔽2)12yes—
294C2×C4×C52−1+412yes—
295C2×C4×C5C2511yescharacteristic
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—24no—
300C2×Dic5—22no—
301C2×Dic5—44no—
302C2×Dic5—42no—
303C2×Dic5C22×D412yes—
304C2×Dic5C22×D412yes—
305C2×Dic5C22×D412yes—
306D20—22no—
307D20—22no—
308D20—48no—
309D20—42no—
310C4×D5—28no—
311C4×D5—24no—
312C4×D5—28no—
313C4×D5—22no—
314C4×D5—22no—
315C4×D5—48no—
316Dic10—24no—
317Dic10—24no—
318Dic10—44no—
319C5×C8—24no—
320#168—24no—
321C22×Q8—21no—
322C22×Q8—52no—
323C22×Q8C22×D511yescharacteristic
324C22×D4—51no—
325C22×D4—101no—
326C23×C4—51no—
327C23×C4—51no—
328C23×C4—51no—
329C23×C4—101no—
330C2×Q16—58no—
331C2×Q16—204no—
332C2×SD16—58no—
333C2×SD16—204no—
334C22×C8—52no—
335C22×C8—58no—
336C22×C8—204no—
337C22×C8C22×D512yes—
338C4⋊Q8—22no—
339C4⋊Q8—54no—
340C4⋊Q8—102no—
341C4⋊Q8C22×D514yes—
342#126—108no—
343#125—54no—
344#125—102no—
345#125—108no—
346#124—54no—
347#124—102no—
348#123—108no—
349C22⋊Q8—54no—
350C22⋊Q8—58no—
351C22⋊Q8—104no—
352C22⋊Q8—102no—
353C22⋊Q8—108no—
354#121—54no—
355#121—102no—
356C4×Q8—22no—
357C4×Q8—58no—
358C4×Q8—104no—
359C4×Q8—108no—
360C4×Q8C22×D514yes—
361C4×D4—54no—
362C4×D4—108no—
363C4×D4—102no—
364#117—108no—
365#117—108no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—101no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—102no—
373C2×C4⋊C4—102no—
374C2×C4⋊C4—101no—
375C2×C4⋊C4—101no—
376C2×C4⋊C4C22×D512yes—
377C2×C4⋊C4C22×D511yescharacteristic
378C2×C22⋊C4—101no—
379C2×C22⋊C4—101no—
380C2×C22⋊C4—101no—
381C2×C22⋊C4—102no—
382C2×C22⋊C4—101no—
383C2×C22⋊C4—101no—
384C2×C42—21no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—54no—
389C2×C42—54no—
390C2×C42—102no—
391C2×C42C22×D511yescharacteristic
392#107—58no—
393#106C22×D518yes—
394C4⋊C8—58no—
395C4⋊C8—54no—
396C4⋊C8C22×D514yes—
397#103—58no—
398#103—58no—
399#103C22×D518yes—
400#102—58no—
401C22×C5#53712yes—
402C22×C5C2×#13512yes—
403C22×C5C2×#13712yes—
404C22×C5C23×D411yescharacteristic
405C2×D5—22no—
406C2×D5—28no—
407C2×D5—22no—
408C2×D5—42no—
409C2×D5—42no—
410C2×D5—42no—
411C2×D5—84no—
412C2×D5C2×#13714yes—
413C4×C5—22no—
414C4×C5—42no—
415C4×C5—44no—
416C4×C5—42no—
417C4×C5#53712yes—
418C4×C5C2×#13514yes—
419C4×C5C2×H(5,𝔽2)11yescharacteristic
420C4×C5C2×2−1+411yescharacteristic
421Dic5—22no—
422Dic5—22no—
423Dic5—22no—
424Dic5—44no—
425Dic5—42no—
426C24—51no—
427C24—101no—
428C2×Q8—21no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—44no—
433C2×Q8—54no—
434C2×Q8—102no—
435C2×Q8—104no—
436C2×Q8—104no—
437C2×Q8C23×D512yes—
438C2×D4—52no—
439C2×D4—104no—
440C2×D4—101no—
441C2×D4—102no—
442C2×D4—102no—
443C2×D4—204no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—54no—
451C22×C4—54no—
452C22×C4—52no—
453C22×C4—52no—
454C22×C4—104no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—104no—
458C22×C4—102no—
459C22×C4—101no—
460C22×C4—104no—
461C22×C4—204no—
462C22×C4D4×D511yescharacteristic
463C22×C4D4×D511yescharacteristic
464C22×C4C23×D511yescharacteristic
465Q16—208no—
466SD16—208no—
467C2×C8—24no—
468C2×C8—54no—
469C2×C8—104no—
470C2×C8—208no—
471C2×C8C23×D514yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—52no—
477C4⋊C4—104no—
478C4⋊C4—104no—
479C4⋊C4—104no—
480C4⋊C4—108no—
481C4⋊C4—108no—
482C4⋊C4—101no—
483C4⋊C4—104no—
484C4⋊C4—104no—
485C4⋊C4C23×D514yes—
486C4⋊C4C23×D512yes—
487C22⋊C4—104no—
488C22⋊C4—104no—
489C22⋊C4—104no—
490C22⋊C4—108no—
491C22⋊C4—104no—
492C22⋊C4—104no—
493C42—24no—
494C42—54no—
495C42—52no—
496C42—54no—
497C42—52no—
498C42—108no—
499C42D4×D512yes—
500C42C23×D512yes—
501C2×C5#324814yescomplemented
502C2×C5C2×#53711yescharacteristic
503C2×C5C22×#13511yescharacteristic
504C2×C5C22×#13711yescharacteristic
505D5—24no—
506D5—82no—
507C23—54no—
508C23—52no—
509C23—102no—
510C23—102no—
511C23—102no—
512C23—202no—
513C23—202no—
514C23—202no—
515C23C2×D4×D511yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—48no—
519Q8—42no—
520Q8—104no—
521Q8—104no—
522Q8—204no—
523D4—102no—
524D4—102no—
525D4—208no—
526D4—202no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—52no—
537C2×C4—108no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—102no—
541C2×C4—108no—
542C2×C4—104no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—104no—
546C2×C4—102no—
547C2×C4—202no—
548C2×C4—204no—
549C2×C4—208no—
550C2×C4#406214yes—
551C2×C4C2×D4×D512yes—
552C2×C4C2×D4×D512yes—
553C2×C4C2×D4×D511yescharacteristic
554C2×C4#414712yes—
555C2×C4#414912yes—
556C2×C4C24×D511yescharacteristic
557C8—24no—
558C8—104no—
559C5C2×#324811yescharacteristic, complemented, Sylow
560C22—54no—
561C22—102no—
562C22—108no—
563C22—102no—
564C22—202no—
565C22—202no—
566C22—202no—
567C22—404no—
568C22#6596312yes—
569C22C2×#406212yes—
570C22D5×#13712yes—
571C22C22×D4×D511yescharacteristic
572C4—22no—
573C4—42no—
574C4—44no—
575C4—42no—
576C4—102no—
577C4—102no—
578C4—102no—
579C4—204no—
580C4—202no—
581C4#6596312yes—
582C4C2×#406214yes—
583C4C2×#414711yescharacteristic
584C4C2×#414911yescharacteristic
585C2—104no—
586C2—402no—
587C2#1061276514yesdirect product factor
588C2C2×#6596311yescharacteristic
589C2C22×#406211yescharacteristic
590C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#475C2
socleC23×C5C22×D4
prime radicalC2×#4041C22
hypercenterC2×#475D5
hypocenterC5C2×#3248
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. #10612765C2direct (N × H)
7. C2×#65146C2semidirect
9. C2×#65130C2semidirect
10. C2×D5×C4⋊C8C2semidirect
11. C2×#65087C2semidirect
12. C2×#65102C2semidirect
23. #66062C22semidirect
39. C5×#475C22semidirect
45. #65376C22semidirect
46. #65320C22semidirect
52. #65146C22semidirect
53. #65130C22semidirect
54. D5×C4⋊C8C22semidirect
55. #65102C22semidirect
56. #65087C22semidirect
142. C2×#475D5semidirect
252. #475C2×D5semidirect
501. C2×C5#3248semidirect
559. C5C2×#3248semidirect
587. C2#10612765direct (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×#659142
C2×C5×#4752
C2×#653762
C2×#653202
#106127652
C2×#651462
C2×#651532
C2×#651302
C2×D5×C4⋊C82
C2×#650872
C2×#651022
C2×#32485