Finite Group Databasebeta

← C2×C5×#5914 #14511410

Subgroups — C2×C5×#5914 #14511410

Order 1920.

1810 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×A4C211yesmaximal, characteristic, complemented
2C2×C5×GL(2,ℤ/4)C212yesmaximal, complemented
3C22×C5×SL(2,ℤ/4)C212yesmaximal, complemented
4C5×#5914C218yesmaximal, direct product factor
5C2×C4×C5×S4C211yesmaximal, characteristic, complemented
6C2×C5×A4⋊Q8C211yesmaximal, characteristic, complemented
7C2×C5×#546—31nomaximal, Hall
8C23×C5×A4C2212yes—
9C22×C5×S4C2211yescharacteristic
10C2×C5×#237—41nomaximal
11C5×D4×A4C2214yescomplemented
12C2×C4×C5×A4C2211yescharacteristic
13C5×GL(2,ℤ/4)C2218yescomplemented
14C2×C5×SL(2,ℤ/4)C2212yes—
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C2×C5×SL(2,ℤ/4)C2218yescomplemented
17C4×C5×S4C2214yescomplemented
18C5×A4⋊Q8C2214yescomplemented
19C2×#5914C511yesmaximal, characteristic, direct product factor, Hall
20C22×C5×#41—31no—
21C23×C5×D4D311yescharacteristic, complemented
22C5×#546—38no—
23C5×#546—38no—
24C2×C5×#124—31no—
25C2×C5×#123—32no—
26C2×C5×C22⋊Q8—31no—
27C2×C5×#121—31no—
28C2×C5×#121—32no—
29C2×C5×#120—32no—
30C2×C4×C5×D4—31no—
31C2×C4×C5×D4—31no—
32C22×C5×C22⋊C4—32no—
33C22×C5×A4—24no—
34C22×C5×A4C2314yes—
35C22×C5×A4C2311yescharacteristic
36C2×C5×S4—22no—
37C2×C5×S4C2312yes—
38C2×C3×C5×D4—41no—
39C2×C5×#67—42no—
40C22×C5×Dic3—42no—
41C5×#237—48no—
42C2×C4×C5×D3—41no—
43C2×C5×Dic6—41no—
44C4×C5×A4C2312yes—
45C5×SL(2,ℤ/4)C2314yes—
46C5×SL(2,ℤ/4)C2312yes—
47C2×D4×A4C2×C511yescharacteristic, complemented
48C2×GL(2,ℤ/4)C2×C512yescomplemented
49C22×SL(2,ℤ/4)C2×C512yescomplemented
50#5914C2×C518yesdirect product factor
51C2×C4×S4C2×C511yescharacteristic, complemented
52C2×A4⋊Q8C2×C511yescharacteristic, complemented
53C25×C5C2×D312yes—
54C2×C5×#41—38no—
55C2×C5×#41—64no—
56C2×C5×#41—62no—
57C2×C5×#41—61no—
58C2×C5×#41—61no—
59C22×C5×Q8—31no—
60C22×C5×D4—31no—
61C22×C5×D4—32no—
62C22×C5×D4—31no—
63C22×C5×D4—34no—
64C22×C5×D4—32no—
65C22×C5×D4—31no—
66C22×C5×D4—32no—
67C22×C5×D4C2×D314yescomplemented
68C23×C4×C5—32no—
69C23×C4×C5—31no—
70C23×C4×C5—32no—
71C23×C4×C5C2×D311yescharacteristic
72C5×#124—38no—
73C5×#123—316no—
74C5×C22⋊Q8—34no—
75C5×C22⋊Q8—34no—
76C5×#121—38no—
77C5×#121—316no—
78C5×#120—38no—
79C5×#120—38no—
80C4×C5×D4—34no—
81C4×C5×D4—38no—
82C4×C5×D4—34no—
83C2×C5×C4⋊C4—31no—
84C2×C5×C4⋊C4—31no—
85C2×C5×C4⋊C4—32no—
86C2×C5×C22⋊C4—32no—
87C2×C5×C22⋊C4—32no—
88C2×C5×C22⋊C4—38no—
89C2×C5×C22⋊C4—32no—
90C2×C5×C22⋊C4—31no—
91C2×C5×C22⋊C4—32no—
92C2×C5×C22⋊C4—31no—
93C2×C5×C22⋊C4—32no—
94C2×C5×C22⋊C4—38no—
95C2×C42×C5—31no—
96C2×#546—31noSylow
97C23×C3×C5—42no—
98C22×C5×D3—41no—
99C2×C5×A4—24no—
100C2×C5×A4#4112yescomplemented
101C2×C5×A4C2411yescharacteristic
102C5×S4—22no—
103C3×C5×D4—44no—
104C2×C3×C4×C5—41no—
105C5×#67—48no—
106C2×C5×Dic3—42no—
107C2×C5×Dic3—41no—
108C2×C5×Dic3—48no—
109C4×C5×D3—44no—
110C5×Dic6—44no—
111C23×A4C22×C512yes—
112C22×S4C22×C511yescharacteristic
113C2×#237—41no—
114D4×A4C22×C514yescomplemented
115C2×C4×A4C22×C511yescharacteristic
116GL(2,ℤ/4)C22×C518yescomplemented
117C2×SL(2,ℤ/4)C22×C512yes—
118C2×SL(2,ℤ/4)C22×C511yescharacteristic
119C2×SL(2,ℤ/4)C22×C518yescomplemented
120C4×S4C22×C514yescomplemented
121A4⋊Q8C22×C514yescomplemented
122C24×C5—24no—
123C24×C5—32no—
124C24×C5—32no—
125C24×C5—34no—
126C24×C5—31no—
127C24×C5—64no—
128C24×C5C22×D314yes—
129C24×C5C22×D311yescharacteristic
130C5×#41—616no—
131C5×#41—68no—
132C5×#41—68no—
133C2×C5×Q8—34no—
134C2×C5×Q8—61no—
135C2×C5×Q8—62no—
136C2×C5×Q8—61no—
137C2×C5×D4—32no—
138C2×C5×D4—34no—
139C2×C5×D4—34no—
140C2×C5×D4—38no—
141C2×C5×D4—34no—
142C2×C5×D4—32no—
143C2×C5×D4—38no—
144C2×C5×D4—32no—
145C2×C5×D4—38no—
146C2×C5×D4—31no—
147C2×C5×D4—62no—
148C2×C5×D4—61no—
149C2×C5×D4—64no—
150C2×C5×D4—62no—
151C2×C5×D4—64no—
152C2×C5×D4—62no—
153C2×C5×D4—62no—
154C2×C5×D4—62no—
155C2×C5×D4—124no—
156C2×C5×D4S411yescharacteristic, complemented
157C22×C4×C5—31no—
158C22×C4×C5—31no—
159C22×C4×C5—32no—
160C22×C4×C5—31no—
161C22×C4×C5—31no—
162C22×C4×C5—32no—
163C22×C4×C5—31no—
164C22×C4×C5—32no—
165C22×C4×C5—38no—
166C22×C4×C5—34no—
167C22×C4×C5—38no—
168C22×C4×C5—61no—
169C22×C4×C5—62no—
170C22×C4×C5—64no—
171C22×C4×C5—62no—
172C22×C4×C5—62no—
173C22×C4×C5—61no—
174C22×C4×C5—124no—
175C22×C4×C5C22×D312yes—
176C5×C4⋊C4—34no—
177C5×C4⋊C4—34no—
178C5×C4⋊C4—38no—
179C5×C22⋊C4—32no—
180C5×C22⋊C4—34no—
181C5×C22⋊C4—38no—
182C5×C22⋊C4—38no—
183C5×C22⋊C4—38no—
184C5×C22⋊C4—34no—
185C5×C22⋊C4—38no—
186C5×C22⋊C4—32no—
187C5×C22⋊C4—34no—
188C42×C5—34no—
189C22×#41—31no—
190C23×D4C5×D311yescharacteristic, complemented
191#546—38no—
192#546—38no—
193C2×#124—31no—
194C2×#123—32no—
195C2×C22⋊Q8—31no—
196C2×#121—31no—
197C2×#121—32no—
198C2×#120—32no—
199C2×C4×D4—31no—
200C2×C4×D4—31no—
201C22×C22⋊C4—32no—
202C22×C3×C5—44no—
203C22×C3×C5—41no—
204C22×C3×C5—84no—
205C2×C5×D3—42no—
206C2×C5×D3—82no—
207C5×A4C2×#4111yescharacteristic, complemented
208C3×C4×C5—42no—
209C5×Dic3—44no—
210C5×Dic3—42no—
211C22×A4—24no—
212C22×A4C23×C514yes—
213C22×A4C23×C511yescharacteristic
214C2×S4—22no—
215C2×S4C23×C512yes—
216C2×C3×D4—41no—
217C2×#67—42no—
218C22×Dic3—42no—
219#237—48no—
220C2×C4×D3—41no—
221C2×Dic6—41no—
222C4×A4C23×C512yes—
223SL(2,ℤ/4)C23×C514yes—
224SL(2,ℤ/4)C23×C512yes—
225C23×C5—24no—
226C23×C5—31no—
227C23×C5—31no—
228C23×C5—32no—
229C23×C5—34no—
230C23×C5—34no—
231C23×C5—34no—
232C23×C5—32no—
233C23×C5—32no—
234C23×C5—64no—
235C23×C5—64no—
236C23×C5—62no—
237C23×C5—64no—
238C23×C5—64no—
239C23×C5—68no—
240C23×C5—64no—
241C23×C5—61no—
242C23×C5—62no—
243C23×C5—62no—
244C23×C5—61no—
245C23×C5—124no—
246C23×C5#23712yescomplemented
247C23×C5C2×S412yes—
248C23×C5C23×D311yescharacteristic
249C5×Q8—64no—
250C5×Q8—64no—
251C5×D4—38no—
252C5×D4—34no—
253C5×D4—64no—
254C5×D4—64no—
255C5×D4—62no—
256C5×D4—68no—
257C5×D4—62no—
258C5×D4—68no—
259C5×D4—128no—
260C5×D4C2×S414yescomplemented
261C2×C4×C5—31no—
262C2×C4×C5—32no—
263C2×C4×C5—32no—
264C2×C4×C5—32no—
265C2×C4×C5—32no—
266C2×C4×C5—32no—
267C2×C4×C5—34no—
268C2×C4×C5—32no—
269C2×C4×C5—34no—
270C2×C4×C5—61no—
271C2×C4×C5—64no—
272C2×C4×C5—61no—
273C2×C4×C5—62no—
274C2×C4×C5—68no—
275C2×C4×C5—62no—
276C2×C4×C5—64no—
277C2×C4×C5—62no—
278C2×C4×C5—68no—
279C2×C4×C5—61no—
280C2×C4×C5—62no—
281C2×C4×C5—64no—
282C2×C4×C5—128no—
283C2×C4×C5C2×S411yescharacteristic
284C25C2×C5×D312yes—
285C2×#41—38no—
286C2×#41—64no—
287C2×#41—62no—
288C2×#41—61no—
289C2×#41—61no—
290C22×Q8—31no—
291C22×D4—31no—
292C22×D4—32no—
293C22×D4—31no—
294C22×D4—34no—
295C22×D4—32no—
296C22×D4—31no—
297C22×D4—32no—
298C22×D4C2×C5×D314yescomplemented
299C23×C4—32no—
300C23×C4—31no—
301C23×C4—32no—
302C23×C4C2×C5×D311yescharacteristic
303#124—38no—
304#123—316no—
305C22⋊Q8—34no—
306C22⋊Q8—34no—
307#121—38no—
308#121—316no—
309#120—38no—
310#120—38no—
311C4×D4—34no—
312C4×D4—38no—
313C4×D4—34no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C22⋊C4—32no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—38no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4—38no—
326C2×C42—31no—
327C2×C3×C5—42no—
328C2×C3×C5—41no—
329C2×C3×C5—84no—
330C5×D3—82no—
331C23×C3—42no—
332C22×D3—41no—
333C2×A4—24no—
334C2×A4C5×#4112yescomplemented
335C2×A4C24×C511yescharacteristic
336S4—22no—
337C3×D4—44no—
338C2×C3×C4—41no—
339#67—48no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—48no—
343C4×D3—44no—
344Dic6—44no—
345C22×C5—24no—
346C22×C5—32no—
347C22×C5—31no—
348C22×C5—32no—
349C22×C5—32no—
350C22×C5—31no—
351C22×C5—31no—
352C22×C5—34no—
353C22×C5—64no—
354C22×C5—64no—
355C22×C5—68no—
356C22×C5—61no—
357C22×C5—64no—
358C22×C5—64no—
359C22×C5—62no—
360C22×C5—62no—
361C22×C5—64no—
362C22×C5—62no—
363C22×C5—124no—
364C22×C5—122no—
365C22×C5—122no—
366C22×C5C2×#23711yescharacteristic, complemented
367C22×C5C22×S414yes—
368C22×C5C22×S411yescharacteristic
369C4×C5—32no—
370C4×C5—62no—
371C4×C5—64no—
372C4×C5—62no—
373C4×C5—62no—
374C4×C5—64no—
375C4×C5C22×S412yes—
376C24—24no—
377C24—32no—
378C24—32no—
379C24—34no—
380C24—31no—
381C24—64no—
382C24C22×C5×D314yes—
383C24C22×C5×D311yescharacteristic
384#41—616no—
385#41—68no—
386#41—68no—
387C2×Q8—34no—
388C2×Q8—61no—
389C2×Q8—62no—
390C2×Q8—61no—
391C2×D4—32no—
392C2×D4—34no—
393C2×D4—34no—
394C2×D4—38no—
395C2×D4—34no—
396C2×D4—32no—
397C2×D4—38no—
398C2×D4—32no—
399C2×D4—38no—
400C2×D4—31no—
401C2×D4—62no—
402C2×D4—61no—
403C2×D4—64no—
404C2×D4—64no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—124no—
410C2×D4C5×S411yescharacteristic, complemented
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—32no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—38no—
420C22×C4—34no—
421C22×C4—38no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—64no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—124no—
429C22×C4C22×C5×D312yes—
430C4⋊C4—34no—
431C4⋊C4—34no—
432C4⋊C4—38no—
433C22⋊C4—32no—
434C22⋊C4—34no—
435C22⋊C4—38no—
436C22⋊C4—38no—
437C22⋊C4—38no—
438C22⋊C4—34no—
439C22⋊C4—38no—
440C22⋊C4—32no—
441C22⋊C4—34no—
442C42—34no—
443C3×C5—41noHall
444C22×C3—41no—
445C22×C3—44no—
446C22×C3—84no—
447C2×D3—42no—
448C2×D3—82no—
449A4C2×C5×#4111yescharacteristic, complemented
450C3×C4—42no—
451Dic3—44no—
452Dic3—42no—
453C2×C5—24no—
454C2×C5—31no—
455C2×C5—31no—
456C2×C5—32no—
457C2×C5—64no—
458C2×C5—122no—
459C2×C5#591412yesdirect product factor
460C2×C5C23×S411yescharacteristic
461C23—24no—
462C23—31no—
463C23—31no—
464C23—32no—
465C23—34no—
466C23—34no—
467C23—34no—
468C23—32no—
469C23—32no—
470C23—64no—
471C23—64no—
472C23—62no—
473C23—64no—
474C23—64no—
475C23—68no—
476C23—64no—
477C23—61no—
478C23—62no—
479C23—62no—
480C23—61no—
481C23—124no—
482C23C5×#23712yescomplemented
483C23C2×C5×S412yes—
484C23C23×C5×D311yescharacteristic
485Q8—64no—
486Q8—64no—
487D4—38no—
488D4—34no—
489D4—64no—
490D4—68no—
491D4—64no—
492D4—62no—
493D4—62no—
494D4—68no—
495D4—128no—
496D4C2×C5×S414yescomplemented
497C2×C4—31no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—32no—
503C2×C4—34no—
504C2×C4—32no—
505C2×C4—34no—
506C2×C4—61no—
507C2×C4—64no—
508C2×C4—61no—
509C2×C4—62no—
510C2×C4—68no—
511C2×C4—62no—
512C2×C4—64no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—68no—
517C2×C4—64no—
518C2×C4—128no—
519C2×C4C2×C5×S411yescharacteristic
520C2×C3—42no—
521C2×C3—41no—
522C2×C3—84no—
523D3—82no—
524C5C2×#591411yescharacteristic, direct product factor, Sylow
525C22—24no—
526C22—32no—
527C22—31no—
528C22—32no—
529C22—32no—
530C22—31no—
531C22—31no—
532C22—34no—
533C22—64no—
534C22—64no—
535C22—68no—
536C22—61no—
537C22—64no—
538C22—64no—
539C22—62no—
540C22—62no—
541C22—64no—
542C22—62no—
543C22—124no—
544C22—122no—
545C22—122no—
546C22C2×C5×#23711yescharacteristic, complemented
547C22C22×C5×S414yes—
548C22C22×C5×S411yescharacteristic
549C4—32no—
550C4—62no—
551C4—64no—
552C4—62no—
553C4—62no—
554C4—64no—
555C4C22×C5×S412yes—
556C3—41noSylow
557C2—24no—
558C2—31no—
559C2—31no—
560C2—32no—
561C2—64no—
562C2—122no—
563C2C5×#591412yesdirect product factor
564C2C23×C5×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC2×A4C24×C5
FrattiniC2C23×C5×S4
FittingC23×C5×D4D3
socleC24×C5C22×D3
hypercenterC2×C5×D4S4
hypocenterA4C2×C5×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×A4C2semidirect
2. C2×C5×GL(2,ℤ/4)C2semidirect
3. C22×C5×SL(2,ℤ/4)C2semidirect
4. C5×#5914C2direct (N × H)
5. C2×C4×C5×S4C2semidirect
6. C2×C5×A4⋊Q8C2semidirect
11. C5×D4×A4C22semidirect
13. C5×GL(2,ℤ/4)C22semidirect
16. C2×C5×SL(2,ℤ/4)C22semidirect
17. C4×C5×S4C22semidirect
18. C5×A4⋊Q8C22semidirect
19. C2×#5914C5direct (N × H)
21. C23×C5×D4D3semidirect
47. C2×D4×A4C2×C5semidirect
48. C2×GL(2,ℤ/4)C2×C5semidirect
49. C22×SL(2,ℤ/4)C2×C5semidirect
50. #5914C2×C5direct (N × H)
51. C2×C4×S4C2×C5semidirect
52. C2×A4⋊Q8C2×C5semidirect
67. C22×C5×D4C2×D3semidirect
100. C2×C5×A4#41semidirect
114. D4×A4C22×C5semidirect
116. GL(2,ℤ/4)C22×C5semidirect
119. C2×SL(2,ℤ/4)C22×C5semidirect
120. C4×S4C22×C5semidirect
121. A4⋊Q8C22×C5semidirect
156. C2×C5×D4S4semidirect
190. C23×D4C5×D3semidirect
207. C5×A4C2×#41semidirect
246. C23×C5#237semidirect
260. C5×D4C2×S4semidirect
298. C22×D4C2×C5×D3semidirect
334. C2×A4C5×#41semidirect
366. C22×C5C2×#237semidirect
410. C2×D4C5×S4semidirect
449. A4C2×C5×#41semidirect
459. C2×C5#5914direct (N × H)
482. C23C5×#237semidirect
496. D4C2×C5×S4semidirect
524. C5C2×#5914direct (N × H)
546. C22C2×C5×#237semidirect
563. C2C5×#5914direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×D4×A42
C2×C5×GL(2,ℤ/4)2
C22×C5×SL(2,ℤ/4)2
C5×#59142
C2×C4×C5×S42
C2×C5×A4⋊Q82
C2×C5×#5463
C2×C5×#2374
C2×#59145