Finite Group Databasebeta

← C5×#85294 #14511032

Subgroups — C5×#85294 #14511032

Order 1920.

582 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 76 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×C22⋊C4C211yesmaximal, characteristic
2C5×#5432C211yesmaximal, characteristic
3C5×#5432C211yesmaximal, characteristic
4C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
6C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8C5×#2480—31nomaximal, Hall
9C23×C5×A4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C5×SL(2,ℤ/4)C2211yescharacteristic
13C2×C5×SL(2,ℤ/4)C2211yescharacteristic
14C2×C5×SL(2,ℤ/4)C2211yescharacteristic
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C5×#879—41nomaximal
17#85294C511yesmaximal, characteristic, direct product factor, Hall
18C2×C5×#126—31no—
19C22×C5×C22⋊C4D311yescharacteristic
20C5×#397—31no—
21C5×#397—31no—
22C5×#397—31no—
23C5×#396—31no—
24C5×#388—31no—
25C5×#388—31no—
26C5×#388—31no—
27C5×#385—31no—
28C5×#385—31no—
29C5×#380—31no—
30C5×#379—31no—
31C5×#379—31no—
32C4×C5×C22⋊C4—31no—
33C22×C5×A4—21no—
34C22×C5×A4—21no—
35C22×C5×A4—21no—
36C22×C5×A4C2311yescharacteristic
37C4×C5×A4—21no—
38C4×C5×A4—21no—
39C5×SL(2,ℤ/4)—21no—
40C5×SL(2,ℤ/4)—21no—
41C5×SL(2,ℤ/4)—21no—
42C5×SL(2,ℤ/4)—21no—
43C3×C5×C22⋊C4—41no—
44C5×#217—41no—
45C5×#217—41no—
46C5×C4⋊Dic3—41no—
47C5×Dic3⋊C4—41no—
48C5×Dic3⋊C4—41no—
49C4×C5×Dic3—41no—
50A4×C22⋊C4C2×C511yescharacteristic
51#5432C2×C511yescharacteristic
52#5432C2×C511yescharacteristic
53C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
54SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
55SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
56C4×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
57C25×C5C2×D311yescharacteristic
58C23×C4×C5C2×D311yescharacteristic
59C23×C4×C5C2×D311yescharacteristic
60C5×#126—61no—
61C5×#126—61no—
62C5×#126—61no—
63C5×#126—61no—
64C2×C5×C4⋊C4—31no—
65C2×C5×C4⋊C4—31no—
66C2×C5×C4⋊C4—31no—
67C2×C5×C4⋊C4—31no—
68C2×C5×C4⋊C4—31no—
69C2×C5×C4⋊C4—31no—
70C2×C5×C22⋊C4—31no—
71C2×C5×C22⋊C4—31no—
72C2×C5×C22⋊C4—31no—
73C2×C5×C22⋊C4—31no—
74C2×C5×C22⋊C4—31no—
75C2×C5×C22⋊C4—31no—
76C2×C5×C22⋊C4—31no—
77C2×C5×C22⋊C4—31no—
78C2×C5×C22⋊C4—31no—
79C2×C5×C22⋊C4—31no—
80C2×C5×C22⋊C4—31no—
81C2×C5×C22⋊C4—31no—
82C2×C5×C22⋊C4—31no—
83C2×C5×C22⋊C4—31no—
84C2×C5×C22⋊C4—31no—
85C2×C5×C22⋊C4—31no—
86C2×C42×C5—31no—
87C2×C42×C5—31no—
88C5×#95—31no—
89C5×#95—31no—
90C5×#95—31no—
91C5×#95—31no—
92C5×#95—31no—
93C5×#95—31no—
94C5×#95—31no—
95C5×#95—31no—
96#2480—31noSylow
97C23×C3×C5—41no—
98C2×C5×A4—41no—
99C2×C5×A4#4111yescharacteristic
100C2×C5×A4#4111yescharacteristic
101C2×C5×A4#4111yescharacteristic
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C5×Dic3—41no—
105C2×C5×Dic3—41no—
106C2×C5×Dic3—41no—
107C2×C5×Dic3—41no—
108C23×A4C22×C511yescharacteristic
109C2×C4×A4C22×C511yescharacteristic
110C2×C4×A4C22×C511yescharacteristic
111C2×SL(2,ℤ/4)C22×C511yescharacteristic
112C2×SL(2,ℤ/4)C22×C511yescharacteristic
113C2×SL(2,ℤ/4)C22×C511yescharacteristic
114C2×SL(2,ℤ/4)C22×C511yescharacteristic
115#879—41no—
116C24×C5—21no—
117C24×C5—21no—
118C24×C5—21no—
119C24×C5—31no—
120C24×C5—31no—
121C24×C5—61no—
122C24×C5—61no—
123C24×C5—61no—
124C24×C5C22×D311yescharacteristic
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—31no—
128C22×C4×C5—31no—
129C22×C4×C5—31no—
130C22×C4×C5—31no—
131C22×C4×C5—31no—
132C22×C4×C5—31no—
133C22×C4×C5—31no—
134C22×C4×C5—31no—
135C22×C4×C5—31no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—31no—
139C22×C4×C5—61no—
140C22×C4×C5—61no—
141C5×C4⋊C4—61no—
142C5×C4⋊C4—61no—
143C5×C4⋊C4—61no—
144C5×C4⋊C4—61no—
145C5×C4⋊C4—61no—
146C5×C4⋊C4—61no—
147C5×C4⋊C4—121no—
148C5×C4⋊C4—121no—
149C5×C4⋊C4—121no—
150C5×C22⋊C4—31no—
151C5×C22⋊C4—31no—
152C5×C22⋊C4—31no—
153C5×C22⋊C4—61no—
154C5×C22⋊C4—61no—
155C5×C22⋊C4—61no—
156C5×C22⋊C4—61no—
157C5×C22⋊C4—61no—
158C5×C22⋊C4—62no—
159C5×C22⋊C4—62no—
160C5×C22⋊C4—61no—
161C5×C22⋊C4—61no—
162C5×C22⋊C4—61no—
163C5×C22⋊C4—61no—
164C5×C22⋊C4—61no—
165C5×C22⋊C4—61no—
166C5×C22⋊C4—61no—
167C5×C22⋊C4—61no—
168C5×C22⋊C4—121no—
169C5×C22⋊C4—121no—
170C5×C22⋊C4—121no—
171C5×C22⋊C4—121no—
172C5×C22⋊C4S411yescharacteristic
173C42×C5—61no—
174C42×C5—61no—
175C42×C5—121no—
176C2×#126—31no—
177C22×C22⋊C4C5×D311yescharacteristic
178#397—31no—
179#397—31no—
180#397—31no—
181#396—31no—
182#388—31no—
183#388—31no—
184#388—31no—
185#385—31no—
186#385—31no—
187#380—31no—
188#379—31no—
189#379—31no—
190C4×C22⋊C4—31no—
191C22×C3×C5—41no—
192C22×C3×C5—81no—
193C22×C3×C5—81no—
194C22×C3×C5—81no—
195C5×A4#12611yescharacteristic, complemented
196C3×C4×C5—81no—
197C3×C4×C5—81no—
198C5×Dic3—81no—
199C5×Dic3—81no—
200C5×Dic3—81no—
201C5×Dic3—81no—
202C22×A4—21no—
203C22×A4—21no—
204C22×A4—21no—
205C22×A4C23×C511yescharacteristic
206C4×A4—21no—
207C4×A4—21no—
208SL(2,ℤ/4)—21no—
209SL(2,ℤ/4)—21no—
210SL(2,ℤ/4)—21no—
211SL(2,ℤ/4)—21no—
212C3×C22⋊C4—41no—
213#217—41no—
214#217—41no—
215C4⋊Dic3—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218C4×Dic3—41no—
219C23×C5—31no—
220C23×C5—31no—
221C23×C5—31no—
222C23×C5—31no—
223C23×C5—32no—
224C23×C5—31no—
225C23×C5—41no—
226C23×C5—61no—
227C23×C5—61no—
228C23×C5—61no—
229C23×C5—61no—
230C23×C5—61no—
231C23×C5—61no—
232C23×C5—61no—
233C23×C5—61no—
234C23×C5—61no—
235C23×C5—61no—
236C23×C5—61no—
237C23×C5—121no—
238C23×C5—121no—
239C23×C5—121no—
240C23×C5—122no—
241C23×C5#23511yescharacteristic
242C23×C5#23711yescharacteristic
243C23×C5#23711yescharacteristic
244C23×C5C2×S411yescharacteristic
245C2×C4×C5—31no—
246C2×C4×C5—31no—
247C2×C4×C5—61no—
248C2×C4×C5—61no—
249C2×C4×C5—61no—
250C2×C4×C5—61no—
251C2×C4×C5—61no—
252C2×C4×C5—61no—
253C2×C4×C5—61no—
254C2×C4×C5—61no—
255C2×C4×C5—61no—
256C2×C4×C5—61no—
257C2×C4×C5—61no—
258C2×C4×C5—61no—
259C2×C4×C5—61no—
260C2×C4×C5—61no—
261C2×C4×C5—61no—
262C2×C4×C5—61no—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—61no—
273C2×C4×C5—61no—
274C2×C4×C5—61no—
275C2×C4×C5—61no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C2×C4×C5—61no—
279C2×C4×C5C2×S411yescharacteristic
280C2×C4×C5C2×S411yescharacteristic
281C25C2×C5×D311yescharacteristic
282C23×C4C2×C5×D311yescharacteristic
283C23×C4C2×C5×D311yescharacteristic
284#126—61no—
285#126—61no—
286#126—61no—
287#126—61no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—31no—
310C2×C42—31no—
311C2×C42—31no—
312#95—31no—
313#95—31no—
314#95—31no—
315#95—31no—
316#95—31no—
317#95—31no—
318#95—31no—
319#95—31no—
320C2×C3×C5—41no—
321C2×C3×C5—41no—
322C2×C3×C5—41no—
323C2×C3×C5—161no—
324C23×C3—41no—
325C2×A4—41no—
326C2×A4C5×#4111yescharacteristic
327C2×A4C5×#4111yescharacteristic
328C2×A4C5×#4111yescharacteristic
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C22×C5—21no—
336C22×C5—21no—
337C22×C5—21no—
338C22×C5—31no—
339C22×C5—31no—
340C22×C5—31no—
341C22×C5—31no—
342C22×C5—31no—
343C22×C5—31no—
344C22×C5—31no—
345C22×C5—31no—
346C22×C5—31no—
347C22×C5—61no—
348C22×C5—62no—
349C22×C5—62no—
350C22×C5—61no—
351C22×C5—61no—
352C22×C5—61no—
353C22×C5—121no—
354C22×C5—121no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5—121no—
358C22×C5—121no—
359C22×C5#87911yescharacteristic, complemented
360C22×C5C22×S411yescharacteristic
361C4×C5—21no—
362C4×C5—21no—
363C4×C5—61no—
364C4×C5—61no—
365C4×C5—121no—
366C4×C5—121no—
367C4×C5—121no—
368C4×C5—121no—
369C4×C5—121no—
370C4×C5—121no—
371C4×C5—121no—
372C4×C5—121no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—31no—
377C24—31no—
378C24—61no—
379C24—61no—
380C24—61no—
381C24C22×C5×D311yescharacteristic
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—61no—
397C22×C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—121no—
405C4⋊C4—121no—
406C4⋊C4—121no—
407C22⋊C4—31no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—61no—
411C22⋊C4—61no—
412C22⋊C4—61no—
413C22⋊C4—61no—
414C22⋊C4—61no—
415C22⋊C4—61no—
416C22⋊C4—61no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C22⋊C4—61no—
422C22⋊C4—61no—
423C22⋊C4—61no—
424C22⋊C4—61no—
425C22⋊C4—121no—
426C22⋊C4—121no—
427C22⋊C4—121no—
428C22⋊C4—121no—
429C22⋊C4C5×S411yescharacteristic
430C42—61no—
431C42—61no—
432C42—121no—
433C3×C5—41noHall
434C22×C3—41no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438A4C5×#12611yescharacteristic, complemented
439C3×C4—81no—
440C3×C4—81no—
441Dic3—81no—
442Dic3—81no—
443Dic3—81no—
444Dic3—81no—
445C2×C5—31no—
446C2×C5—31no—
447C2×C5—31no—
448C2×C5—31no—
449C2×C5—41no—
450C2×C5—121no—
451C2×C5#591211yescharacteristic
452C2×C5#591411yescharacteristic
453C2×C5#591411yescharacteristic
454C23—31no—
455C23—31no—
456C23—31no—
457C23—31no—
458C23—32no—
459C23—31no—
460C23—41no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—61no—
465C23—61no—
466C23—61no—
467C23—61no—
468C23—61no—
469C23—61no—
470C23—61no—
471C23—61no—
472C23—121no—
473C23—121no—
474C23—121no—
475C23—122no—
476C23C5×#23511yescharacteristic
477C23C5×#23711yescharacteristic
478C23C5×#23711yescharacteristic
479C23C2×C5×S411yescharacteristic
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4C2×C5×S411yescharacteristic
515C2×C4C2×C5×S411yescharacteristic
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—161no—
520C5#8529411yescharacteristic, direct product factor, Sylow
521C22—21no—
522C22—21no—
523C22—21no—
524C22—31no—
525C22—31no—
526C22—31no—
527C22—31no—
528C22—31no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—31no—
533C22—61no—
534C22—62no—
535C22—62no—
536C22—61no—
537C22—61no—
538C22—61no—
539C22—121no—
540C22—121no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22C5×#87911yescharacteristic, complemented
546C22C22×C5×S411yescharacteristic
547C4—21no—
548C4—21no—
549C4—61no—
550C4—61no—
551C4—121no—
552C4—121no—
553C4—121no—
554C4—121no—
555C4—121no—
556C4—121no—
557C4—121no—
558C4—121no—
559C3—41noSylow
560C2—31no—
561C2—31no—
562C2—31no—
563C2—31no—
564C2—41no—
565C2—121no—
566C2C5×#591211yescharacteristic
567C2C5×#591411yescharacteristic
568C2C5×#591411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C22⋊C4D3
socleC24×C5C22×D3
prime radicalC23×C5×A4C22
hypercenterC5×C22⋊C4S4
hypocenterA4C5×#126
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C5×C4⋊SL(2,ℤ/4)C2semidirect
5. C5×SL(2,ℤ/4)⋊C4C2semidirect
6. C5×SL(2,ℤ/4)⋊C4C2semidirect
7. C4×C5×SL(2,ℤ/4)C2semidirect
17. #85294C5direct (N × H)
53. C4⋊SL(2,ℤ/4)C2×C5semidirect
54. SL(2,ℤ/4)⋊C4C2×C5semidirect
55. SL(2,ℤ/4)⋊C4C2×C5semidirect
56. C4×SL(2,ℤ/4)C2×C5semidirect
195. C5×A4#126semidirect
359. C22×C5#879semidirect
438. A4C5×#126semidirect
520. C5#85294direct (N × H)
545. C22C5×#879semidirect

Maximal subgroups

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

subgroupindex
C5×A4×C22⋊C42
C5×#54322
C5×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C4×C5×SL(2,ℤ/4)2
C5×#24803
C5×#8794
#852945