Finite Group Databasebeta

← C5×#41×S4 #14511421

Subgroups — C5×#41×S4 #14511421

Order 1920.

1706 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1706 conjugacy classes of subgroups — too many to draw. Only its 208 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
1C5×A4×#41C211yesmaximal, characteristic, complemented
2C5×#5919C211yesmaximal, characteristic, complemented
3C5×Q8×S4C211yesmaximal, characteristic, complemented
4C5×#5914C213yesmaximal, complemented
5C5×D4×S4C213yesmaximal, complemented
6C5×#5912C213yesmaximal, complemented
7C2×C4×C5×S4C213yesmaximal, complemented
8C5×D4×#41—31nomaximal, Hall
9C22×C5×S4C2213yes—
10C5×D3×#41—41nomaximal
11C5×Q8×A4C2211yescharacteristic, complemented
12C5×D4×A4C2213yescomplemented
13C2×C4×C5×A4C2213yescomplemented
14C5×GL(2,ℤ/4)C2216yescomplemented
15C2×C5×SL(2,ℤ/4)C2213yescomplemented
16C5×C4⋊S4C2213yescomplemented
17C4×C5×S4C2211yescharacteristic
18C4×C5×S4C2213yes—
19C4×C5×S4C2213yescomplemented
20C4×C5×S4C2213yescomplemented
21C5×A4⋊Q8C2213yescomplemented
22#41×S4C511yesmaximal, characteristic, direct product factor, Hall
23C22×C5×#41—31no—
24C22×C5×#41D311yescharacteristic, complemented
25C5×#550—31no—
26C5×D4×Q8—31no—
27C5×#548—31no—
28C5×#548—31no—
29C5×#547—33no—
30C5×#546—33no—
31C5×#546—33no—
32C5×D42—33no—
33C5×#532—33no—
34C5×#525—33no—
35C5×#525—33no—
36C4×C5×#41—31no—
37C2×C4×C5×D4—33no—
38C22×C5×A4C2313yes—
39C2×C5×S4—26no—
40C2×C5×S4C2313yes—
41C2×C5×S4C2311yescharacteristic
42C3×C5×#41—41no—
43C5×#239—41no—
44C5×D3×Q8—41no—
45C5×#237—43no—
46C5×D3×D4—43no—
47C5×#235—43no—
48C2×C4×C5×D3—43no—
49C4×C5×A4C2313yes—
50C4×C5×A4C2311yescharacteristic
51C5×SL(2,ℤ/4)C2313yes—
52C5×SL(2,ℤ/4)C2311yescharacteristic
53A4×#41C2×C511yescharacteristic, complemented
54#5919C2×C511yescharacteristic, complemented
55Q8×S4C2×C511yescharacteristic, complemented
56#5914C2×C513yescomplemented
57D4×S4C2×C513yescomplemented
58#5912C2×C513yescomplemented
59C2×C4×S4C2×C513yescomplemented
60C2×C5×#41—31no—
61C2×C5×#41—33no—
62C2×C5×#41—33no—
63C2×C5×#41—31no—
64C2×C5×#41—33no—
65C2×C5×#41—33no—
66C2×C5×#41—31no—
67C2×C5×#41—61no—
68C2×C5×#41—63no—
69C2×C5×#41—63no—
70C2×C5×#41—61no—
71C22×C5×Q8—31no—
72C22×C5×Q8C2×D311yescharacteristic, complemented
73C22×C5×D4—33no—
74C22×C5×D4—33no—
75C22×C5×D4C2×D313yescomplemented
76C23×C4×C5—33no—
77C23×C4×C5C2×D313yescomplemented
78C5×C4⋊Q8—33no—
79C5×#127—33no—
80C5×#124—33no—
81C5×#124—33no—
82C5×#123—36no—
83C5×#123—36no—
84C5×C22⋊Q8—33no—
85C5×C22⋊Q8—33no—
86C5×C22⋊Q8—33no—
87C5×C22⋊Q8—33no—
88C5×#121—33no—
89C5×#121—33no—
90C5×#121—33no—
91C5×#121—36no—
92C5×#121—36no—
93C5×#121—33no—
94C5×#120—36no—
95C5×#120—36no—
96C4×C5×Q8—31no—
97C4×C5×Q8—31no—
98C4×C5×D4—33no—
99C4×C5×D4—33no—
100C4×C5×D4—33no—
101C4×C5×D4—33no—
102C4×C5×D4—33no—
103C4×C5×D4—33no—
104C4×C5×D4—36no—
105C4×C5×D4—31no—
106C4×C5×D4—33no—
107C4×C5×D4—33no—
108C4×C5×D4—33no—
109C5×#117—33no—
110C2×C5×C4⋊C4—33no—
111C2×C5×C22⋊C4—33no—
112C2×C5×C22⋊C4—33no—
113C2×C42×C5—33no—
114D4×#41—31noSylow
115C22×C5×D3—43no—
116C2×C5×A4—23no—
117C2×C5×A4C2411yescharacteristic
118C5×S4—23no—
119C5×S4#4112yesdirect product factor
120C3×C5×Q8—41no—
121C3×C5×D4—43no—
122C2×C3×C4×C5—43no—
123C5×#67—46no—
124C2×C5×Dic3—43no—
125C5×D12—43no—
126C4×C5×D3—41no—
127C4×C5×D3—43no—
128C4×C5×D3—43no—
129C4×C5×D3—43no—
130C5×Dic6—43no—
131C22×S4C22×C513yes—
132D3×#41—41no—
133Q8×A4C22×C511yescharacteristic, complemented
134D4×A4C22×C513yescomplemented
135C2×C4×A4C22×C513yescomplemented
136GL(2,ℤ/4)C22×C516yescomplemented
137C2×SL(2,ℤ/4)C22×C513yescomplemented
138C4⋊S4C22×C513yescomplemented
139C4×S4C22×C511yescharacteristic
140C4×S4C22×C513yes—
141C4×S4C22×C513yescomplemented
142C4×S4C22×C513yescomplemented
143A4⋊Q8C22×C513yescomplemented
144C24×C5—33no—
145C24×C5C22×D313yes—
146C5×#41—31no—
147C5×#41—33no—
148C5×#41—33no—
149C5×#41—61no—
150C5×#41—63no—
151C5×#41—63no—
152C5×#41—63no—
153C5×#41—63no—
154C5×#41—63no—
155C5×#41—66no—
156C5×#41—63no—
157C5×#41—63no—
158C5×#41—61no—
159C5×#41—63no—
160C5×#41—63no—
161C5×#41S411yescharacteristic, direct product factor
162C2×C5×Q8—31no—
163C2×C5×Q8—33no—
164C2×C5×Q8—33no—
165C2×C5×Q8—63no—
166C2×C5×Q8—61no—
167C2×C5×D4—31no—
168C2×C5×D4—33no—
169C2×C5×D4—36no—
170C2×C5×D4—33no—
171C2×C5×D4—33no—
172C2×C5×D4—33no—
173C2×C5×D4—36no—
174C2×C5×D4—33no—
175C2×C5×D4—33no—
176C2×C5×D4—36no—
177C2×C5×D4—66no—
178C2×C5×D4—66no—
179C2×C5×D4—66no—
180C2×C5×D4—63no—
181C2×C5×D4—63no—
182C22×C4×C5—33no—
183C22×C4×C5—33no—
184C22×C4×C5—33no—
185C22×C4×C5—33no—
186C22×C4×C5—33no—
187C22×C4×C5—33no—
188C22×C4×C5—31no—
189C22×C4×C5—33no—
190C22×C4×C5—33no—
191C22×C4×C5—33no—
192C22×C4×C5—33no—
193C22×C4×C5—63no—
194C22×C4×C5—63no—
195C22×C4×C5—63no—
196C22×C4×C5—63no—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D313yes—
199C5×C4⋊C4—31no—
200C5×C4⋊C4—33no—
201C5×C4⋊C4—33no—
202C5×C4⋊C4—33no—
203C5×C4⋊C4—33no—
204C5×C4⋊C4—33no—
205C5×C4⋊C4—36no—
206C5×C22⋊C4—31no—
207C5×C22⋊C4—33no—
208C5×C22⋊C4—33no—
209C5×C22⋊C4—33no—
210C5×C22⋊C4—33no—
211C5×C22⋊C4—33no—
212C5×C22⋊C4—36no—
213C5×C22⋊C4—36no—
214C5×C22⋊C4—36no—
215C5×C22⋊C4—31no—
216C5×C22⋊C4—33no—
217C42×C5—31no—
218C42×C5—33no—
219C42×C5—33no—
220C42×C5—33no—
221C22×#41—31no—
222C22×#41C5×D311yescharacteristic, complemented
223#550—31no—
224D4×Q8—31no—
225#548—31no—
226#548—31no—
227#547—33no—
228#546—33no—
229#546—33no—
230D42—33no—
231#532—33no—
232#525—33no—
233#525—33no—
234C4×#41—31no—
235C2×C4×D4—33no—
236C22×C3×C5—43no—
237C2×C5×D3—41no—
238C2×C5×D3—43no—
239C2×C5×D3—86no—
240C5×A4C2×#4111yescharacteristic, complemented
241C3×C4×C5—43no—
242C3×C4×C5—41no—
243C5×Dic3—43no—
244C5×Dic3—41no—
245C22×A4C23×C513yes—
246C2×S4—26no—
247C2×S4C23×C513yes—
248C2×S4C23×C511yescharacteristic
249C3×#41—41no—
250#239—41no—
251D3×Q8—41no—
252#237—43no—
253D3×D4—43no—
254#235—43no—
255C2×C4×D3—43no—
256C4×A4C23×C513yes—
257C4×A4C23×C511yescharacteristic
258SL(2,ℤ/4)C23×C513yes—
259SL(2,ℤ/4)C23×C511yescharacteristic
260C23×C5—23no—
261C23×C5—31no—
262C23×C5—33no—
263C23×C5—33no—
264C23×C5—33no—
265C23×C5—63no—
266C23×C5—63no—
267C23×C5—66no—
268C23×C5—63no—
269C23×C5—123no—
270C23×C5C23×D311yescharacteristic
271C5×Q8—33no—
272C5×Q8—61no—
273C5×Q8—63no—
274C5×Q8—63no—
275C5×Q8C2×S411yescharacteristic, complemented
276C5×D4—32no—
277C5×D4—32no—
278C5×D4—36no—
279C5×D4—33no—
280C5×D4—63no—
281C5×D4—66no—
282C5×D4—63no—
283C5×D4—63no—
284C5×D4—66no—
285C5×D4—66no—
286C5×D4—63no—
287C5×D4—66no—
288C5×D4—63no—
289C5×D4—126no—
290C5×D4C2×S413yescomplemented
291C2×C4×C5—31no—
292C2×C4×C5—33no—
293C2×C4×C5—33no—
294C2×C4×C5—33no—
295C2×C4×C5—31no—
296C2×C4×C5—33no—
297C2×C4×C5—31no—
298C2×C4×C5—33no—
299C2×C4×C5—31no—
300C2×C4×C5—33no—
301C2×C4×C5—33no—
302C2×C4×C5—31no—
303C2×C4×C5—33no—
304C2×C4×C5—63no—
305C2×C4×C5—61no—
306C2×C4×C5—63no—
307C2×C4×C5—63no—
308C2×C4×C5—63no—
309C2×C4×C5—66no—
310C2×C4×C5—63no—
311C2×C4×C5—63no—
312C2×C4×C5—63no—
313C2×C4×C5—63no—
314C2×C4×C5—61no—
315C2×C4×C5—63no—
316C2×C4×C5—123no—
317C2×C4×C5C2×S413yescomplemented
318C2×#41—31no—
319C2×#41—33no—
320C2×#41—33no—
321C2×#41—31no—
322C2×#41—33no—
323C2×#41—33no—
324C2×#41—31no—
325C2×#41—61no—
326C2×#41—63no—
327C2×#41—63no—
328C2×#41—61no—
329C22×Q8—31no—
330C22×Q8C2×C5×D311yescharacteristic, complemented
331C22×D4—33no—
332C22×D4—33no—
333C22×D4C2×C5×D313yescomplemented
334C23×C4—33no—
335C23×C4C2×C5×D313yescomplemented
336C4⋊Q8—33no—
337#127—33no—
338#124—33no—
339#124—33no—
340#123—36no—
341#123—36no—
342C22⋊Q8—33no—
343C22⋊Q8—33no—
344C22⋊Q8—33no—
345C22⋊Q8—33no—
346#121—33no—
347#121—33no—
348#121—33no—
349#121—36no—
350#121—36no—
351#121—33no—
352#120—36no—
353#120—36no—
354C4×Q8—31no—
355C4×Q8—31no—
356C4×D4—33no—
357C4×D4—33no—
358C4×D4—33no—
359C4×D4—33no—
360C4×D4—33no—
361C4×D4—33no—
362C4×D4—36no—
363C4×D4—31no—
364C4×D4—33no—
365C4×D4—33no—
366C4×D4—33no—
367#117—33no—
368C2×C4⋊C4—33no—
369C2×C22⋊C4—33no—
370C2×C22⋊C4—33no—
371C2×C42—33no—
372C2×C3×C5—41no—
373C2×C3×C5—83no—
374C5×D3—42no—
375C5×D3—83no—
376C22×D3—43no—
377C2×A4—23no—
378C2×A4C24×C511yescharacteristic
379S4—23no—
380S4C5×#4112yesdirect product factor
381C3×Q8—41no—
382C3×D4—43no—
383C2×C3×C4—43no—
384#67—46no—
385C2×Dic3—43no—
386D12—43no—
387C4×D3—41no—
388C4×D3—43no—
389C4×D3—43no—
390C4×D3—43no—
391Dic6—43no—
392C22×C5—31no—
393C22×C5—31no—
394C22×C5—32no—
395C22×C5—33no—
396C22×C5—63no—
397C22×C5—63no—
398C22×C5—63no—
399C22×C5—63no—
400C22×C5—61no—
401C22×C5—63no—
402C22×C5—63no—
403C22×C5—61no—
404C22×C5—63no—
405C22×C5—126no—
406C22×C5—126no—
407C22×C5D3×#4111yescharacteristic, complemented
408C22×C5C22×S413yes—
409C4×C5—32no—
410C4×C5—31no—
411C4×C5—33no—
412C4×C5—63no—
413C4×C5—61no—
414C4×C5—63no—
415C4×C5—61no—
416C4×C5—63no—
417C4×C5C22×S413yes—
418C4×C5C22×S411yescharacteristic
419C24—33no—
420C24C22×C5×D313yes—
421#41—31no—
422#41—33no—
423#41—33no—
424#41—61no—
425#41—63no—
426#41—63no—
427#41—63no—
428#41—63no—
429#41—63no—
430#41—66no—
431#41—63no—
432#41—63no—
433#41—61no—
434#41—63no—
435#41—63no—
436#41C5×S411yescharacteristic, direct product factor
437C2×Q8—31no—
438C2×Q8—33no—
439C2×Q8—33no—
440C2×Q8—63no—
441C2×Q8—61no—
442C2×D4—31no—
443C2×D4—33no—
444C2×D4—36no—
445C2×D4—33no—
446C2×D4—33no—
447C2×D4—33no—
448C2×D4—36no—
449C2×D4—33no—
450C2×D4—33no—
451C2×D4—36no—
452C2×D4—66no—
453C2×D4—66no—
454C2×D4—63no—
455C2×D4—66no—
456C2×D4—63no—
457C22×C4—33no—
458C22×C4—33no—
459C22×C4—33no—
460C22×C4—33no—
461C22×C4—33no—
462C22×C4—33no—
463C22×C4—31no—
464C22×C4—33no—
465C22×C4—33no—
466C22×C4—33no—
467C22×C4—33no—
468C22×C4—63no—
469C22×C4—63no—
470C22×C4—63no—
471C22×C4—63no—
472C22×C4C22×C5×D311yescharacteristic
473C22×C4C22×C5×D313yes—
474C4⋊C4—31no—
475C4⋊C4—33no—
476C4⋊C4—33no—
477C4⋊C4—33no—
478C4⋊C4—33no—
479C4⋊C4—33no—
480C4⋊C4—36no—
481C22⋊C4—31no—
482C22⋊C4—33no—
483C22⋊C4—33no—
484C22⋊C4—33no—
485C22⋊C4—33no—
486C22⋊C4—33no—
487C22⋊C4—36no—
488C22⋊C4—36no—
489C22⋊C4—36no—
490C22⋊C4—31no—
491C22⋊C4—33no—
492C42—31no—
493C42—33no—
494C42—33no—
495C42—33no—
496C3×C5—41noHall
497C22×C3—43no—
498C2×D3—41no—
499C2×D3—43no—
500C2×D3—86no—
501A4C2×C5×#4111yescharacteristic, complemented
502C3×C4—43no—
503C3×C4—41no—
504Dic3—43no—
505Dic3—41no—
506C2×C5—23no—
507C2×C5—31no—
508C2×C5—31no—
509C2×C5—63no—
510C2×C5—62no—
511C2×C5—123no—
512C2×C5C23×S411yescharacteristic
513C23—23no—
514C23—31no—
515C23—33no—
516C23—33no—
517C23—33no—
518C23—63no—
519C23—63no—
520C23—66no—
521C23—63no—
522C23—123no—
523C23C23×C5×D311yescharacteristic
524Q8—33no—
525Q8—61no—
526Q8—63no—
527Q8—63no—
528Q8C2×C5×S411yescharacteristic, complemented
529D4—32no—
530D4—32no—
531D4—36no—
532D4—33no—
533D4—63no—
534D4—63no—
535D4—66no—
536D4—66no—
537D4—63no—
538D4—66no—
539D4—63no—
540D4—66no—
541D4—63no—
542D4—126no—
543D4C2×C5×S413yescomplemented
544C2×C4—31no—
545C2×C4—33no—
546C2×C4—31no—
547C2×C4—33no—
548C2×C4—33no—
549C2×C4—33no—
550C2×C4—31no—
551C2×C4—33no—
552C2×C4—31no—
553C2×C4—33no—
554C2×C4—33no—
555C2×C4—31no—
556C2×C4—33no—
557C2×C4—63no—
558C2×C4—61no—
559C2×C4—63no—
560C2×C4—63no—
561C2×C4—63no—
562C2×C4—66no—
563C2×C4—63no—
564C2×C4—63no—
565C2×C4—61no—
566C2×C4—63no—
567C2×C4—63no—
568C2×C4—63no—
569C2×C4—123no—
570C2×C4C2×C5×S413yescomplemented
571C2×C3—41no—
572C2×C3—83no—
573D3—42no—
574D3—83no—
575C5#41×S411yescharacteristic, direct product factor, Sylow
576C22—31no—
577C22—31no—
578C22—32no—
579C22—33no—
580C22—63no—
581C22—63no—
582C22—63no—
583C22—63no—
584C22—61no—
585C22—63no—
586C22—63no—
587C22—61no—
588C22—63no—
589C22—126no—
590C22—126no—
591C22C5×D3×#4111yescharacteristic, complemented
592C22C22×C5×S413yes—
593C4—32no—
594C4—31no—
595C4—33no—
596C4—63no—
597C4—61no—
598C4—63no—
599C4—61no—
600C4—63no—
601C4C22×C5×S411yescharacteristic
602C4C22×C5×S413yes—
603C3—41noSylow
604C2—23no—
605C2—31no—
606C2—31no—
607C2—63no—
608C2—62no—
609C2—123no—
610C2C23×C5×S411yescharacteristic
subgroupG/H
centerC4×C5C22×S4
commutator, cosocleC2×A4C24×C5
FrattiniC2C23×C5×S4
FittingC22×C5×#41D3
socleC23×C5C23×D3
hypercenterC5×#41S4
hypocenterA4C2×C5×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C4×C5 ◁ C5×#41

Complements

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

NH ≅ G/Ntype
1. C5×A4×#41C2semidirect
2. C5×#5919C2semidirect
3. C5×Q8×S4C2semidirect
4. C5×#5914C2semidirect
5. C5×D4×S4C2semidirect
6. C5×#5912C2semidirect
7. C2×C4×C5×S4C2semidirect
11. C5×Q8×A4C22semidirect
12. C5×D4×A4C22semidirect
13. C2×C4×C5×A4C22semidirect
14. C5×GL(2,ℤ/4)C22semidirect
15. C2×C5×SL(2,ℤ/4)C22semidirect
16. C5×C4⋊S4C22semidirect
19. C4×C5×S4C22semidirect
20. C4×C5×S4C22semidirect
21. C5×A4⋊Q8C22semidirect
22. #41×S4C5direct (N × H)
24. C22×C5×#41D3semidirect
53. A4×#41C2×C5semidirect
54. #5919C2×C5semidirect
55. Q8×S4C2×C5semidirect
56. #5914C2×C5semidirect
57. D4×S4C2×C5semidirect
58. #5912C2×C5semidirect
59. C2×C4×S4C2×C5semidirect
72. C22×C5×Q8C2×D3semidirect
75. C22×C5×D4C2×D3semidirect
77. C23×C4×C5C2×D3semidirect
119. C5×S4#41direct (N × H)
133. Q8×A4C22×C5semidirect
134. D4×A4C22×C5semidirect
135. C2×C4×A4C22×C5semidirect
136. GL(2,ℤ/4)C22×C5semidirect
137. C2×SL(2,ℤ/4)C22×C5semidirect
138. C4⋊S4C22×C5semidirect
141. C4×S4C22×C5semidirect
142. C4×S4C22×C5semidirect
143. A4⋊Q8C22×C5semidirect
161. C5×#41S4direct (N × H)
222. C22×#41C5×D3semidirect
240. C5×A4C2×#41semidirect
275. C5×Q8C2×S4semidirect
290. C5×D4C2×S4semidirect
317. C2×C4×C5C2×S4semidirect
330. C22×Q8C2×C5×D3semidirect
333. C22×D4C2×C5×D3semidirect
335. C23×C4C2×C5×D3semidirect
380. S4C5×#41direct (N × H)
407. C22×C5D3×#41semidirect
436. #41C5×S4direct (N × H)
501. A4C2×C5×#41semidirect
528. Q8C2×C5×S4semidirect
543. D4C2×C5×S4semidirect
570. C2×C4C2×C5×S4semidirect
575. C5#41×S4direct (N × H)
591. C22C5×D3×#41semidirect

Maximal subgroups

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

subgroupindex
C5×A4×#412
C5×#59192
C5×Q8×S42
C5×#59142
C5×D4×S42
C5×#59122
C2×C4×C5×S42
C5×D4×#413
C5×D3×#414
#41×S45