Finite Group Databasebeta

← C2×C5×#5912 #14511408

Subgroups — C2×C5×#5912 #14511408

Order 1920.

1790 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C5×A4C211yesmaximal, characteristic, complemented
2C2×C5×GL(2,ℤ/4)C212yesmaximal, complemented
3C5×#5912C218yesmaximal, direct product factor
4C2×C5×C4⋊S4C211yesmaximal, characteristic, complemented
5C2×C4×C5×S4C212yesmaximal, complemented
6C2×C5×A4⋊Q8C211yesmaximal, characteristic, complemented
7C2×C5×#525—31nomaximal, Hall
8C23×C5×A4C2211yescharacteristic
9C22×C5×S4C2212yes—
10C2×C5×#235—41nomaximal
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C2×C4×C5×A4C2214yescomplemented
14C5×GL(2,ℤ/4)C2218yescomplemented
15C2×C5×SL(2,ℤ/4)C2212yes—
16C5×C4⋊S4C2214yescomplemented
17C4×C5×S4C2218yescomplemented
18C5×A4⋊Q8C2214yescomplemented
19C2×#5912C511yesmaximal, characteristic, direct product factor, Hall
20C22×C5×#41—31no—
21C24×C4×C5D311yescharacteristic, complemented
22C5×#525—38no—
23C5×#525—38no—
24C2×C5×#123—32no—
25C2×C5×C22⋊Q8—31no—
26C2×C5×C22⋊Q8—31no—
27C2×C5×#121—31no—
28C2×C5×#121—31no—
29C2×C5×#120—32no—
30C2×C4×C5×D4—32no—
31C2×C4×C5×D4—32no—
32C2×C5×#117—31no—
33C22×C5×A4—22no—
34C22×C5×A4C2312yes—
35C22×C5×A4C2311yescharacteristic
36C2×C5×S4—24no—
37C2×C5×S4C2314yes—
38C22×C3×C4×C5—41no—
39C2×C5×#67—42no—
40C5×#235—48no—
41C2×C5×D12—41no—
42C2×C4×C5×D3—42no—
43C2×C5×Dic6—41no—
44C4×C5×A4C2312yes—
45C4×C5×A4C2312yes—
46C5×SL(2,ℤ/4)C2314yes—
47C22×C4×A4C2×C511yescharacteristic, complemented
48C2×GL(2,ℤ/4)C2×C512yescomplemented
49#5912C2×C518yesdirect product factor
50C2×C4⋊S4C2×C511yescharacteristic, complemented
51C2×C4×S4C2×C512yescomplemented
52C2×A4⋊Q8C2×C511yescharacteristic, complemented
53C25×C5C2×D311yescharacteristic
54C2×C5×#41—38no—
55C2×C5×#41—61no—
56C2×C5×#41—61no—
57C2×C5×#41—61no—
58C2×C5×#41—64no—
59C2×C5×#41—61no—
60C22×C5×Q8—31no—
61C22×C5×D4—32no—
62C22×C5×D4—32no—
63C22×C5×D4—31no—
64C22×C5×D4—32no—
65C23×C4×C5—31no—
66C23×C4×C5—31no—
67C23×C4×C5—31no—
68C23×C4×C5—31no—
69C23×C4×C5—34no—
70C23×C4×C5—32no—
71C23×C4×C5C2×D311yescharacteristic
72C23×C4×C5C2×D311yescharacteristic
73C23×C4×C5C2×D314yescomplemented
74C5×#123—316no—
75C5×C22⋊Q8—34no—
76C5×C22⋊Q8—38no—
77C5×C22⋊Q8—34no—
78C5×#121—34no—
79C5×#121—38no—
80C5×#121—34no—
81C5×#120—38no—
82C5×#120—38no—
83C4×C5×D4—38no—
84C4×C5×D4—316no—
85C4×C5×D4—38no—
86C5×#117—38no—
87C2×C5×C4⋊C4—31no—
88C2×C5×C4⋊C4—31no—
89C2×C5×C4⋊C4—32no—
90C2×C5×C4⋊C4—32no—
91C2×C5×C22⋊C4—32no—
92C2×C5×C22⋊C4—32no—
93C2×C5×C22⋊C4—32no—
94C2×C5×C22⋊C4—32no—
95C2×C5×C22⋊C4—32no—
96C2×C42×C5—32no—
97C2×#525—31noSylow
98C23×C3×C5—41no—
99C22×C5×D3—42no—
100C2×C5×A4—22no—
101C2×C5×A4#4112yescomplemented
102C2×C5×A4C2411yescharacteristic
103C5×S4—24no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—44no—
107C5×#67—48no—
108C2×C5×Dic3—42no—
109C5×D12—44no—
110C4×C5×D3—48no—
111C5×Dic6—44no—
112C23×A4C22×C511yescharacteristic
113C22×S4C22×C512yes—
114C2×#235—41no—
115C2×C4×A4C22×C511yescharacteristic
116C2×C4×A4C22×C511yescharacteristic
117C2×C4×A4C22×C514yescomplemented
118GL(2,ℤ/4)C22×C518yescomplemented
119C2×SL(2,ℤ/4)C22×C512yes—
120C4⋊S4C22×C514yescomplemented
121C4×S4C22×C518yescomplemented
122A4⋊Q8C22×C514yescomplemented
123C24×C5—22no—
124C24×C5—31no—
125C24×C5—31no—
126C24×C5—32no—
127C24×C5—32no—
128C24×C5—62no—
129C24×C5C22×D312yes—
130C24×C5C22×D311yescharacteristic
131C5×#41—68no—
132C5×#41—68no—
133C5×#41—68no—
134C5×#41—68no—
135C2×C5×Q8—34no—
136C2×C5×Q8—61no—
137C2×C5×Q8—62no—
138C2×C5×Q8—61no—
139C2×C5×D4—34no—
140C2×C5×D4—38no—
141C2×C5×D4—38no—
142C2×C5×D4—34no—
143C2×C5×D4—34no—
144C2×C5×D4—61no—
145C2×C5×D4—62no—
146C2×C5×D4—64no—
147C2×C5×D4—68no—
148C2×C5×D4—64no—
149C2×C5×D4—62no—
150C2×C5×D4—64no—
151C2×C5×D4—62no—
152C2×C5×D4—61no—
153C22×C4×C5—31no—
154C22×C4×C5—31no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5—31no—
158C22×C4×C5—34no—
159C22×C4×C5—34no—
160C22×C4×C5—31no—
161C22×C4×C5—31no—
162C22×C4×C5—32no—
163C22×C4×C5—34no—
164C22×C4×C5—32no—
165C22×C4×C5—34no—
166C22×C4×C5—34no—
167C22×C4×C5—32no—
168C22×C4×C5—32no—
169C22×C4×C5—32no—
170C22×C4×C5—38no—
171C22×C4×C5—62no—
172C22×C4×C5—62no—
173C22×C4×C5—61no—
174C22×C4×C5—62no—
175C22×C4×C5—62no—
176C22×C4×C5—64no—
177C22×C4×C5—62no—
178C22×C4×C5S411yescharacteristic, complemented
179C22×C4×C5C22×D312yes—
180C22×C4×C5C22×D312yes—
181C5×C4⋊C4—34no—
182C5×C4⋊C4—34no—
183C5×C4⋊C4—38no—
184C5×C4⋊C4—38no—
185C5×C22⋊C4—38no—
186C5×C22⋊C4—38no—
187C5×C22⋊C4—34no—
188C5×C22⋊C4—38no—
189C5×C22⋊C4—38no—
190C5×C22⋊C4—34no—
191C42×C5—38no—
192C22×#41—31no—
193C24×C4C5×D311yescharacteristic, complemented
194#525—38no—
195#525—38no—
196C2×#123—32no—
197C2×C22⋊Q8—31no—
198C2×C22⋊Q8—31no—
199C2×#121—31no—
200C2×#121—31no—
201C2×#120—32no—
202C2×C4×D4—32no—
203C2×C4×D4—32no—
204C2×#117—31no—
205C22×C3×C5—41no—
206C22×C3×C5—42no—
207C22×C3×C5—82no—
208C2×C5×D3—44no—
209C2×C5×D3—84no—
210C5×A4C2×#4111yescharacteristic, complemented
211C3×C4×C5—42no—
212C3×C4×C5—42no—
213C5×Dic3—44no—
214C22×A4—22no—
215C22×A4C23×C512yes—
216C22×A4C23×C511yescharacteristic
217C2×S4—24no—
218C2×S4C23×C514yes—
219C22×C3×C4—41no—
220C2×#67—42no—
221#235—48no—
222C2×D12—41no—
223C2×C4×D3—42no—
224C2×Dic6—41no—
225C4×A4C23×C512yes—
226C4×A4C23×C512yes—
227SL(2,ℤ/4)C23×C514yes—
228C23×C5—22no—
229C23×C5—31no—
230C23×C5—31no—
231C23×C5—32no—
232C23×C5—32no—
233C23×C5—34no—
234C23×C5—32no—
235C23×C5—32no—
236C23×C5—31no—
237C23×C5—34no—
238C23×C5—62no—
239C23×C5—62no—
240C23×C5—62no—
241C23×C5—61no—
242C23×C5—64no—
243C23×C5—62no—
244C23×C5—64no—
245C23×C5—62no—
246C23×C5—64no—
247C23×C5—64no—
248C23×C5—62no—
249C23×C5#23512yescomplemented
250C23×C5C2×S411yescharacteristic
251C23×C5C23×D311yescharacteristic
252C5×Q8—64no—
253C5×Q8—64no—
254C5×D4—64no—
255C5×D4—68no—
256C5×D4—64no—
257C5×D4—616no—
258C5×D4—68no—
259C5×D4—64no—
260C5×D4—68no—
261C5×D4—64no—
262C2×C4×C5—31no—
263C2×C4×C5—31no—
264C2×C4×C5—34no—
265C2×C4×C5—32no—
266C2×C4×C5—32no—
267C2×C4×C5—32no—
268C2×C4×C5—32no—
269C2×C4×C5—34no—
270C2×C4×C5—34no—
271C2×C4×C5—32no—
272C2×C4×C5—32no—
273C2×C4×C5—34no—
274C2×C4×C5—34no—
275C2×C4×C5—34no—
276C2×C4×C5—61no—
277C2×C4×C5—64no—
278C2×C4×C5—61no—
279C2×C4×C5—68no—
280C2×C4×C5—62no—
281C2×C4×C5—64no—
282C2×C4×C5—64no—
283C2×C4×C5—62no—
284C2×C4×C5—68no—
285C2×C4×C5C2×S411yescharacteristic
286C2×C4×C5C2×S411yescharacteristic
287C2×C4×C5C2×S414yescomplemented
288C25C2×C5×D311yescharacteristic
289C2×#41—38no—
290C2×#41—61no—
291C2×#41—61no—
292C2×#41—61no—
293C2×#41—64no—
294C2×#41—61no—
295C22×Q8—31no—
296C22×D4—32no—
297C22×D4—32no—
298C22×D4—31no—
299C22×D4—32no—
300C23×C4—31no—
301C23×C4—31no—
302C23×C4—31no—
303C23×C4—31no—
304C23×C4—34no—
305C23×C4—32no—
306C23×C4C2×C5×D311yescharacteristic
307C23×C4C2×C5×D311yescharacteristic
308C23×C4C2×C5×D314yescomplemented
309#123—316no—
310C22⋊Q8—34no—
311C22⋊Q8—38no—
312C22⋊Q8—34no—
313#121—34no—
314#121—38no—
315#121—34no—
316#120—38no—
317#120—38no—
318C4×D4—38no—
319C4×D4—316no—
320C4×D4—38no—
321#117—38no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—32no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—32no—
331C2×C42—32no—
332C2×C3×C5—42no—
333C2×C3×C5—41no—
334C2×C3×C5—82no—
335C5×D3—84no—
336C23×C3—41no—
337C22×D3—42no—
338C2×A4—22no—
339C2×A4C5×#4112yescomplemented
340C2×A4C24×C511yescharacteristic
341S4—24no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—44no—
345#67—48no—
346C2×Dic3—42no—
347D12—44no—
348C4×D3—48no—
349Dic6—44no—
350C22×C5—22no—
351C22×C5—32no—
352C22×C5—31no—
353C22×C5—32no—
354C22×C5—32no—
355C22×C5—31no—
356C22×C5—31no—
357C22×C5—34no—
358C22×C5—32no—
359C22×C5—62no—
360C22×C5—62no—
361C22×C5—64no—
362C22×C5—61no—
363C22×C5—62no—
364C22×C5—64no—
365C22×C5—64no—
366C22×C5—64no—
367C22×C5—62no—
368C22×C5—64no—
369C22×C5—124no—
370C22×C5—124no—
371C22×C5C2×#23511yescharacteristic, complemented
372C22×C5C22×S412yes—
373C22×C5C22×S411yescharacteristic
374C4×C5—32no—
375C4×C5—32no—
376C4×C5—64no—
377C4×C5—64no—
378C4×C5—64no—
379C4×C5C22×S412yes—
380C4×C5C22×S412yes—
381C24—22no—
382C24—31no—
383C24—31no—
384C24—32no—
385C24—32no—
386C24—62no—
387C24C22×C5×D312yes—
388C24C22×C5×D311yescharacteristic
389#41—68no—
390#41—68no—
391#41—68no—
392#41—68no—
393C2×Q8—34no—
394C2×Q8—61no—
395C2×Q8—62no—
396C2×Q8—61no—
397C2×D4—34no—
398C2×D4—38no—
399C2×D4—38no—
400C2×D4—34no—
401C2×D4—34no—
402C2×D4—61no—
403C2×D4—62no—
404C2×D4—64no—
405C2×D4—62no—
406C2×D4—64no—
407C2×D4—68no—
408C2×D4—64no—
409C2×D4—62no—
410C2×D4—61no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—34no—
417C22×C4—34no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—38no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4C5×S411yescharacteristic, complemented
437C22×C4C22×C5×D312yes—
438C22×C4C22×C5×D312yes—
439C4⋊C4—34no—
440C4⋊C4—34no—
441C4⋊C4—38no—
442C4⋊C4—38no—
443C22⋊C4—38no—
444C22⋊C4—38no—
445C22⋊C4—34no—
446C22⋊C4—38no—
447C22⋊C4—38no—
448C22⋊C4—34no—
449C42—38no—
450C3×C5—41noHall
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—82no—
454C2×D3—44no—
455C2×D3—84no—
456A4C2×C5×#4111yescharacteristic, complemented
457C3×C4—42no—
458C3×C4—42no—
459Dic3—44no—
460C2×C5—22no—
461C2×C5—31no—
462C2×C5—31no—
463C2×C5—32no—
464C2×C5—62no—
465C2×C5—124no—
466C2×C5#591212yesdirect product factor
467C2×C5C23×S411yescharacteristic
468C23—22no—
469C23—31no—
470C23—31no—
471C23—32no—
472C23—32no—
473C23—34no—
474C23—32no—
475C23—32no—
476C23—31no—
477C23—34no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—61no—
482C23—64no—
483C23—62no—
484C23—64no—
485C23—62no—
486C23—64no—
487C23—64no—
488C23—62no—
489C23C5×#23512yescomplemented
490C23C2×C5×S411yescharacteristic
491C23C23×C5×D311yescharacteristic
492Q8—64no—
493Q8—64no—
494D4—64no—
495D4—68no—
496D4—68no—
497D4—64no—
498D4—616no—
499D4—64no—
500D4—68no—
501D4—64no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—34no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—34no—
510C2×C4—34no—
511C2×C4—32no—
512C2×C4—32no—
513C2×C4—34no—
514C2×C4—34no—
515C2×C4—34no—
516C2×C4—61no—
517C2×C4—64no—
518C2×C4—61no—
519C2×C4—68no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—68no—
525C2×C4C2×C5×S411yescharacteristic
526C2×C4C2×C5×S411yescharacteristic
527C2×C4C2×C5×S414yescomplemented
528C2×C3—42no—
529C2×C3—41no—
530C2×C3—82no—
531D3—84no—
532C5C2×#591211yescharacteristic, direct product factor, Sylow
533C22—22no—
534C22—32no—
535C22—31no—
536C22—32no—
537C22—32no—
538C22—31no—
539C22—31no—
540C22—34no—
541C22—32no—
542C22—62no—
543C22—62no—
544C22—64no—
545C22—61no—
546C22—62no—
547C22—64no—
548C22—64no—
549C22—64no—
550C22—62no—
551C22—64no—
552C22—124no—
553C22—124no—
554C22C2×C5×#23511yescharacteristic, complemented
555C22C22×C5×S412yes—
556C22C22×C5×S411yescharacteristic
557C4—32no—
558C4—32no—
559C4—64no—
560C4—64no—
561C4—64no—
562C4C22×C5×S412yes—
563C4C22×C5×S412yes—
564C3—41noSylow
565C2—22no—
566C2—31no—
567C2—31no—
568C2—32no—
569C2—62no—
570C2—124no—
571C2C5×#591212yesdirect product factor
572C2C23×C5×S411yescharacteristic
subgroupG/H
centerC2×C4×C5C2×S4
commutator, cosocleC2×A4C24×C5
FrattiniC2C23×C5×S4
FittingC24×C4×C5D3
socleC24×C5C22×D3
hypercenterC22×C4×C5S4
hypocenterA4C2×C5×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5×A4C2semidirect
2. C2×C5×GL(2,ℤ/4)C2semidirect
3. C5×#5912C2direct (N × H)
4. C2×C5×C4⋊S4C2semidirect
5. C2×C4×C5×S4C2semidirect
6. C2×C5×A4⋊Q8C2semidirect
13. C2×C4×C5×A4C22semidirect
14. C5×GL(2,ℤ/4)C22semidirect
16. C5×C4⋊S4C22semidirect
17. C4×C5×S4C22semidirect
18. C5×A4⋊Q8C22semidirect
19. C2×#5912C5direct (N × H)
21. C24×C4×C5D3semidirect
47. C22×C4×A4C2×C5semidirect
48. C2×GL(2,ℤ/4)C2×C5semidirect
49. #5912C2×C5direct (N × H)
50. C2×C4⋊S4C2×C5semidirect
51. C2×C4×S4C2×C5semidirect
52. C2×A4⋊Q8C2×C5semidirect
73. C23×C4×C5C2×D3semidirect
101. C2×C5×A4#41semidirect
117. C2×C4×A4C22×C5semidirect
118. GL(2,ℤ/4)C22×C5semidirect
120. C4⋊S4C22×C5semidirect
121. C4×S4C22×C5semidirect
122. A4⋊Q8C22×C5semidirect
178. C22×C4×C5S4semidirect
193. C24×C4C5×D3semidirect
210. C5×A4C2×#41semidirect
249. C23×C5#235semidirect
287. C2×C4×C5C2×S4semidirect
308. C23×C4C2×C5×D3semidirect
339. C2×A4C5×#41semidirect
371. C22×C5C2×#235semidirect
436. C22×C4C5×S4semidirect
456. A4C2×C5×#41semidirect
466. C2×C5#5912direct (N × H)
489. C23C5×#235semidirect
527. C2×C4C2×C5×S4semidirect
532. C5C2×#5912direct (N × H)
554. C22C2×C5×#235semidirect
571. C2C5×#5912direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C2×C5×GL(2,ℤ/4)2
C5×#59122
C2×C5×C4⋊S42
C2×C4×C5×S42
C2×C5×A4⋊Q82
C2×C5×#5253
C2×C5×#2354
C2×#59125