Finite Group Databasebeta

← C2×#10618003 #13007071

Subgroups — C2×#10618003 #13007071

Order 1280.

6566 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#125C211yesmaximal, characteristic, complemented
2#10618003C2132yesmaximal, direct product factor
3C2×#66033C212yesmaximal, complemented
4C2×#66034C211yesmaximal, characteristic, complemented
5C2×#66027C214yesmaximal, complemented
6C2×#66026C214yesmaximal, complemented
7C2×C5⋊#564C211yesmaximal, characteristic, complemented
8C2×#66030C214yesmaximal, complemented
9C2×#65848C214yesmaximal, complemented
10C2×#65849C214yesmaximal, complemented
11C2×#65845C214yesmaximal, complemented
12C2×#65813C212yesmaximal, complemented
13C2×C5×#125C2214yes—
14C2×C5×#125C2211yescharacteristic
15C2×C5×#125C2212yes—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#125C22116yescomplemented
18C22×C5×C4⋊C4C2214yes—
19C22×C5×C4⋊C4C2212yes—
20C22×C42×C5C2211yescharacteristic
21C2×#4076C2214yes—
22C2×#4076C2212yes—
23C2×#4075C2214yes—
24C2×C4×C22⋊D5C2214yes—
25C2×C4×C22⋊D5C2214yes—
26C2×C4×C22⋊D5C2214yes—
27C2×#4070C2214yes—
28C2×#4070C2212yes—
29#66034C22116yescomplemented
30#66033C22132yescomplemented
31#66030C22164yescomplemented
32#66027C22164yescomplemented
33#66026C22164yescomplemented
34C5⋊#564C22116yescomplemented
35#65849C22164yescomplemented
36#65848C22164yescomplemented
37C2×#4044C2214yes—
38C2×#4044C2214yes—
39C2×#4044C2214yes—
40C2×#4044C2214yes—
41#65845C22164yescomplemented
42C2×#4043C2214yes—
43C2×#4042C2214yes—
44C2×#4042C2214yes—
45C2×#4042C2214yes—
46C2×#4042C2214yes—
47C2×#4041C2214yes—
48C2×#4040C2214yes—
49C2×#4040C2214yes—
50C2×#4040C2214yes—
51C2×#4040C2214yes—
52C2×#4039C2214yes—
53C2×#4038C2214yes—
54C2×D5×C4⋊C4C2214yes—
55C2×#4036C2214yes—
56C2×#4035C2214yes—
57C2×#4035C2214yes—
58C2×#4035C2214yes—
59C2×#4035C2214yes—
60C2×#4034C2214yes—
61C2×#4033C2214yes—
62#65813C22132yescomplemented
63C2×#4022C2214yes—
64C2×#4021C2212yes—
65C2×C4×D20C2212yes—
66C2×#4018C2214yes—
67C2×#4016C2212yes—
68C2×C4×Dic10C2212yes—
69C2×#3553—51nomaximal, Sylow
70C23×C4×C5C2311yescharacteristic
71C23×C4×C5C2312yes—
72C23×C4×C5C2314yes—
73C22×C22⋊D5C2314yes—
74C22×D20C2312yes—
75C22×D20C2314yes—
76C22×C4×D5C2314yes—
77C22×C4×D5C2314yes—
78C22×C4×D5C2314yes—
79C22×Dic10C2312yes—
80C22×Dic10C2314yes—
81C5×#125C2318yes—
82C5×#125C23132yes—
83C5×#125C23116yes—
84C5×#125C2318yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C4⋊C4C2314yes—
87C2×C5×C4⋊C4C2312yes—
88C2×C5×C4⋊C4C2314yes—
89C2×C5×C4⋊C4C2314yes—
90C2×C5×C4⋊C4C2314yes—
91C2×C5×C4⋊C4C2312yes—
92C2×C5×C4⋊C4C23132yes—
93C2×C5×C4⋊C4C23116yes—
94C2×C42×C5C2312yes—
95C2×C42×C5C2311yescharacteristic
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2318yes—
98C2×C22⋊Dic5C2314yes—
99#4076C23116yes—
100#4076C23132yes—
101#4075C23132yes—
102C4×C22⋊D5C23132yes—
103C4×C22⋊D5C23132yes—
104C4×C22⋊D5C23132yes—
105C2×#685C2314yes—
106C2×#685C2314yes—
107C2×#685C2314yes—
108C2×#685C2314yes—
109C2×#685C2314yes—
110C2×#685C2314yes—
111C2×#685C2314yes—
112C2×#685C2314yes—
113C2×C4⋊Dic5C2312yes—
114C2×C4⋊Dic5C2312yes—
115C2×C4⋊Dic5C2314yes—
116C2×C4⋊Dic5C2314yes—
117#4070C23116yes—
118#4070C23132yes—
119C2×#683C2314yes—
120C2×#683C2314yes—
121C2×#683C2314yes—
122C2×#683C2314yes—
123C2×#683C2314yes—
124C2×#683C2314yes—
125C2×#683C2314yes—
126C2×#683C2314yes—
127C2×C4×Dic5C2314yes—
128C2×C4×Dic5C2314yes—
129C2×C4×Dic5C2314yes—
130#4044C23132yes—
131#4044C23132yes—
132#4044C23132yes—
133#4044C23132yes—
134#4043C23132yes—
135#4042C23132yes—
136#4042C23132yes—
137#4042C23132yes—
138#4042C23132yes—
139#4041C23132yes—
140#4040C23132yes—
141#4040C23132yes—
142#4040C23132yes—
143#4040C23132yes—
144#4039C23132yes—
145#4038C23132yes—
146D5×C4⋊C4C23132yes—
147#4036C23132yes—
148#4035C23132yes—
149#4035C23132yes—
150#4035C23132yes—
151#4035C23132yes—
152#4034C23132yes—
153#4033C23132yes—
154#4022C23132yes—
155#4021C23116yes—
156C4×D20C23116yes—
157#4018C23132yes—
158#4016C23116yes—
159C4×Dic10C23116yes—
160#3553—532no—
161C2×#564—51no—
162C2×#563—52no—
163C2×#562—51no—
164C2×#554—54no—
165C2×#553—54no—
166C2×#552—54no—
167C2×#541—54no—
168C2×#539—54no—
169C2×#539—54no—
170C2×#529—52no—
171C22×#125D511yescharacteristic, complemented
172C24×C5C2411yescharacteristic
173C23×D5C2414yes—
174C22×C4×C5—24no—
175C22×C4×C5—48no—
176C22×C4×C5—44no—
177C22×C4×C5C2412yes—
178C22×C4×C5C2412yes—
179C22×C4×C5C2411yescharacteristic
180C22×C4×C5C2411yescharacteristic
181C22×C4×C5C2414yes—
182C22×C4×C5C2414yes—
183C22×C4×C5C2414yes—
184C22×C4×C5C2418yes—
185C22×C4×C5C24116yes—
186C2×C22⋊D5—216no—
187C2×C22⋊D5C24116yes—
188C22×Dic5C2414yes—
189C22×Dic5C2414yes—
190C22×Dic5C2414yes—
191C2×D20—44no—
192C2×D20—48no—
193C2×D20C2418yes—
194C2×D20C24116yes—
195C2×C4×D5—48no—
196C2×C4×D5—48no—
197C2×C4×D5—48no—
198C2×C4×D5C24116yes—
199C2×C4×D5C24116yes—
200C2×C4×D5C24116yes—
201C2×Dic10—44no—
202C2×Dic10—48no—
203C2×Dic10C2418yes—
204C2×Dic10C24116yes—
205C5×C4⋊C4C24116yes—
206C5×C4⋊C4C24116yes—
207C5×C4⋊C4C2418yes—
208C5×C4⋊C4C24116yes—
209C5×C4⋊C4C24116yes—
210C5×C4⋊C4C24116yes—
211C5×C4⋊C4C2418yes—
212C42×C5C2418yes—
213C42×C5C2414yes—
214C42×C5C2414yes—
215C22⋊Dic5C24116yes—
216#685C24116yes—
217#685C24116yes—
218#685C24116yes—
219#685C24116yes—
220#685C24116yes—
221#685C24116yes—
222#685C24116yes—
223#685C24116yes—
224C4⋊Dic5C2418yes—
225C4⋊Dic5C2418yes—
226C4⋊Dic5C24116yes—
227C4⋊Dic5C24116yes—
228#683C24116yes—
229#683C24116yes—
230#683C24116yes—
231#683C24116yes—
232#683C24116yes—
233#683C24116yes—
234#683C24116yes—
235#683C24116yes—
236C4×Dic5C24116yes—
237C4×Dic5C24116yes—
238C4×Dic5C24116yes—
239#564—516no—
240#563—532no—
241#562—516no—
242#554—564no—
243#553—564no—
244#552—564no—
245#541—564no—
246#539—564no—
247#539—564no—
248C2×C4⋊Q8—54no—
249#529—532no—
250C2×#126—54no—
251C2×#126—54no—
252C2×#126—54no—
253C2×#126—54no—
254C2×#126—54no—
255C2×#125—54no—
256C2×#125—54no—
257C2×#125—54no—
258C2×#125—54no—
259C2×#125—54no—
260C2×#125—52no—
261C2×#125C2×D514yes—
262C2×#125C2×D512yes—
263C2×#125C2×D511yescharacteristic
264C2×#125C2×D511yescharacteristic
265C2×#125C2×D5116yescomplemented
266C2×#124—52no—
267C2×#123—54no—
268C2×#123—54no—
269C2×#123—54no—
270C2×#123—54no—
271C2×#123—54no—
272C2×C22⋊Q8—52no—
273C2×C22⋊Q8—54no—
274C2×C22⋊Q8—54no—
275C2×C22⋊Q8—54no—
276C2×C22⋊Q8—54no—
277C2×C22⋊Q8—54no—
278C2×C22⋊Q8—54no—
279C2×#121—52no—
280C2×#121—54no—
281C2×#121—54no—
282C2×C4×Q8—54no—
283C2×C4×Q8—52no—
284C2×C4×D4—54no—
285C2×C4×D4—54no—
286C2×C4×D4—54no—
287C2×C4×D4—54no—
288C2×C4×D4—52no—
289C2×#117—54no—
290C2×#117—54no—
291C22×C4⋊C4—54no—
292C22×C4⋊C4C2×D512yes—
293C22×C4⋊C4C2×D514yes—
294C22×C42C2×D511yescharacteristic
295C23×C5—24no—
296C23×C52−1+414yes—
297C23×C5C2512yes—
298C23×C5C2511yescharacteristic
299C22×D5—28no—
300C22×D5—28no—
301C22×D5—28no—
302C22×D5C2518yes—
303C2×C4×C5—28no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—28no—
307C2×C4×C5—24no—
308C2×C4×C5—24no—
309C2×C4×C5—28no—
310C2×C4×C5—416no—
311C2×C4×C5—48no—
312C2×C4×C5C2514yes—
313C2×C4×C5C2518yes—
314C2×C4×C5C2518yes—
315C2×C4×C5C2512yes—
316C2×C4×C5C2512yes—
317C2×C4×C5C2514yes—
318C22⋊D5—232no—
319C2×Dic5—28no—
320C2×Dic5—28no—
321C2×Dic5—28no—
322C2×Dic5C2518yes—
323C2×Dic5C2518yes—
324C2×Dic5C2518yes—
325D20—48no—
326D20—416no—
327C4×D5—416no—
328C4×D5—416no—
329C4×D5—416no—
330Dic10—48no—
331Dic10—416no—
332C22×Q8—54no—
333C22×Q8—52no—
334C22×D4—54no—
335C22×D4—54no—
336C22×D4—52no—
337C23×C4—54no—
338C23×C4—54no—
339C23×C4—54no—
340C23×C4C22×D511yescharacteristic
341C23×C4C22×D514yes—
342C23×C4C22×D512yes—
343C4⋊Q8—532no—
344#126—532no—
345#126—532no—
346#126—532no—
347#126—532no—
348#126—532no—
349#125—532no—
350#125—532no—
351#125—532no—
352#125—532no—
353#125—532no—
354#125—516no—
355#125C22×D5132yes—
356#125C22×D5116yes—
357#125C22×D518yes—
358#125C22×D518yes—
359#124—516no—
360#123—532no—
361#123—532no—
362#123—532no—
363#123—532no—
364#123—532no—
365C22⋊Q8—516no—
366C22⋊Q8—532no—
367C22⋊Q8—532no—
368C22⋊Q8—532no—
369C22⋊Q8—532no—
370C22⋊Q8—532no—
371C22⋊Q8—532no—
372#121—516no—
373#121—532no—
374#121—532no—
375C4×Q8—532no—
376C4×Q8—516no—
377C4×D4—532no—
378C4×D4—532no—
379C4×D4—532no—
380C4×D4—532no—
381C4×D4—516no—
382#117—532no—
383#117—532no—
384C2×C4⋊C4—54no—
385C2×C4⋊C4—54no—
386C2×C4⋊C4—54no—
387C2×C4⋊C4—54no—
388C2×C4⋊C4—54no—
389C2×C4⋊C4—52no—
390C2×C4⋊C4—54no—
391C2×C4⋊C4—54no—
392C2×C4⋊C4—54no—
393C2×C4⋊C4—52no—
394C2×C4⋊C4—54no—
395C2×C4⋊C4—54no—
396C2×C4⋊C4—532no—
397C2×C4⋊C4C22×D514yes—
398C2×C4⋊C4C22×D514yes—
399C2×C4⋊C4C22×D514yes—
400C2×C4⋊C4C22×D512yes—
401C2×C4⋊C4C22×D514yes—
402C2×C4⋊C4C22×D514yes—
403C2×C4⋊C4C22×D512yes—
404C2×C4⋊C4C22×D5116yes—
405C2×C4⋊C4C22×D5132yes—
406C2×C22⋊C4—54no—
407C2×C22⋊C4—54no—
408C2×C22⋊C4—54no—
409C2×C22⋊C4—54no—
410C2×C22⋊C4—54no—
411C2×C22⋊C4—54no—
412C2×C22⋊C4—54no—
413C2×C22⋊C4—54no—
414C2×C22⋊C4—54no—
415C2×C42—54no—
416C2×C42—54no—
417C2×C42—54no—
418C2×C42C22×D511yescharacteristic
419C2×C42C22×D512yes—
420C2×C42C22×D511yescharacteristic
421C2×C42C22×D518yes—
422C22×C5—24no—
423C22×C5—28no—
424C22×C5C2×2−1+414yes—
425C22×C5C2×2−1+412yes—
426C22×C5#58514yes—
427C22×C5C2611yescharacteristic
428C2×D5—28no—
429C2×D5—28no—
430C2×D5—28no—
431C2×D5—416no—
432C4×C5—22no—
433C4×C5—22no—
434C4×C5—28no—
435C4×C5—24no—
436C4×C5—24no—
437C4×C5—28no—
438Dic5—28no—
439Dic5—28no—
440Dic5—28no—
441C24—54no—
442C24C23×D511yescharacteristic
443C2×Q8—516no—
444C2×Q8—58no—
445C2×Q8—208no—
446C2×Q8—204no—
447C2×D4—516no—
448C2×D4—516no—
449C2×D4—58no—
450C2×D4—1016no—
451C2×D4—208no—
452C2×D4—204no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—54no—
457C22×C4—54no—
458C22×C4—54no—
459C22×C4—516no—
460C22×C4—516no—
461C22×C4—516no—
462C22×C4—208no—
463C22×C4—208no—
464C22×C4—208no—
465C22×C4C23×D514yes—
466C22×C4C23×D512yes—
467C22×C4C23×D514yes—
468C22×C4C23×D514yes—
469C22×C4C23×D511yescharacteristic
470C22×C4C23×D511yescharacteristic
471C22×C4C23×D512yes—
472C22×C4C23×D518yes—
473C22×C4C23×D5116yes—
474C4⋊C4—516no—
475C4⋊C4—516no—
476C4⋊C4—516no—
477C4⋊C4—516no—
478C4⋊C4—516no—
479C4⋊C4—58no—
480C4⋊C4—516no—
481C4⋊C4—516no—
482C4⋊C4—516no—
483C4⋊C4—58no—
484C4⋊C4—516no—
485C4⋊C4—516no—
486C4⋊C4C23×D5116yes—
487C4⋊C4C23×D5116yes—
488C4⋊C4C23×D5116yes—
489C4⋊C4C23×D518yes—
490C4⋊C4C23×D5116yes—
491C4⋊C4C23×D5116yes—
492C4⋊C4C23×D518yes—
493C22⋊C4—516no—
494C22⋊C4—516no—
495C22⋊C4—516no—
496C22⋊C4—516no—
497C22⋊C4—516no—
498C22⋊C4—516no—
499C22⋊C4—516no—
500C22⋊C4—516no—
501C22⋊C4—516no—
502C42—516no—
503C42—516no—
504C42—516no—
505C42C23×D514yes—
506C42C23×D518yes—
507C42C23×D514yes—
508C2×C5—24no—
509C2×C5#355314yescomplemented
510C2×C5C22×2−1+411yescharacteristic
511C2×C5C2×#58512yes—
512D5—48no—
513C23—24no—
514C23—58no—
515C23—108no—
516C23—108no—
517C23—108no—
518C23#414714yes—
519C23C24×D512yes—
520C23C24×D511yescharacteristic
521Q8—2016no—
522Q8—208no—
523D4—1032no—
524D4—2016no—
525D4—208no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—416no—
535C2×C4—58no—
536C2×C4—58no—
537C2×C4—58no—
538C2×C4—108no—
539C2×C4—108no—
540C2×C4—108no—
541C2×C4—2016no—
542C2×C4—2016no—
543C2×C4—2016no—
544C2×C4C24×D514yes—
545C2×C4C24×D518yes—
546C2×C4C24×D518yes—
547C2×C4C24×D512yes—
548C2×C4C24×D512yes—
549C2×C4C24×D514yes—
550C5C2×#355311yescharacteristic, complemented, Sylow
551C22—24no—
552C22—28no—
553C22—108no—
554C22—108no—
555C22—108no—
556C22—2016no—
557C22C2×#414714yes—
558C22C2×#414712yes—
559C22#6628314yes—
560C22C25×D511yescharacteristic
561C4—22no—
562C4—22no—
563C4—24no—
564C4—28no—
565C4—28no—
566C4—24no—
567C4—108no—
568C4—108no—
569C4—108no—
570C2—24no—
571C2—208no—
572C2#1061800314yesdirect product factor
573C2C22×#414711yescharacteristic
574C2C2×#6628312yes—
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC22×C5×#125C2
socleC23×C5C25
hypercenterC22×#125D5
hypocenterC5C2×#3553
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C5×#125C2semidirect
2. #10618003C2direct (N × H)
3. C2×#66033C2semidirect
4. C2×#66034C2semidirect
5. C2×#66027C2semidirect
6. C2×#66026C2semidirect
7. C2×C5⋊#564C2semidirect
8. C2×#66030C2semidirect
9. C2×#65848C2semidirect
10. C2×#65849C2semidirect
11. C2×#65845C2semidirect
12. C2×#65813C2semidirect
17. C2×C5×#125C22semidirect
29. #66034C22semidirect
30. #66033C22semidirect
31. #66030C22semidirect
32. #66027C22semidirect
33. #66026C22semidirect
34. C5⋊#564C22semidirect
35. #65849C22semidirect
36. #65848C22semidirect
41. #65845C22semidirect
62. #65813C22semidirect
171. C22×#125D5semidirect
265. C2×#125C2×D5semidirect
509. C2×C5#3553semidirect
550. C5C2×#3553semidirect
572. C2#10618003direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1252
#106180032
C2×#660332
C2×#660342
C2×#660272
C2×#660262
C2×C5⋊#5642
C2×#660302
C2×#658482
C2×#658492
C2×#658452
C2×#658132
C2×#35535