Finite Group Databasebeta

← C2×#10618876 #13006981

Subgroups — C2×#10618876 #13006981

Order 1280.

7086 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#555C211yesmaximal, characteristic, complemented
2#10618876C2132yesmaximal, direct product factor
3C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
4C2×#66056C211yesmaximal, characteristic, complemented
5C2×#66009C214yesmaximal, complemented
6C2×#66006C214yesmaximal, complemented
7C2×#65945C214yesmaximal, complemented
8C2×#65937C214yesmaximal, complemented
9C2×#65891C212yesmaximal, complemented
10C2×#65877C212yesmaximal, complemented
11C2×#65863C214yesmaximal, complemented
12C2×#65853C214yesmaximal, complemented
13C22×Q8×D5C2212yes—
14C22×#711C2214yes—
15C5×#555C22116yescomplemented
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#124C2214yes—
18C2×C5×#121C2214yes—
19C2×C4×C5×D4C2212yes—
20C2×C5×#117C2214yes—
21C2×#4090C2212yes—
22C2×#4090C2214yes—
23C2×#4084C2212yes—
24C2×#4082C2212yes—
25C2×#4082C2214yes—
26C2×#4081C2218yes—
27C2×D4×Dic5C2214yes—
28C2×C4×C22⋊D5C2214yes—
29C2×#4072C2214yes—
30C22×C4⋊Dic5C2214yes—
31C2×#4070C2214yes—
32D5×C4⋊Q8C22116yescomplemented
33#66056C22116yescomplemented
34#66009C22164yescomplemented
35#66006C22164yescomplemented
36#65945C22164yescomplemented
37#65937C22164yescomplemented
38#65891C22132yescomplemented
39#65877C22132yescomplemented
40#65863C22164yescomplemented
41#65853C22164yescomplemented
42C2×#4043C2212yes—
43C2×D5×C4⋊C4C2214yes—
44C2×#4036C2212yes—
45C2×#4036C2214yes—
46C2×#4034C2214yes—
47C2×#4034C2214yes—
48C2×#4033C2214yes—
49C2×#4029C2218yes—
50C2×#4029C2214yes—
51C2×#4025C2218yes—
52C2×#4025C2214yes—
53C2×#4024C2218yes—
54C2×#4024C2218yes—
55C2×#4024C2218yes—
56C2×#4018C2214yes—
57C2×C42×D5C2211yescharacteristic
58C2×#4016C2218yes—
59C2×#4015C2214yes—
60C2×#4015C2211yescharacteristic
61C2×C4×Dic10C2212yes—
62C2×#3544—51nomaximal, Sylow
63C22×C5×Q8C2312yes—
64C22×C5×D4C2312yes—
65C22×C5×D4C2314yes—
66C23×C4×C5C2314yes—
67C22×C22⋊D5C2314yes—
68C23×Dic5C2318yes—
69C2×Q8×D5—48no—
70C2×Q8×D5C23116yes—
71C2×#711—416no—
72C2×#711C23132yes—
73C22×C4×D5C2311yescharacteristic
74C22×C4×D5C2312yes—
75C22×C4×D5C2314yes—
76C22×Dic10C2312yes—
77C22×Dic10C2312yes—
78C22×Dic10C2314yes—
79C22×Dic10C2314yes—
80C5×C4⋊Q8C2318yes—
81C5×#124C23132yes—
82C5×#121C23132yes—
83C4×C5×D4C23116yes—
84C5×#117C23132yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C22⋊C4C2318yes—
88C2×C5×C22⋊C4C2314yes—
89C2×C42×C5C2311yescharacteristic
90C2×C42×C5C2314yes—
91C2×C22⋊Dic5C2314yes—
92C2×C22⋊Dic5C2314yes—
93C2×C22⋊Dic5C2318yes—
94C2×C22⋊Dic5C2314yes—
95#4090C23132yes—
96#4090C23116yes—
97#4084C23116yes—
98#4082C23116yes—
99#4082C23132yes—
100#4081C23164yes—
101D4×Dic5C23132yes—
102C4×C22⋊D5C23132yes—
103C2×#685C2314yes—
104#4072C23132yes—
105C2×C4⋊Dic5C2312yes—
106C2×C4⋊Dic5C2314yes—
107C2×C4⋊Dic5C2318yes—
108C2×C4⋊Dic5C2314yes—
109C2×C4⋊Dic5C2314yes—
110C2×C4⋊Dic5C23132yes—
111#4070C23132yes—
112C2×#683C2314yes—
113C2×#683C2314yes—
114C2×#683C2318yes—
115C2×#683C2318yes—
116C2×#683C2318yes—
117C2×C4×Dic5C2314yes—
118C2×C4×Dic5C2311yescharacteristic
119C2×C4×Dic5C2312yes—
120C2×C4×Dic5C2314yes—
121#4043C23116yes—
122D5×C4⋊C4C23132yes—
123#4036C23116yes—
124#4036C23132yes—
125#4034C23132yes—
126#4034C23132yes—
127#4033C23132yes—
128#4029C23164yes—
129#4029C23132yes—
130#4025C23164yes—
131#4025C23132yes—
132#4024C23164yes—
133#4024C23164yes—
134#4024C23164yes—
135#4018C23132yes—
136C42×D5C2318yes—
137#4016C23164yes—
138#4015C2318yes—
139#4015C23132yes—
140C4×Dic10C23116yes—
141#3544—532no—
142C2×#563—54no—
143C2×Q82—51no—
144C2×#555D511yescharacteristic, complemented
145C2×#554—52no—
146C2×#552—54no—
147C2×#547—54no—
148C2×#544—54no—
149C2×#542—54no—
150C2×#536—54no—
151C22×C4⋊Q8—51no—
152C2×#529—52no—
153C24×C5C2414yes—
154C23×D5C2411yescharacteristic
155C2×C5×Q8—44no—
156C2×C5×Q8C2418yes—
157C2×C5×D4—28no—
158C2×C5×D4—48no—
159C2×C5×D4C2418yes—
160C2×C5×D4C24116yes—
161C22×C4×C5—216no—
162C22×C4×C5C2414yes—
163C22×C4×C5C2411yescharacteristic
164C22×C4×C5C2414yes—
165C22×C4×C5C24116yes—
166C22×C4×C5C2412yes—
167C2×C22⋊D5—216no—
168C2×C22⋊D5C24116yes—
169C22×Dic5—416no—
170C22×Dic5C2411yescharacteristic
171C22×Dic5C2414yes—
172C22×Dic5C2412yes—
173C22×Dic5C2414yes—
174C22×Dic5C2414yes—
175C22×Dic5C24132yes—
176Q8×D5—432no—
177#711—464no—
178C2×C4×D5—216no—
179C2×C4×D5—24no—
180C2×C4×D5—28no—
181C2×C4×D5C24116yes—
182C2×C4×D5C2414yes—
183C2×C4×D5C2418yes—
184C2×Dic10—28no—
185C2×Dic10—48no—
186C2×Dic10—48no—
187C2×Dic10—44no—
188C2×Dic10C2418yes—
189C2×Dic10C24116yes—
190C2×Dic10C2418yes—
191C2×Dic10C24116yes—
192C5×C4⋊C4C2418yes—
193C5×C4⋊C4C24116yes—
194C5×C22⋊C4C24132yes—
195C5×C22⋊C4C24116yes—
196C42×C5C24116yes—
197C42×C5C2414yes—
198C22⋊Dic5C24116yes—
199C22⋊Dic5C24116yes—
200C22⋊Dic5C24132yes—
201C22⋊Dic5C24116yes—
202#685C24116yes—
203C4⋊Dic5C2418yes—
204C4⋊Dic5C24116yes—
205C4⋊Dic5C24132yes—
206C4⋊Dic5C24116yes—
207C4⋊Dic5C24116yes—
208#683C24116yes—
209#683C24116yes—
210#683C24132yes—
211#683C24132yes—
212#683C24132yes—
213C4×Dic5C24116yes—
214C4×Dic5C2414yes—
215C4×Dic5C2418yes—
216C4×Dic5C24116yes—
217C22×#41—54no—
218C23×Q8—52no—
219#563—564no—
220Q82—516no—
221#555C2×D5116yescomplemented
222#554—532no—
223#552—564no—
224#547—564no—
225#544—564no—
226#542—564no—
227#536—564no—
228C2×C4⋊Q8—52no—
229C2×C4⋊Q8—54no—
230C2×C4⋊Q8—516no—
231C2×C4⋊Q8—54no—
232C2×C4⋊Q8—54no—
233C2×C4⋊Q8—51no—
234C2×C4⋊Q8—54no—
235C2×C4⋊Q8C2×D511yescharacteristic
236#529—532no—
237C2×#126—54no—
238C2×#126—58no—
239C2×#125—52no—
240C2×#125—54no—
241C2×#125—58no—
242C2×#124—52no—
243C2×#124—54no—
244C2×#124C2×D514yes—
245C2×#123—58no—
246C2×#123—54no—
247C2×#123—58no—
248C2×C22⋊Q8—54no—
249C2×C22⋊Q8—52no—
250C2×C22⋊Q8—58no—
251C2×C22⋊Q8—58no—
252C2×C22⋊Q8—58no—
253C2×#121—52no—
254C2×#121C2×D514yes—
255C2×C4×Q8—54no—
256C2×C4×Q8—52no—
257C2×C4×D4—54no—
258C2×C4×D4—54no—
259C2×C4×D4C2×D512yes—
260C2×#117—54no—
261C2×#117—54no—
262C2×#117C2×D514yes—
263C22×C4⋊C4—54no—
264C22×C4⋊C4—54no—
265C22×C42—51no—
266C23×C5—28no—
267C23×C5—28no—
268C23×C5—28no—
269C23×C5C2518yes—
270C23×C5C2511yescharacteristic
271C22×D5—24no—
272C22×D5H(5,𝔽2)14yes—
273C22×D5C2512yes—
274C5×Q8—48no—
275C5×D4—216no—
276C5×D4—416no—
277C2×C4×C5—28no—
278C2×C4×C5—22no—
279C2×C4×C5—28no—
280C2×C4×C5—232no—
281C2×C4×C52−1+418yes—
282C2×C4×C5C2512yes—
283C2×C4×C5C2518yes—
284C2×C4×C5C2518yes—
285C2×C4×C5C2514yes—
286C22⋊D5—232no—
287C2×Dic5—22no—
288C2×Dic5—28no—
289C2×Dic5—28no—
290C2×Dic5—28no—
291C2×Dic5—24no—
292C2×Dic5—432no—
293C2×Dic5C2518yes—
294C2×Dic5C2514yes—
295C2×Dic5C2512yes—
296C2×Dic5C2518yes—
297C2×Dic5C2518yes—
298C4×D5—232no—
299C4×D5—28no—
300C4×D5—216no—
301Dic10—216no—
302Dic10—416no—
303Dic10—416no—
304Dic10—48no—
305C2×#41—532no—
306C2×#41—2016no—
307C22×Q8—516no—
308C22×Q8—52no—
309C22×Q8—52no—
310C22×Q8—54no—
311C22×Q8—54no—
312C22×Q8—208no—
313C22×Q8C22×D512yes—
314C22×D4—54no—
315C22×D4C22×D512yes—
316C22×D4C22×D514yes—
317C23×C4—58no—
318C23×C4—54no—
319C23×C4—52no—
320C23×C4—51no—
321C23×C4C22×D514yes—
322C4⋊Q8—516no—
323C4⋊Q8—532no—
324C4⋊Q8—532no—
325C4⋊Q8—532no—
326C4⋊Q8—58no—
327C4⋊Q8—532no—
328C4⋊Q8C22×D518yes—
329#126—532no—
330#126—564no—
331#125—516no—
332#125—532no—
333#125—564no—
334#124—516no—
335#124—532no—
336#124C22×D5132yes—
337#123—564no—
338#123—532no—
339#123—564no—
340C22⋊Q8—532no—
341C22⋊Q8—516no—
342C22⋊Q8—564no—
343C22⋊Q8—564no—
344C22⋊Q8—564no—
345#121—516no—
346#121C22×D5132yes—
347C4×Q8—532no—
348C4×Q8—516no—
349C4×D4—532no—
350C4×D4—532no—
351C4×D4C22×D5116yes—
352#117—532no—
353#117—532no—
354#117C22×D5132yes—
355C2×C4⋊C4—54no—
356C2×C4⋊C4—52no—
357C2×C4⋊C4—58no—
358C2×C4⋊C4—58no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—58no—
361C2×C4⋊C4—54no—
362C2×C4⋊C4—54no—
363C2×C4⋊C4—54no—
364C2×C4⋊C4—54no—
365C2×C4⋊C4—58no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4C22×D514yes—
368C2×C4⋊C4C22×D512yes—
369C2×C22⋊C4—54no—
370C2×C22⋊C4—54no—
371C2×C22⋊C4—58no—
372C2×C22⋊C4—54no—
373C2×C22⋊C4—54no—
374C2×C22⋊C4C22×D518yes—
375C2×C22⋊C4C22×D514yes—
376C2×C42—52no—
377C2×C42—54no—
378C2×C42—54no—
379C2×C42—51no—
380C2×C42—58no—
381C2×C42C22×D514yes—
382C2×C42C22×D511yescharacteristic
383C22×C5—28no—
384C22×C5—28no—
385C22×C5—28no—
386C22×C5—416no—
387C22×C5C2×H(5,𝔽2)12yes—
388C22×C5C2×2−1+414yes—
389C22×C5C2611yescharacteristic
390C2×D5—24no—
391C2×D5—28no—
392C2×D5C2×H(5,𝔽2)14yes—
393C4×C5—28no—
394C4×C5—22no—
395C4×C5—28no—
396C4×C5C2×2−1+418yes—
397Dic5—22no—
398Dic5—28no—
399Dic5—28no—
400Dic5—28no—
401Dic5—24no—
402C24—51no—
403C24C23×D514yes—
404#41—2064no—
405C2×Q8—44no—
406C2×Q8—58no—
407C2×Q8—58no—
408C2×Q8—516no—
409C2×Q8—516no—
410C2×Q8—108no—
411C2×Q8—208no—
412C2×Q8—204no—
413C2×Q8—208no—
414C2×Q8—2032no—
415C2×Q8C23×D518yes—
416C2×D4—28no—
417C2×D4—48no—
418C2×D4—516no—
419C2×D4—1016no—
420C2×D4C23×D518yes—
421C2×D4C23×D5116yes—
422C22×C4—216no—
423C22×C4—54no—
424C22×C4—52no—
425C22×C4—51no—
426C22×C4—54no—
427C22×C4—54no—
428C22×C4—532no—
429C22×C4—58no—
430C22×C4—54no—
431C22×C4—516no—
432C22×C4—104no—
433C22×C4—1016no—
434C22×C4—108no—
435C22×C4—2016no—
436C22×C4C23×D5116yes—
437C22×C4C23×D511yescharacteristic
438C22×C4C23×D514yes—
439C22×C4C23×D514yes—
440C22×C4C23×D512yes—
441C4⋊C4—516no—
442C4⋊C4—58no—
443C4⋊C4—532no—
444C4⋊C4—532no—
445C4⋊C4—532no—
446C4⋊C4—516no—
447C4⋊C4—516no—
448C4⋊C4—516no—
449C4⋊C4—516no—
450C4⋊C4—532no—
451C4⋊C4C23×D5116yes—
452C4⋊C4C23×D518yes—
453C22⋊C4—516no—
454C22⋊C4—516no—
455C22⋊C4—532no—
456C22⋊C4—516no—
457C22⋊C4—516no—
458C22⋊C4C23×D5132yes—
459C22⋊C4C23×D5116yes—
460C42—58no—
461C42—54no—
462C42—516no—
463C42—516no—
464C42C23×D514yes—
465C42C23×D5116yes—
466C2×C5—48no—
467C2×C5#354414yescomplemented
468C2×C5C22×H(5,𝔽2)11yescharacteristic
469C2×C5C22×2−1+412yes—
470D5—24no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—54no—
475C23—52no—
476C23—104no—
477C23C24×D518yes—
478C23C24×D511yescharacteristic
479Q8—48no—
480Q8—1016no—
481Q8—2016no—
482Q8—208no—
483Q8—2016no—
484D4—216no—
485D4—416no—
486D4—1032no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—232no—
491C2×C4—58no—
492C2×C4—54no—
493C2×C4—52no—
494C2×C4—58no—
495C2×C4—58no—
496C2×C4—108no—
497C2×C4—1032no—
498C2×C4—104no—
499C2×C4—108no—
500C2×C4—108no—
501C2×C4—102no—
502C2×C4—108no—
503C2×C4—1016no—
504C2×C4—2032no—
505C2×C4#415018yes—
506C2×C4C24×D512yes—
507C2×C4C24×D518yes—
508C2×C4C24×D518yes—
509C2×C4C24×D514yes—
510C5C2×#354411yescharacteristic, complemented, Sylow
511C22—28no—
512C22—28no—
513C22—28no—
514C22—416no—
515C22—54no—
516C22—104no—
517C22—108no—
518C22C2×#415014yes—
519C22D5×H(5,𝔽2)12yes—
520C22C25×D511yescharacteristic
521C4—28no—
522C4—28no—
523C4—22no—
524C4—104no—
525C4—108no—
526C4—108no—
527C4—102no—
528C4—108no—
529C4C2×#415018yes—
530C2—48no—
531C2—104no—
532C2#1061887614yesdirect product factor
533C2C22×#415012yes—
534C2C2×D5×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×#555C2
socleC23×C5C25
hypercenterC2×#555D5
hypocenterC5C2×#3544
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×C5×#555C2semidirect
2. #10618876C2direct (N × H)
3. C2×D5×C4⋊Q8C2semidirect
4. C2×#66056C2semidirect
5. C2×#66009C2semidirect
6. C2×#66006C2semidirect
7. C2×#65945C2semidirect
8. C2×#65937C2semidirect
9. C2×#65891C2semidirect
10. C2×#65877C2semidirect
11. C2×#65863C2semidirect
12. C2×#65853C2semidirect
15. C5×#555C22semidirect
32. D5×C4⋊Q8C22semidirect
33. #66056C22semidirect
34. #66009C22semidirect
35. #66006C22semidirect
36. #65945C22semidirect
37. #65937C22semidirect
38. #65891C22semidirect
39. #65877C22semidirect
40. #65863C22semidirect
41. #65853C22semidirect
144. C2×#555D5semidirect
221. #555C2×D5semidirect
467. C2×C5#3544semidirect
510. C5C2×#3544semidirect
532. C2#10618876direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5552
#106188762
C2×D5×C4⋊Q82
C2×#660562
C2×#660092
C2×#660062
C2×#659452
C2×#659372
C2×#658912
C2×#658772
C2×#658632
C2×#658532
C2×#35445