Finite Group Databasebeta

← C2×#10613607 #12970570

Subgroups — C2×#10613607 #12970570

Order 1280.

2146 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#66014C211yesmaximal, characteristic
3C2×C5×#497C211yesmaximal, characteristic, complemented
4C2×#65475C212yesmaximal, complemented
5C2×#65471C212yesmaximal, complemented
6C2×#65459C212yesmaximal
7C2×#65451C212yesmaximal, complemented
8C2×#65371C211yesmaximal, characteristic, complemented
9C2×#65349C211yesmaximal, characteristic, complemented
10#10613607C2116yesmaximal, direct product factor
11C2×#64992C211yesmaximal, characteristic, complemented
12C2×#64994C211yesmaximal, characteristic, complemented
13C22×Q8×D5C2212yes—
14C22×C5×Q16C2212yes—
15C22×C5×SD16C2212yes—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#124C2211yescharacteristic
18C2×#4092C2211yescharacteristic
19C2×Q8×Dic5C2211yescharacteristic
20C2×#4090C2212yes—
21C22×#689C2212yes—
22C2×#4084C2211yescharacteristic
23C2×D4×Dic5C2211yescharacteristic
24C22×#687C2212yes—
25D5×C4⋊Q8—24no—
26D5×C4⋊Q8C2218yescomplemented
27#66014—24no—
28#66014C2218yes—
29C2×D5×C4⋊C4—21no—
30C2×D5×C4⋊C4—21no—
31C2×#4034—21no—
32C2×#4034—21no—
33C2×#4029—21no—
34C2×#4027—21no—
35C2×#4025—21no—
36C2×#4023—21no—
37C2×C42×D5C2211yescharacteristic
38C2×#4016C2211yescharacteristic
39C2×#4015C2211yescharacteristic
40C5×#497C2218yescomplemented
41C2×C5×#97C2211yescharacteristic
42C2×#3967C2212yes—
43C2×#3967C2212yes—
44#65475C22116yescomplemented
45#65471C22116yescomplemented
46#65459C22116yes—
47#65451C22116yescomplemented
48C2×#3952C2212yes—
49C2×#3948C2212yes—
50C2×#3948C2212yes—
51C2×#3946C2212yes—
52#65371C2218yescomplemented
53#65349C2218yescomplemented
54C2×#3935C2211yescharacteristic
55#64994C2218yescomplemented
56#64992C2218yescomplemented
57C2×#3218—51nomaximal, Sylow
58C22×C5×Q8C2311yescharacteristic
59C22×C5×Q8C2312yes—
60C22×C5×D4C2311yescharacteristic
61C22×C22⋊D5—21no—
62C23×Dic5—21no—
63C2×Q8×D5—24no—
64C2×Q8×D5—44no—
65C2×Q8×D5—44no—
66C2×Q8×D5C2318yes—
67C22×C4×D5—22no—
68C22×C4×D5D411yescharacteristic
69C22×C4×D5D411yescharacteristic
70C22×C4×D5C2311yescharacteristic
71C22×Dic10—22no—
72C22×Dic10C2312yes—
73C2×C5×Q16—44no—
74C2×C5×Q16C2318yes—
75C2×C5×SD16—44no—
76C2×C5×SD16C2318yes—
77C22×C5×C8C2312yes—
78C5×C4⋊Q8—22no—
79C5×C4⋊Q8C2314yes—
80C5×#124—22no—
81C5×#124C2314yes—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—21no—
86C2×C42×C5C2311yescharacteristic
87C2×C22⋊Dic5—21no—
88C2×C22⋊Dic5—21no—
89#4092—22no—
90#4092C2314yes—
91Q8×Dic5—22no—
92Q8×Dic5C2314yes—
93#4090—24no—
94#4090C2318yes—
95C2×#689—44no—
96C2×#689C2318yes—
97#4084—22no—
98#4084C2314yes—
99D4×Dic5—22no—
100D4×Dic5C2314yes—
101C2×#687—44no—
102C2×#687C2318yes—
103C2×#685—21no—
104C2×C4⋊Dic5—21no—
105C2×C4⋊Dic5—21no—
106C2×C4⋊Dic5—21no—
107C2×C4⋊Dic5C2311yescharacteristic
108C2×C4⋊Dic5C2311yescharacteristic
109C2×#683—21no—
110C2×#683—21no—
111C2×#683—21no—
112C2×#683—22no—
113C2×#683—22no—
114C2×C4×Dic5—21no—
115C2×C4×Dic5D411yescharacteristic
116C2×C4×Dic5D411yescharacteristic
117C2×C4×Dic5C2311yescharacteristic
118C22×#168C2312yes—
119D5×C4⋊C4—28no—
120D5×C4⋊C4—28no—
121#4034—28no—
122#4034—28no—
123#4029—28no—
124#4027—28no—
125#4025—28no—
126#4023—28no—
127C42×D5D414yes—
128C42×D5C2314yes—
129#4016—22no—
130#4016C2314yes—
131#4015—22no—
132#4015C2314yes—
133C5×#97C2314yes—
134#3967C2318yes—
135#3967C2318yes—
136#3952C2318yes—
137#3948C2318yes—
138#3948C2318yes—
139#3946C2318yes—
140#3935C2314yes—
141C22×C4⋊Q8—51no—
142C2×#529—51no—
143#3218—516no—
144C2×#497—51no—
145C2×#497D511yescharacteristic, complemented
146C2×#491—51no—
147C2×#491—51no—
148C2×#462—52no—
149C2×#461—52no—
150C2×#451—52no—
151C2×#448—52no—
152C2×#432—51no—
153C24×C5—21no—
154C23×D5C2×D411yescharacteristic
155C2×C5×Q8—21no—
156C2×C5×Q8—22no—
157C2×C5×Q8—24no—
158C2×C5×Q8—24no—
159C2×C5×Q8—24no—
160C2×C5×Q8C2412yes—
161C2×C5×Q8C2414yes—
162C2×C5×D4—21no—
163C2×C5×D4—24no—
164C2×C5×D4C2412yes—
165C22×C4×C5—21no—
166C22×C4×C5—22no—
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2411yescharacteristic
170C2×C22⋊D5—24no—
171C2×C22⋊D5—44no—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C22×Dic5—22no—
175C22×Dic5—24no—
176C22×Dic5—44no—
177C22×Dic5C2×D411yescharacteristic
178C22×Dic5C2×D411yescharacteristic
179C22×Dic5C2×D411yescharacteristic
180Q8×D5—48no—
181Q8×D5—48no—
182Q8×D5—416no—
183C2×C4×D5—24no—
184C2×C4×D5—24no—
185C2×C4×D5—24no—
186C2×C4×D5—28no—
187C2×C4×D5—48no—
188C2×C4×D5C2×D414yes—
189C2×C4×D5C2×D414yes—
190C2×C4×D5C2×D412yes—
191C2×C4×D5C2412yes—
192C2×Dic10—24no—
193C2×Dic10—24no—
194C2×Dic10—22no—
195C2×Dic10—28no—
196C2×Dic10—48no—
197C2×Dic10C2414yes—
198C5×Q16—48no—
199C5×SD16—48no—
200C2×C5×C8—24no—
201C2×C5×C8C2414yes—
202C5×C4⋊C4—24no—
203C5×C4⋊C4—24no—
204C5×C22⋊C4—24no—
205C5×C22⋊C4—24no—
206C42×C5C2×D412yes—
207C42×C5C2412yes—
208C22⋊Dic5—24no—
209C22⋊Dic5—24no—
210#689—48no—
211#687—48no—
212#685—24no—
213C4⋊Dic5—24no—
214C4⋊Dic5—24no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—24no—
218C4⋊Dic5C2412yes—
219C4⋊Dic5C2412yes—
220#683—24no—
221#683—24no—
222#683—24no—
223#683—28no—
224#683—28no—
225C4×Dic5—24no—
226C4×Dic5C2×D412yes—
227C4×Dic5C2×D414yes—
228C4×Dic5C2×D414yes—
229C4×Dic5C2412yes—
230C2×#168—24no—
231C2×#168C2414yes—
232C23×Q8—52no—
233C22×Q16—52no—
234C22×Q16C2×D512yes—
235C22×SD16—52no—
236C22×SD16C2×D512yes—
237C2×C4⋊Q8—52no—
238C2×C4⋊Q8—58no—
239C2×C4⋊Q8—51no—
240C2×C4⋊Q8—101no—
241C2×C4⋊Q8—101no—
242C2×C4⋊Q8—104no—
243C2×C4⋊Q8C2×D511yescharacteristic
244#529—58no—
245#529—104no—
246C2×#126—101no—
247C2×#125—51no—
248C2×#124C2×D511yescharacteristic
249C2×#123—101no—
250C2×C22⋊Q8—51no—
251C2×#121—51no—
252C2×C4×Q8—51no—
253C2×C4×D4—51no—
254C2×C4×D4—101no—
255C2×#117—101no—
256C22×C4⋊C4—101no—
257C22×C4⋊C4—101no—
258C22×C42—51no—
259#497—58no—
260#497C2×D518yescomplemented
261#491—58no—
262#491—58no—
263#462—516no—
264#461—516no—
265#451—516no—
266#448—516no—
267#432—58no—
268C2×C4⋊C8—52no—
269C2×#103—52no—
270C2×#103—52no—
271C2×#103—52no—
272C2×#103—52no—
273C2×C22⋊C8—52no—
274C2×#97—51no—
275C2×#97C2×D511yescharacteristic
276C23×C5—22no—
277C23×C5—42no—
278C23×C5—42no—
279C23×C5—42no—
280C23×C5C22×D411yescharacteristic
281C22×D5—22no—
282C22×D5—22no—
283C22×D5#13714yes—
284C22×D5C22×D412yes—
285C5×Q8—24no—
286C5×Q8—24no—
287C5×Q8—24no—
288C5×Q8—44no—
289C5×Q8—42no—
290C5×D4—24no—
291C5×D4—42no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—44no—
297C2×C4×C5—42no—
298C2×C4×C5C22×D412yes—
299C2×C4×C5C22×D412yes—
300C2×C4×C5C22×D411yescharacteristic
301C2×C4×C5H(5,𝔽2)14yes—
302C2×C4×C5C2511yescharacteristic
303C22⋊D5—48no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—24no—
309C2×Dic5—44no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—48no—
313C2×Dic5#13714yes—
314C2×Dic5C22×D412yes—
315C2×Dic5C22×D412yes—
316C2×Dic5C22×D412yes—
317C4×D5—28no—
318C4×D5—24no—
319C4×D5—24no—
320C4×D5—28no—
321C4×D5—416no—
322Dic10—24no—
323Dic10—24no—
324Dic10—416no—
325Dic10—44no—
326C5×C8—24no—
327#168—24no—
328C22×Q8—58no—
329C22×Q8—52no—
330C22×Q8—102no—
331C22×Q8—104no—
332C22×Q8—204no—
333C22×Q8—204no—
334C22×Q8C22×D511yescharacteristic
335C22×Q8C22×D512yes—
336C22×D4—101no—
337C22×D4C22×D511yescharacteristic
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—102no—
342C23×C4—101no—
343C2×Q16—44no—
344C2×Q16—58no—
345C2×Q16—204no—
346C2×Q16C22×D518yes—
347C2×SD16—44no—
348C2×SD16—58no—
349C2×SD16—204no—
350C2×SD16C22×D518yes—
351C22×C8—52no—
352C22×C8C22×D512yes—
353C4⋊Q8—22no—
354C4⋊Q8—58no—
355C4⋊Q8—54no—
356C4⋊Q8—108no—
357C4⋊Q8—108no—
358C4⋊Q8—104no—
359C4⋊Q8—102no—
360C4⋊Q8C22×D514yes—
361#126—108no—
362#125—54no—
363#125—102no—
364#124—22no—
365#124C22×D514yes—
366#123—108no—
367C22⋊Q8—54no—
368C22⋊Q8—102no—
369#121—54no—
370#121—102no—
371C4×Q8—54no—
372C4×Q8—102no—
373C4×D4—54no—
374C4×D4—102no—
375C4×D4—108no—
376#117—108no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—51no—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—108no—
382C2×C4⋊C4—108no—
383C2×C4⋊C4—102no—
384C2×C4⋊C4—102no—
385C2×C4⋊C4—101no—
386C2×C4⋊C4—101no—
387C2×C4⋊C4—101no—
388C2×C4⋊C4—101no—
389C2×C4⋊C4—101no—
390C2×C4⋊C4—101no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—101no—
394C2×C22⋊C4—101no—
395C2×C22⋊C4—101no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—54no—
400C2×C42—54no—
401C2×C42—101no—
402C2×C42C22×D511yescharacteristic
403C4⋊C8—58no—
404#103—58no—
405#103—58no—
406#103—58no—
407#103—58no—
408C22⋊C8—58no—
409#97—54no—
410#97C22×D514yes—
411C22×C5—42no—
412C22×C5—42no—
413C22×C5—42no—
414C22×C5—84no—
415C22×C5#53512yes—
416C22×C5C2×#13712yes—
417C22×C5C2×#13712yes—
418C22×C5C23×D411yescharacteristic
419C2×D5—22no—
420C2×D5—28no—
421C2×D5—22no—
422C2×D5C2×#13714yes—
423C4×C5—22no—
424C4×C5—22no—
425C4×C5—44no—
426C4×C5—42no—
427C4×C5#53512yes—
428C4×C5C2×H(5,𝔽2)12yes—
429Dic5—22no—
430Dic5—22no—
431Dic5—44no—
432Dic5—42no—
433Dic5—42no—
434Dic5C2×#13714yes—
435C24—21no—
436C24—51no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—54no—
443C2×Q8—108no—
444C2×Q8—102no—
445C2×Q8—104no—
446C2×Q8—104no—
447C2×Q8—208no—
448C2×Q8—208no—
449C2×Q8—208no—
450C2×Q8—2016no—
451C2×Q8C23×D514yes—
452C2×Q8C23×D512yes—
453C2×D4—21no—
454C2×D4—24no—
455C2×D4—104no—
456C2×D4—204no—
457C2×D4C23×D512yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—51no—
461C22×C4—51no—
462C22×C4—51no—
463C22×C4—54no—
464C22×C4—54no—
465C22×C4—52no—
466C22×C4—52no—
467C22×C4—104no—
468C22×C4—104no—
469C22×C4—108no—
470C22×C4—104no—
471C22×C4—104no—
472C22×C4—101no—
473C22×C4—101no—
474C22×C4—102no—
475C22×C4—208no—
476C22×C4—204no—
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480Q16—48no—
481Q16—208no—
482SD16—48no—
483SD16—208no—
484C2×C8—24no—
485C2×C8—54no—
486C2×C8—104no—
487C2×C8C23×D514yes—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—52no—
491C4⋊C4—52no—
492C4⋊C4—108no—
493C4⋊C4—108no—
494C4⋊C4—104no—
495C4⋊C4—104no—
496C4⋊C4—101no—
497C4⋊C4—101no—
498C4⋊C4—104no—
499C4⋊C4—104no—
500C4⋊C4—104no—
501C4⋊C4—104no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—104no—
505C22⋊C4—104no—
506C22⋊C4—104no—
507C42—54no—
508C42—52no—
509C42—54no—
510C42—52no—
511C42—104no—
512C42D4×D512yes—
513C42C23×D512yes—
514C2×C5—82no—
515C2×C5#321814yescomplemented
516C2×C5C2×#53511yescharacteristic
517C2×C5C22×#13711yescharacteristic
518C2×C5C22×#13711yescharacteristic
519D5—24no—
520C23—22no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—54no—
525C23—52no—
526C23—102no—
527C23—102no—
528C23C2×D4×D511yescharacteristic
529Q8—24no—
530Q8—24no—
531Q8—24no—
532Q8—44no—
533Q8—42no—
534Q8—104no—
535Q8—104no—
536Q8—2016no—
537Q8—204no—
538D4—24no—
539D4—42no—
540D4—208no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—54no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—108no—
552C2×C4—108no—
553C2×C4—102no—
554C2×C4—102no—
555C2×C4—104no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—104no—
560C2×C4—2016no—
561C2×C4—208no—
562C2×C4—204no—
563C2×C4—202no—
564C2×C4—202no—
565C2×C4C2×D4×D512yes—
566C2×C4C2×D4×D512yes—
567C2×C4C2×D4×D511yescharacteristic
568C2×C4#414414yes—
569C2×C4C24×D511yescharacteristic
570C8—24no—
571C8—104no—
572C5C2×#321811yescharacteristic, complemented, Sylow
573C22—42no—
574C22—42no—
575C22—42no—
576C22—54no—
577C22—84no—
578C22—108no—
579C22—102no—
580C22—102no—
581C22#6604912yes—
582C22D5×#13712yes—
583C22D5×#13712yes—
584C22C22×D4×D511yescharacteristic
585C4—22no—
586C4—22no—
587C4—44no—
588C4—42no—
589C4—54no—
590C4—102no—
591C4—102no—
592C4—202no—
593C4—202no—
594C4—204no—
595C4#6604912yes—
596C4C2×#414412yes—
597C2—82no—
598C2—104no—
599C2#1061360714yesdirect product factor
600C2C2×#6604911yescharacteristic
601C2C2×D5×#13711yescharacteristic
602C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#497C2
socleC23×C5C22×D4
prime radicalC2×#4084C22
hypercenterC2×#497D5
hypocenterC5C2×#3218
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊Q8C2semidirect
3. C2×C5×#497C2semidirect
4. C2×#65475C2semidirect
5. C2×#65471C2semidirect
7. C2×#65451C2semidirect
8. C2×#65371C2semidirect
9. C2×#65349C2semidirect
10. #10613607C2direct (N × H)
11. C2×#64992C2semidirect
12. C2×#64994C2semidirect
26. D5×C4⋊Q8C22semidirect
40. C5×#497C22semidirect
44. #65475C22semidirect
45. #65471C22semidirect
47. #65451C22semidirect
52. #65371C22semidirect
53. #65349C22semidirect
55. #64994C22semidirect
56. #64992C22semidirect
145. C2×#497D5semidirect
260. #497C2×D5semidirect
515. C2×C5#3218semidirect
572. C5C2×#3218semidirect
599. C2#10613607direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#660142
C2×C5×#4972
C2×#654752
C2×#654712
C2×#654592
C2×#654512
C2×#653712
C2×#653492
#106136072
C2×#649922
C2×#649942
C2×#32185